Supplementary Angles
What are supplementary angles.
Answer: Supplementary angles are angles whose sum is 180 °
No matter how large or small angles 1 and 2 on the left become, the two angles remain supplementary which means that they add up to 180°.
Do supplementary angles need to be next to each other (ie adjacent)?
Answer: No!
Supplementary angles do not need to be adjacent angles (angles next to one another).
Both pairs of angles pictured below are supplementary.
Angles that are supplementary and adjacent are known as a linear pair .
Interactive Supplementary Angles
Click and drag around the points below to explore and discover the rule for vertical angles on your own.
You can click and drag points A, B, and C.
(Full Size Interactive Supplementary Angles )
| ∠ A | 90 |
|---|---|
| ∠ B | 90 |
Practice Problems
If $$m \angle 1 =32 $$°, what is the $$m \angle 2 ? $$
$$ m \angle 1 + m \angle 2 = 180° \\ 32° + m \angle 2 = 180° \\ m \angle 2 = 180°-32° \\ m \angle 2 = 148° $$
$$ \angle c $$ and $$ \angle F $$ are supplementary. If $$m \angle C$$ is 25°, what is the $$m \angle F$$?
$$ m \angle c + m \angle F = 180° \\ 25° + m \angle F = 180° \\ m \angle F = 180°-25° = 155° $$
If the ratio of two supplementary angles is $$ 2:1 $$, what is the measure of the larger angle?
First, since this is a ratio problem, we will let the larger angle be 2x and the smaller angle x . We know that $$ 2x + 1x = 180$$ , so now, let's first solve for x:
$$ 3x = 180° \\ x = \frac{180°}{3} = 60° $$
Now, the larger angle is the 2x which is 2(60) = 120 degrees Answer: 120 degrees
If the ratio of two supplementary angles is 8:1, what is the measure of the smaller angle?
First, since this is a ratio problem, we will let the larger angle be 8x and the smaller angle x . We know that 8x + 1x = 180 , so now, let's first solve for x:
$$ 9x = 180° \\ x = \frac{180°}{9} = 20° $$
Now, the smaller angle is the 1x which is 1(20°) = 20° Answer: 20°
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Angle Pair: Supplementary Angles
Supplementary angles.
If the measures of two angles sum up to [latex]180^\circ[/latex], they are called supplementary angles . You’ll notice that when this pair of angles are adjacent, they form a straight angle. Each angle is called a supplement of the other.
Take for instance the diagram above, [latex]\angle AXN[/latex] and [latex]\angle NXF[/latex] are supplementary. If we add their angle measures ([latex]120^\circ + 60^\circ [/latex]), we get [latex]180^\circ[/latex].
But the angles don’t have to be adjacent nor share a common side and vertex to be considered as supplementary angles.
[latex]\angle H[/latex] and [latex]\angle S[/latex] are supplementary. Why? Because even though they are non-adjacent angles, the sum of their measures is [latex]180^\circ[/latex].
[latex]35^\circ + 145^\circ = 180^\circ [/latex]
Example Problems Involving Supplementary Angles
Let’s delve more into the relationship of this angle pair by going through some examples.
Example 1: Are [latex]\angle ERW[/latex] and [latex]\angle WRQ[/latex] supplementary?
We have to add the angle measures of both angles in order to find out if they sum up to [latex]180^\circ[/latex].
And they do! Therefore, [latex]\angle ERW[/latex] and [latex]\angle WRQ[/latex] are supplementary angles .
Example 2: If [latex]\angle CYM[/latex] and [latex]\angle LKG[/latex] are supplementary, what is the measure of [latex]\angle CYM[/latex]?
We are only given the measure of [latex]\angle LKG[/latex]. However, since we know that supplementary angles add up to [latex]180^\circ[/latex], we can simply use subtraction in order to find the measure of [latex]\angle CYM[/latex].
Thus, the measure of [latex]\angle CYM[/latex] is [latex]\textbf{127}^\circ[/latex].
Example 3: [latex]\angle JBU[/latex] and [latex]\angle UBT[/latex] are supplementary. Find the missing angle measure.
This problem is similar to our previous example. The only difference is that the two angles are adjacent to each other. However, the concept stays the same. We can find the missing measure by subtracting the given measure of [latex]\angle UBT[/latex] from [latex]180^\circ[/latex].
The missing angle measure or the measure of [latex]\angle JBU[/latex] is [latex]\textbf{42}^\circ[/latex].
This makes sense because if we add both angle measures, we get [latex]180^\circ[/latex].
[latex]138^\circ + 42^\circ = 180^\circ [/latex]
This proves that both angles are indeed supplementary.
Example 4: What is the value of [latex]x[/latex]?
Just by looking at the diagram, we can tell that [latex]\angle PVH[/latex] and [latex]\angle HVA[/latex] are supplementary. Together, the angle pair form a straight angle while adjacent to each other. A straight angle measures [latex]180^\circ[/latex] and so are supplementary angles.
Both of the angle measures are given but one is expressed in the form of an algebraic expression. It may look challenging but it’s really not. Since we know that they are supplementary, we will set up our equation such that the sum of the angle measures is [latex]180^\circ[/latex]. Then we solve for [latex]x[/latex].
So, the value of [latex]x[/latex] is [latex]\textbf{9}[/latex].
To check if we got the correct answer, let’s plug in the value of [latex]x[/latex] into our original equation. If both sides of the equation equal to [latex]180[/latex], then we got the correct value for [latex]x[/latex].
Perfect! [latex]9[/latex] indeed is the correct value for [latex]x[/latex]. While checking, we also found out that the measure of [latex]\angle HVA[/latex] is [latex]61^\circ[/latex].
Example 5: Suppose [latex]\angle Q[/latex] and [latex]\angle F[/latex] are supplementary. Find the measures of the two angles.
Here we are given two supplementary angles whose measures are expressed in algebraic expressions. Let’s go ahead and set up our equation then solve for the variable [latex]x[/latex].
Now that we know the value of [latex]x[/latex], we can use this to find the measure of each angle. We’ll simply replace [latex]x[/latex] with [latex]7[/latex] on each of the algebraic expressions then simplify.
So the measure of [latex]\angle Q[/latex] is [latex]\textbf{156}^\circ[/latex] and the measure of [latex]\angle F[/latex] is [latex]\textbf{24}^\circ[/latex].
If we add both angle measures, we get [latex]180^\circ[/latex] which means our answers are correct.
[latex]156^\circ + 24^\circ = 180^\circ [/latex]
Example 6: Two supplementary angles are such that the measure of one angle is 3 times the measure of the other. Determine the measure of each angle.
Let [latex]x^\circ[/latex] be the measure of the first angle. Since the second angle measures 3 times than the first, then it will be [latex]3x ^\circ[/latex]. Keep in mind that the angles are supplementary so the right side of the equation must be [latex]180 ^\circ[/latex].
Using the value of [latex]x[/latex], the measure of the second angle will be [latex]3x = 3\left( {45} \right) = 135[/latex].
Therefore, the measures of the angles are [latex]\textbf{45} ^\circ[/latex] and [latex]\textbf{135} ^\circ[/latex] which when added sum up to [latex]180 ^\circ[/latex] .
You may also be interested in these related math lessons or tutorials:
Alternate Exterior Angles
Alternate Interior Angles
Complementary Angles
Corresponding Angles
Vertical Angles
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Supplementary angles
Here you will learn about supplementary angles, including how to find missing angles by applying knowledge of supplementary angles.
Students will first learn about supplementary angles as a part of measurement and data in 4th grade. They will expand that knowledge as they progress through middle school.
What are supplementary angles?
Supplementary angles are two angles that add up to 180 degrees. They can be adjacent or not adjacent. Adjacent angles are angles that share a common arm and a vertex.
| Adjacent supplementary angles | Non-adjacent supplementary angles |
|---|---|
Adjacent supplementary angles make a straight line.
You can use what you know about supplementary angles to decompose angles and find the measurement of an unknown angle.
Common Core State Standards
How does this relate to 4th grade math?
- Grade 4: Measurement and Data (4.MD.C.7) Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, for example, by using an equation with a symbol for the unknown angle measure.
![[FREE] Angles Worksheet (Grade 4)](https://thirdspacelearning.com/wp-content/uploads/2023/08/Angles-check-for-understanding-quiz-listing-image.png "problem solving in supplementary angles")
[FREE] Angles Worksheet (Grade 4)
Use this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support!
How to find the measure of supplementary angles
In order to find the measure of supplementary angles you need to:
Determine which angles are supplementary.
Identify given angle measurements and the unknown angle or angles.
Find the missing angle.
Clearly state the answer.
Supplementary angles examples
Example 1: finding missing angle measures (adjacent angles).
The two angles shown are supplementary. Find the measure of angle x.
The question states that angles x and y are supplementary and equal 180^{\circ}.
2 Identify given angle measurements and the unknown angle or angles.
Angle y is 17^{\circ}.
The unknown angle is angle x, and when it’s added to 17^{\circ} will equal 180^{\circ}.
3 Find the missing angle.
To find the missing angle, you will subtract 17 from 180.
4 Clearly state the answer using angle terminology.
The measurement of angle x is 163^{\circ}.
Example 2: finding missing angle measures (adjacent angles)
\angle ABC and \angle CBD are supplementary angles. \angle ABC has a measure of 109^{\circ}. What is the measure of \angle CBD?
The question states that \angle ABC and \angle CBD are supplementary angles, which means they equal 180 degrees.
\angle A B C + \angle C B D=180^{\circ}
The question states that \angle A B C=109^{\circ}.
\angle CBD is unknown, and can be called x, and when added to 109^{\circ} will equal 180^{\circ}.
In order to find the measure of \angle CBD, you will subtract 109 from 180.
x=71^{\circ}
The measurement of angle CBD is 71^{\circ}.
Example 3: finding missing angle (non-adjacent angles)
The pair of angles are supplementary. Find the measure of angle F.
The question states that angles D and F are supplementary and equal 180^{\circ} .
Angle D has a measurement of 21^{\circ}.
The measurement of angle F is unknown, but when added to angle D, will equal 180^{\circ}.
21^{\circ}+F=180^{\circ}
In order to find the measure of the second angle, \angle F, you will subtract 21 from 180.
\angle F=159
The measurement of \angle F is 159^{\circ}.
Example 4: finding missing angle (not adjacent)
\angle L and \angle M are supplementary angles. \angle M measures at 98^{\circ}. What is the measure of \angle L?
As stated in the question, \angle L and \angle M are supplementary angles, which means when added together will equal 180^{\circ}.
L + M=180^{\circ}
\angle M measures at 98^{\circ}.
The measurement of angle L is unknown, but when added to \angle M, will equal 180^{\circ}.
L+98^{\circ}=180^{\circ}
You will subtract 98 from 180 to find the measure of \angle L.
L=82^{\circ}
The measurement of \angle L is 82^{\circ}.
Example 5: finding missing angle (no diagram)
Two angles ‘x and y' are supplementary and one of them is 147^{\circ}. What is the size of the other angle?
The question states that angles x and y are supplementary angles, meaning when added together they will equal 180^{\circ}.
x+y=180^{\circ}
Because it is not stated which angle is 147 degrees, you can assume that \angle x= 147^{\circ}.
This would mean that \angle y is unknown, but when added to \angle x would equal 180^{\circ}.
147^{\circ}+y=180^{\circ}
You will subtract 147 from 180 to find the measure of \angle y.
y=33^{\circ}
The measurement of \angle y is 33^{\circ}.
Example 6: finding missing angle
Two angles ‘x and y' are supplementary and one of them is a right angle. What is the size of the other angle?
The question states that angles x and y are supplementary, meaning when added together they will equal 180^{\circ}.
Because it is not stated which angle is a right angle, you can assume that \angle x is a right angle. A right angle will always measure at 90^{\circ}.
90^{\circ}+y=180^{\circ}
You will subtract 90 from 180 to find the measure of \angle y.
y=90^{\circ}
The measurement of \angle y is 90^{\circ}. \; \angle y is also a right angle.
Teaching tips for supplementary angles
- Rather than having students practice finding and decomposing supplementary angles on multiple skill worksheets, provide them with a variety of practice problems, activities, and/or projects that have a real-world context. This will deepen their understanding of this skill.
Easy mistakes to make
- Mixing up supplementary angles and complementary angles Students may mix up supplementary and complementary angles, thinking that complementary angles sum to 180^{\circ} and that supplementary angles sum to 90^{\circ}. However, complementary angles have a sum of 90^{\circ} and supplementary angles have a sum of 180^{\circ}.
- Assuming supplementary angles must have a common vertex Supplementary angles can be either adjacent or not adjacent. Not adjacent supplementary angles do not share a common vertex, but their angles will still add up to 180^{\circ}.
Related angles lessons
- Acute angle
- Obtuse angle
- Right angle
- Adjacent angles
- Complementary angles
- Geometry theorems
- Vertical angle theorem
- Straight angle
- Angles point
- Pentagon angles
Practice supplementary angles questions
1. The two angles shown are supplementary. Find the measure of \angle pmn.
The two angles are supplementary, so they must have a sum of 180.
You will subtract 124 from 180 to find the measure of the missing angle.
The missing angle measures at 56^{\circ}.
2. The two angles shown are supplementary. Which of the following could be the measure of \angle srt?
\angle srt is an acute angle, which means that it’s measurement is less than 90^{\circ}.
An angle with the measure of 114^{\circ} would be an obtuse angle. \angle srt can not equal 114^{\circ}.
An angle with the measure of 90^{\circ} would be a right angle. \angle srt is not a right angle.
An angle with the measure of 10^{\circ} would be an acute angle, however, it would be a very skinny acute angle. \angle srt is too large to equal 10^{\circ}.
\angle srt is an acute angle, with a measurement of 45^{\circ}.
2. \angle a and \angle b are supplementary angles. \angle a measures at 14^{\circ}. What is the measure of \angle b?
You will subtract 14 from 180 to find the measure of the missing angle.
The missing angle measures at 166^{\circ}.
4. \angle f and \angle g are supplementary angles. \angle f measures at 113^{\circ}. What is the measure of \angle g?
You will subtract 113 from 180 to find the measure of the missing angle.
The missing angle measures at 67^{\circ}.
5. Two angles 'x and y' are supplementary and one of them is 47^{\circ}. What is the size of the other angle?
You will subtract 47 from 180 to find the measure of the missing angle.
The missing angle measures at 133^{\circ}.
6. Two angles 'x and y' are supplementary and one of them is 123^{\circ}. What is the size of the other angle?
To find the measure of the missing angle, you can subtract 123 from 180.
The missing angle measures at 57^{\circ}.
Supplementary angles FAQs
Supplementary angles add up to 180 degrees, or a straight angle. Complementary angles add up to 90 degrees, or a right angle.
No, three angles will never be supplementary, even if the sum of their measurement is 180^{\circ}. Supplementary angles will only occur in pairs.
\angle a + \angle b = 180^{\circ}; two angles are supplementary if angle \; a and angle \; b equal 180^{\circ}.
The next lessons are
- Angles in parallel lines
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Complementary Angles And Supplementary Angles
Related Pages Pairs Of Angles Types Of Angles More Geometry Lessons
In these lessons, we will learn
- about complementary angles Click here
- about supplementary angles Click here
- how to solve problems involving complementary and supplementary angles Click here
The following table gives a summary of complementary and supplementary angles. Scroll down the page if you need more explanations about complementary and supplementary angles, videos and worksheets.
What Are Complementary Angles
Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees (right angle). One of the complementary angles is said to be the complement of the other.
The two angles do not need to be together or adjacent. They just need to add up to 90 degrees. If the two complementary angles are adjacent then they will form a right angle.
| ∠ is the complement of ∠ | |
| In a right triangle, the two acute angles are complementary. This is because the sum of angles in a triangle is 180˚ and the right angle is 90˚. Therefore, the other two angles must add up to 90˚. |
Example: x and y are complementary angles. Given x = 35˚, find the value y.
Solution: x + y = 90˚ 35˚ + y = 90˚ y = 90˚ – 35˚ = 55˚
Worksheets for Complementary Angles
What Are Supplementary Angles
Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees (straight line) . One of the supplementary angles is said to be the supplement of the other.
The two angles do not need to be together or adjacent. They just need to add up to 180 degrees. If the two supplementary angles are adjacent then they will form a straight line.
Example: x and y are supplementary angles. Given x = 72˚, find the value y.
Solution: x + y = 180˚ 72˚ + y = 180˚ y = 180˚ –72˚ = 108˚
Worksheets for Supplementary Angles
A mnemonic to help you remember: The C in C omplementary stands for C orner, 90˚
The S in S upplementary stands for S traight, 180˚
Video Lessons
Have a look at the following videos for further explanations of complementary angles and supplementary angles:
How To Identify And Differentiate Complementary And Supplementary Angles?
This video describes complementary and supplementary angles with a few example problems. It will also explain a neat trick to remember the difference between complementary and supplementary angles.
- Find x if these angles are complementary angles.
- Find n and the missing angle if these angles are complementary angles.
- Find x if these angles are supplementary angles.
- Find y and the missing angles if these angles are supplementary angles.
How To Find The Measure Of Complementary Angles Using Algebra?
Step 1: Make sure that the angles are complementary. Step 2: Setup a solvable equation. Step 3: Solve the equation.
Example: ∠1 = 8x + 6<br ∠2 = 19x + 3 ∠1 and ∠2 are complementary. Solve for x.
Complementary And Supplementary Angles Word Problem
Complementary Word Problem How to solve a word problem about its angle and its complement?
Example: The measure of an angle is 43° more than its complement. Find the measure of each angle.
Complementary And Supplementary Angles
What it means for angles to be complementary and supplementary and do a few problems to find complements and supplements for different angles.
Find the measure of the complementary angle for each of the following angles: a) 7° b) 18° c) 72°
Find the measure of the supplementary angle for each of the following angles: a) 124° b) 75°
Complementary And Supplementary Angles Word Problems
Create a system of linear equations to find the measure of an angle knowing information about its complement and supplement.
Example: The supplement of angle y measures 12x + 4 and the complement of the angle measures 6x. What is the measure of the angle?
Word Problems On Complementary And Supplementary Angles
- The measure of an angle is 14 degrees less than the measure of its complement. Find the measures of the two angles.
- The measure of an angle is 6 degrees more than twice the measure of its supplement. Find the measures of the two angles.
- The measure of the supplement of an angle is 20 degrees less than 4 times the measure of the angle. Find the measures of the two angles.
- The supplement of an angle is 12 more than 3 times the complement. Find the angle, the complement and the supplement.
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MathBootCamps
Supplementary angles and examples.
Supplementary angles are angles whose measures sum to 180°. In the lesson below, we will review this idea along with taking a look at some example problems. [adsenseWide]
Example problems with supplementary angles
Let’s look at a few examples of how you would work with the concept of supplementary angles.
The two angles lie along a straight line, so they are supplementary. Therefore: \(x + 118 = 180\). Solving this equation: \(\begin{align}x + 118 &= 180\\ x &= \boxed{62} \end{align} \)
The angles \(A\) and \(B\) are supplementary. If \(m\angle A = (2x)^{\circ}\) and \(m\angle B = (2x-2)^{\circ}\), what is the value of \(x\)?
Since the angles are supplementary, their measures add to 180°. In other words: \(2x + (2x – 2) = 180\). Solving this equation gives the value of \(x\).
\(\begin{align}2x + (2x – 2) &= 180\\ 4x – 2 &= 180\\ 4x &= 182\\ x &= \boxed{45.5} \end{align} \)
The previous example could have asked for some different information. Let’s look at a similar example that asks a slightly different question.
The angles \(A\) and \(B\) are supplementary. If \(m\angle A = (2x+5)^{\circ}\) and \(m\angle B = (x-20)^{\circ}\), what is \(m \angle A\)?
This time you are being asked for the measure of the angle and not just \(x\). But, the value of \(x\) is needed to find the measure of the angle. So, first set up an equation and find \(x\).
\(\begin{align}2x+5 + x – 20 &= 180\\ 3x-15 &= 180 \\ 3x &= 195\\ x&= 65\end{align}\)
The measure of angle \(A\) is then: \(m\angle A = (2x+5)^{\circ}\) and \(x = 65\)
\(m\angle A = (2(65)+5)^{\circ} = \boxed{135^{\circ}} \)
There isn’t much to working with supplementary angles. You just have to remember that their sum is 180° and that any set of angles lying along a straight line will also be supplementary.
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Supplementary Angles
These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°:
Notice that together they make a straight angle .
But the angles don't have to be together.
These two are supplementary because 60° + 120° = 180°
Play With It ...
(Drag the points)
When the two angles add to 180°, we say they "Supplement" each other. Supplement comes from Latin supplere , to complete or "supply" what is needed
Spelling: be careful, it is not "Suppl i mentary Angle"
Complementary vs Supplementary
A related idea is Complementary Angles , they add up to 90°
How to remember which is which? Well, alphabetically they are:
- Complementary add to 90°
- Supplementary add to 180°
You can also think:
- " S " of S upplementary is for " S traight" (180° is a straight line)
Or you can think:
- when you are right you get a compliment (sounds like compl e ment)
- "supplement" (like a vitamin supplement) is something extra, so is bigger
Complementary and Supplementary Angles
Before we solve the worked-out problems on complementary and supplementary angles we will recall the definition of complementary angles and supplementary angles.
Complementary Angles: Two angles are called complementary angles, if their sum is one right angle i.e. 90°.
Each angle is called the complement of the other. Example, 20° and 70° are complementary angles, because 20° + 70° = 90°.
Clearly, 20° is the complement of 70° and 70° is the complement of 20°. Thus, the complement of angle 53° = 90° - 53° = 37°.
Supplementary Angles: Two angles are called supplementary angles, if their sum is two right angles i.e. 180°.
Each angle is called the supplement of the other. Example, 30° and 150° are supplementary angles, because 30° + 150° = 180°.
Clearly, 30° is the supplement of 150° and 150° is the supplement of 30°. Thus, the supplement of angle 105° = 180° - 105° = 75°.
Solved problems on complementary and supplementary angles: 1. Find the complement of the angle 2/3 of 90°. Solution: Convert 2/3 of 90°
2/3 × 90° = 60°
Complement of 60° = 90° - 60° = 30°
Therefore, complement of the angle 2/3 of 90° = 30°
2. Find the supplement of the angle 4/5 of 90°. Solution: Convert 4/5 of 90°
4/5 × 90° = 72°
Supplement of 72° = 180° - 72° = 108°
Therefore, supplement of the angle 4/5 of 90° = 108°
3. The measure of two complementary angles are (2x - 7)° and (x + 4)°. Find the value of x. Solution: According to the problem, (2x - 7)° and (x + 4)°, are complementary angles’ so we get;
(2x - 7)° + (x + 4)° = 90°
or, 2x - 7° + x + 4° = 90°
or, 2x + x - 7° + 4° = 90°
or, 3x - 3° = 90°
or, 3x - 3° + 3° = 90° + 3°
or, 3x = 93°
or, x = 93°/3°
or, x = 31°
Therefore, the value of x = 31°.
4. The measure of two supplementary angles are (3x + 15)° and (2x + 5)°. Find the value of x. Solution: According to the problem, (3x + 15)° and (2x + 5)°, are complementary angles’ so we get;
(3x + 15)° + (2x + 5)° = 180°
or, 3x + 15° + 2x + 5° = 180°
or, 3x + 2x + 15° + 5° = 180°
or, 5x + 20° = 180°
or, 5x + 20° - 20° = 180° - 20°
or, 5x = 160°
or, x = 160°/5°
or, x = 32°
Therefore, the value of x = 32°.
5. The difference between the two complementary angles is 180°. Find the measure of the angle. Solution: Let one angle be of measure x°.
Then complement of x° = (90 - x)
Difference = 18°
Therefore, (90° - x) – x = 18°
or, 90° - 2x = 18°
or, 90° - 90° - 2x = 18° - 90°
or, -2x = -72°
or, x = 72°/2°
or, x = 36°
Also, 90° - x
= 90° - 36°
Therefore, the two angles are 36°, 54°.
6. POQ is a straight line and OS stands on PQ. Find the value of x and the measure of ∠ POS, ∠ SOR and ∠ ROQ.
Solution: POQ is a straight line.
Therefore, ∠POS + ∠SOR + ∠ROQ = 180°
or, (5x + 4°) + (x - 2°) + (3x + 7°) = 180°
or, 5x + 4° + x - 2° + 3x + 7° = 180°
or, 5x + x + 3x + 4° - 2° + 7° = 180°
or, 9x + 9° = 180°
or, 9x + 9° - 9° = 180° - 9°
or, 9x = 171°
or, x = 171/9
or, x = 19° Put the value of x = 19°
Therefore, x - 2
= 17° Again, 3x + 7
= 3 × 19° + 7°
= 64° And again, 5x + 4
= 5 × 19° + 4°
Therefore, the measure of the three angles is 17°, 64°, 99°. These are the above solved examples on complementary and supplementary angles explained step-by-step with detailed explanation.
● Lines and Angles
Fundamental Geometrical Concepts
Classification of Angles
Related Angles
Some Geometric Terms and Results
Complementary Angles
Supplementary Angles
Adjacent Angles
Linear Pair of Angles
Vertically Opposite Angles
Parallel Lines
Transversal Line
Parallel and Transversal Lines
7th Grade Math Problems 8th Grade Math Practice From Complementary and Supplementary Angles to HOME PAGE
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- Complementary Angles Supplementary Angles
Supplementary And Complementary Angles
Supplementary angles and complementary angles are defined with respect to the addition of two angles. If the sum of two angles is 180 degrees then they are said to be supplementary angles , which form a linear angle together. Whereas if the sum of two angles is 90 degrees, then they are said to be complementary angles, and they form a right angle together.
When two line segments or lines meet at a common point (called a vertex), at the point of intersection an angle is formed. When a ray is rotated about its endpoint, then the measure of its rotation in an anti-clockwise direction is the angle formed between its initial and final position.
In fig. 1 if the ray OP is rotated in the direction of the ray OQ, then the measure of its rotation represents the angle formed by it. In this case, the measure of rotation which is the angle formed between the initial side and the terminal side is represented by Ɵ.
Complementary Angles
When the sum of two angles is 90°, then the angles are known as complementary angles . In other words, if two angles add up to form a right angle, then these angles are referred to as complementary angles. Here we say that the two angles complement each other.
How to Find Complementary Angles?
Suppose if one angle is x then the other angle will be 90 ° – x. Hence, we use these complementary angles for trigonometry ratios , where one ratio is complementary to another ratio by 90 degrees such as;
- sin (90° – A) = cos A and cos (90° – A) = sin A
- tan (90° – A) = cot A and cot (90° – A) = tan A
- sec (90° – A) = cosec A and cosec (90° – A) = sec A
Hence, you can see here the trigonometric ratio of the angles gets changed if they complement each other.
In the above figure, the measure of angle BOD is 60 ° and angle AOD measures 30 ° . By adding both of these angles we get a right angle, therefore ∠BOD and ∠AOD are complementary angles.
The following angles in Fig. 3 given below are complementary to each other as the measure of the sum of both the angles is 90 ° . ∠POQ and ∠ABC are complementary and are called complements of each other.
For example: To find the complement of 2x + 52°, subtract the given angle from 90 degrees.
90 o – (2x + 52 o ) = 90 o – 2x – 52 o = -2x + 38 o
The complement of 2x + 52 o is 38 o – 2x.
Supplementary Angles
When the sum of two angles is 180°, then the angles are known as supplementary angles. In other words, if two angles add up, to form a straight angle, then those angles are referred to as supplementary angles .
How to Find Supplementary Angles?
The two angles form a linear angle, such that, if one angle is x, then the other the angle is 180 ° – x. The linearity here proves that the properties of the angles remain the same. Take the examples of trigonometric ratios such as;
- Sin (180 – A) = Sin A
- Cos (180 – A) = – Cos A (quadrant is changed)
- Tan (180 – A) = – Tan A
In Fig. 4 given above, the measure of ∠AOC is 60 o and ∠AOB measures 120 o . By adding both of these angles we get a straight angle. Therefore, ∠AOC and ∠AOB are supplementary angles, and both of these angles are known as a supplement to each other.
Also, learn:
- Adjacent Angles Vertical
- Lines And Angles
Difference between Complementary and Supplementary Angles
| Sum is equal to 90 degrees | Sum is equal to 180 degrees |
| Two angles complement each other | Two angles supplement each other |
| These angles do not form linear pair of angles | These angles form linear pair of angles |
| Meant only for right angles | Meant only for straight angles |
How to remember easily the difference between Complementary angles and supplementary angles?
- “ C “ letter of C omplementary stands for “ C orner” (A right angle, 90 ° )
- “ S “ letter of S upplementary stands for “ S traight” ( a straight line, 180 ° )
Video Lesson on Types of Angles
Solved Examples
The example problems on supplementary and complementary angles are given below:
Find the complement of 40 degrees.
Solution:
As the given angle is 40 degrees, then,
The complement is 50 degrees.
We know that sum of complementary angles = 90 degrees
So, 40° + 50° = 90°
Find the supplement of the angle 1/3 of 210°.
Step 1: Convert 1/3 of 210°
That is, (1/3) x 210° = 70°
Step 2: Supplement of 70° = 180° – 70° = 110°
Therefore, the supplement of the angle 1/3 of 210° is 110°
Example 3:
The measures of the two angles are (x + 25)° and (3x + 15)°. Find the value of x if angles are supplementary angles.
We know that, Sum of Supplementary angles = 180 degrees
(x + 25)° + (3x + 15)° = 180°
4x + 40° = 180°
4x = 140°
x = 35°
The value of x is 35 degrees.
The difference between the two complementary angles is 52°. Find both the angles.
Let, First angle = m degrees, then,
Second angle = (90 – m)degrees {as per the definition of complementary angles}
Difference between angles = 52°
(90° – m) – m = 52°
90° – 2m = 52°
– 2m = 52° – 90°
Again, second angle = 90° – 19° = 71°
Therefore, the required angles are 19°, 71°.
Practice Questions
- Check if 65° and 35° are complementary angles.
- Are 80° and 100° supplementary?
- Find the complement of 54°.
- Find the supplement of 99°.
- If one angle measures 50° and is supplementary to another angle. Then find the value of another angle. Also, state what type of angle it is?
Frequently Asked Questions – FAQs
What are complementary angles give example., what are supplementary angles give examples., how to find complementary angles, what is the complementary angle of 40 degrees, how to find supplementary angles.
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My question is in Maths Class 7 : “What is the supplement of 2x-13?”
Supplementary angles sum up to 180 degrees. Therefore, 2x-13 = 180 2x = 180+13 x = 193/2 x = 96.5 degrees
What is the complementary angle of 60° ?
The complementary angle of 60° is 30° 60°+30° = 90°
What is the complement of 40°
The complement of 40° is 50° Since 90° – 40° = 50°
Let x and y are the two angles, supplementary to each other. As per the statement, say, y = 2x Therefore, x+y = 180 degrees (supplementary angles) x+2x = 180 3x = 180 x = 180/3 = 60 Hence, x = 60 degrees, y = 2x = 2(60) = 120 degrees
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Possibilities of a supplementary angle
Practice questions, supplementary angles – explanation & examples, what are supplementary angles.
Supplementary angles are pairs angles such that the sum of their angles is equal to 180 degrees.
Although the angle measurement of straight is equal to 180 degrees, a straight angle can’t be called a supplementary angle because the angle only appears in a single form. For angles to be called supplementary, they must add up to 180° and appear in pairs.
- An acute and obtuse angle
A supplementary angle can be composed of one acute angle and another obtuse angle.
Illustration:
∠ θ and ∠ β are supplementary angles because they add up to 180 degrees. ∠ θ is an acute angle, while ∠ β is an obtuse angle.
∠ θ and ∠ β are also adjacent angles because they share a common vertex and arm.
An acute angle is an angle whose measure of degree is more than zero degrees but less than 90 degrees.
On the other hand, an obtuse angle is an angle whose measure of degree is more than 90 degrees but less than 180 degrees.
Common examples of supplementary angles of this type include:
⟹ 120° and 60°
⟹ 30° and 150°
⟹ 100° + 80°
⟹ 140° and 40°
⟹ 160° and 20° etc.
- Two right angles
A supplementary angle can be made up of two right angles. A right angle is an angle that is exactly 90 degrees.
- Non-adjacent supplementary angles
Two pairs of supplementary angles don’t have to be in the same figure.
The two angles in the above separate figures are complementary, i.e., 140 0 + 40 0 = 180 0
How to Find Supplementary Angles?
We can calculate supplementary angles by subtracting the given one angle from 180 degrees. To find the other angle, use the following formula:
- ∠x = 180° – ∠y or ∠y = 180° – ∠x where ∠x or ∠y is the given angle.
Let’s work on the following examples.
Check whether the angles 127° and 53° are a pair of supplementary angles.
127° + 53° = 180°
Hence, 127° and 53° are pairs of supplementary angles.
Check if the two angles, 170°, and 19° are supplementary angles.
170° + 19° = 189°
Since 189°≠ 180°, therefore, 170° and 19° are not supplementary angles.
Given two supplementary angles as: (x – 2) ° and (x + 5) °, determine the value of x.
The sum of the angles must be equal to 180 degrees: (x – 2) + (2x + 5) = 180
⟹ x – 2 + 2x + 5 = 180
⟹ x + 2x – 2 + 5 = 180
⟹ 3x + 3 = 180
⟹ 3x + 3 – 3 = 180 — 3
⟹ 3x = 180 — 3
Divide both sides by 3 to get x as;
x = 59° Therefore, the value of x is 59°.
Calculate the value of θ in the figure below.
⟹ (5θ + 4°) + (θ – 2°) + (3θ + 7°) = 180°
⟹ 5θ + 4° + θ – 2° + 3θ + 7° = 180°
⟹ 5θ + θ + 3θ + 4° – 2° + 7° = 180°
⟹ 9θ + 9° = 180°
⟹ 9θ + 9° – 9° = 180° – 9°
⟹ 9θ = 171°
⟹ θ = 171/9
The ratio of a pair of supplementary angles is 1:8. Find the two measures of the two angles?
Let r be the common ratio.
One angle will be r, and the other will be 8r
Therefore, r + 8r = 180.
Substitute r = 20 in the initial equations.
Hence, one angle is 20 degrees, and the other is 160 degrees.
Therefore, the angles 20 degrees and 160 degrees are the two supplementary angles.
Determine the supplement angle of (x + 10) °.
⟹ (x + 10) ° = 180 ° – (x + 10) °
= 180° – 10° – x°
= (170 – x) °
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Understanding Supplementary Angles in Geometry
In the fascinating world of geometry, angles play a crucial role in defining shapes and relationships. Among the various types of angles, supplementary angles hold a special significance. Understanding supplementary angles is essential for grasping the intricacies of geometric figures and solving related problems.
What are Supplementary Angles?
Supplementary angles are two angles that add up to 180 degrees. Imagine a straight line. Now, if you draw a ray from any point on that line, you will create two angles that share the same vertex (the point where the ray meets the line) and whose measures add up to 180 degrees. These two angles are supplementary angles.
Identifying Supplementary Angles
Identifying supplementary angles is relatively straightforward. Look for two angles that share a common vertex and whose non-shared sides form a straight line. Here are some key points to remember:
- Supplementary angles always add up to 180 degrees.
- They share a common vertex.
- Their non-shared sides form a straight line.
Calculating Supplementary Angles
If you know the measure of one angle, you can easily calculate the measure of its supplement. Simply subtract the known angle from 180 degrees. For example, if one angle measures 70 degrees, its supplement would be 180 degrees – 70 degrees = 110 degrees.
Examples of Supplementary Angles
Let’s consider some real-world examples of supplementary angles:
- When you open a door, the angle between the door and the door frame forms a supplementary angle with the angle between the door and the wall.
- The angle formed by the hands of a clock at 3:00 is supplementary to the angle formed at 9:00.
- The angle of a ramp leading up to a building is supplementary to the angle of the ramp leading down.
Importance of Supplementary Angles
Understanding supplementary angles is essential for:
- Solving geometric problems involving triangles, quadrilaterals, and other polygons.
- Analyzing angles in real-world situations, such as architecture, engineering, and design.
- Developing a strong foundation in geometry and trigonometry.
Supplementary angles are a fundamental concept in geometry, playing a crucial role in understanding and manipulating shapes. By mastering the identification and calculation of supplementary angles, you can unlock a deeper understanding of geometric relationships and solve a wide range of problems.
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Appendix B: Geometry
Using properties of angles to solve problems, learning outcomes.
- Find the supplement of an angle
- Find the complement of an angle
Are you familiar with the phrase ‘do a [latex]180[/latex]?’ It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is [latex]180[/latex] degrees. See the image below.
[latex]\angle A[/latex] is the angle with vertex at [latex]\text{point }A[/latex].
We measure angles in degrees, and use the symbol [latex]^ \circ[/latex] to represent degrees. We use the abbreviation [latex]m[/latex] to for the measure of an angle. So if [latex]\angle A[/latex] is [latex]\text{27}^ \circ [/latex], we would write [latex]m\angle A=27[/latex].
If the sum of the measures of two angles is [latex]\text{180}^ \circ[/latex], then they are called supplementary angles. In the images below, each pair of angles is supplementary because their measures add to [latex]\text{180}^ \circ [/latex]. Each angle is the supplement of the other.
The sum of the measures of supplementary angles is [latex]\text{180}^ \circ [/latex].
The sum of the measures of complementary angles is [latex]\text{90}^ \circ[/latex].
Supplementary and Complementary Angles
If the sum of the measures of two angles is [latex]\text{180}^\circ [/latex], then the angles are supplementary .
If angle [latex]A[/latex] and angle [latex]B[/latex] are supplementary, then [latex]m\angle{A}+m\angle{B}=180^\circ[/latex].
If the sum of the measures of two angles is [latex]\text{90}^\circ[/latex], then the angles are complementary .
If angle [latex]A[/latex] and angle [latex]B[/latex] are complementary, then [latex]m\angle{A}+m\angle{B}=90^\circ[/latex].
In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.
In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.
Use a Problem Solving Strategy for Geometry Applications.
- Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
- Identify what you are looking for.
- Name what you are looking for and choose a variable to represent it.
- Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.
The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.
An angle measures [latex]\text{40}^ \circ[/latex].
1. Find its supplement
2. Find its complement
| 1. | |
| Step 1. the problem. Draw the figure and label it with the given information. | |
| Step 2. what you are looking for. | The supplement of a [latex]40°[/latex]angle. |
| Step 3. Choose a variable to represent it. | Let [latex]s=[/latex]the measure of the supplement. |
| Step 4. Write the appropriate formula for the situation and substitute in the given information. | [latex]m\angle A+m\angle B=180[/latex] [latex]s+40=180[/latex] |
| Step 5. the equation. | [latex]s=140[/latex] |
| Step 6. [latex]140+40\stackrel{?}{=}180[/latex] [latex]180=180\checkmark[/latex] | |
| Step 7. the question. | The supplement of the [latex]40°[/latex]angle is [latex]140°[/latex]. |
| 2. | ||||||||||||||||||||
| Step 1. the problem. Draw the figure and label it with the given information. | ||||||||||||||||||||
| Step 2. what you are looking for. |
Write the appropriate formula for the situation and substitute in the given information. [latex]m\angle A+m\angle B=90[/latex] Step 5. Solve the equation. [latex]c+40=90[/latex] [latex]c=50[/latex] Step 6. Check: [latex]50+40\stackrel{?}{=}90[/latex] In the following video we show more examples of how to find the supplement and complement of an angle. Did you notice that the words complementary and supplementary are in alphabetical order just like [latex]90[/latex] and [latex]180[/latex] are in numerical order? Two angles are supplementary. The larger angle is [latex]\text{30}^ \circ[/latex] more than the smaller angle. Find the measure of both angles.
  In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation. PROBLEMS ON COMPLEMENTARY AND SUPPLEMENTARY ANGLESC omplementary Angles : The two angles whose sum is 90˚  Supplementary Angles : The two angles whose sum is 180˚  In each case. find the missing angle : Example 1 :  In the diagram shown above, ∠ ABD and∠DBC are complementary angles. ∠ABD + ∠DBC = 90° Substitute ∠ABD = a° and ∠DBC = 25°. a° + 25° = 90° Subtract 25° from both sides. Example 2 :  In the diagram shown above, ∠PQS and∠SQR are complementary angles. ∠PQS + ∠SQR = 180° Substitute ∠PQS = 147° and ∠SQR = b°. 147° + b° = 180° Subtract 147° from both sides. Example 3 :  In the diagram shown above, ∠MNQ, ∠QNP, and ∠PNO are complementary angles. ∠MNQ + ∠QNP + ∠PNO = 90° Substitute ∠MNQ = 62°, ∠QNP = c° and ∠PNO = c°. 62° + c° + c° = 90° 62° + 2c° = 90° Subtract 62° from both sides. Divide both sides by 2. Example 4 :  In the diagram shown above, ∠ABE, ∠EBD and ∠DBC are supplementary angles. ∠ABE + ∠EBD + ∠DBC = 180° Substitute ∠ABE = d°, ∠EBD = 90° and ∠DBC = 30°. d° + 90° + 30° = 180° d° + 120° = 180° Subtract 120° from both sides. Example 5 :  m∠MNQ + m∠QNP + m∠PNO = 90° Substitute ∠MNQ = 12°, ∠QNP = e° and ∠PNO = 33°. 12° + e° + 33° = 90° 45° + e° = 90° Subtract 45° from both sides. Example 6 :  Substitute ∠MNQ = f°, ∠QNP = f° and ∠PNO = 38°. f° + f° + 38° = 90° 2f° + 38° = 90° Subtract 38° from both sides. Example 7 :  In the diagram shown above, ∠ABF, ∠FBE, ∠EBD and ∠DBC are supplementary angles. ∠ABF + ∠FBE + ∠EBD + ∠DBC = 180° Substitute ∠ABF = 32°, ∠EBD = 90° and∠FBE = ∠DBC = g°. 32° + g° + 90° + g° = 180° 2g° + 122° = 180° Subtract 122° from both sides. Example 8 :  ∠ABF, ∠FBE, ∠EBD and ∠DBC are supplementary angles. Substitute ∠ABF = h°, ∠FBE = 48°, ∠EBD = h°, ∠DBC = 74° h° + 48° + h° + 74° = 180° 2h° + 122° = 180° Example 9 :  Substitute ∠ABE = ∠EBD = ∠DBC = x°. x° + x° + x° = 180° Divide both sides by 3. Example 10 :  Substitute ∠ABF = 52°, ∠FBE = ∠EBD = y°, ∠DBC = 50°. 52° + y° + y° + 50° = 180° 2y° + 102° = 180° Subtract 102° from both sides. Kindly mail your feedback to [email protected] We always appreciate your feedback. © All rights reserved. onlinemath4all.com
Recent ArticlesSat math practice videos (part - 2). Jun 18, 24 11:18 AM SAT Math Videos on Exponents (Part - 1)Jun 18, 24 02:38 AM Honors Algebra 2 Problems with Solutions (Part - 1)Jun 18, 24 01:07 AM  Supplementary Angles – Definition, Types, Facts, Examples, FAQsWhat are supplementary angles, properties of supplementary angles, how to find supplementary angles, solved examples on supplementary angles, practice problems on supplementary angles, frequently asked questions about supplementary angles. Two angles are said to be supplementary when the sum of angle measures is equal to 18 0 . Note that the two angles need not be adjacent to be supplementary. So, what do supplementary angles look like? Take a look!
 Supplementary Angles: DefinitionSupplementary angles can be defined as a pair of angles that add up to 180°. Related Worksheets
What Are Adjacent and Non-adjacent Supplementary Angles?The supplementary angles can be both non-adjacent or adjacent. What does a supplementary angle look like? Let us move forward to learn about the two types of supplementary angles. Adjacent Supplementary Angles: The adjacent supplementary angles are the supplementary angles that share a common vertex and a common side. The angles together form a straight angle (and thus, a straight line). They are also called ‘angles in a linear pair’ or ‘linear pair of angles.’ Example: Consider the following diagram. The angles ∠XOZ and ∠YOZ are adjacent supplementary angles. They share a common vertex O and a common side OZ. Also, 60° + 120° = 180°  Non-Adjacent Supplementary Angles: The non-adjacent supplementary angles are the angles which are not adjacent, but the sum of their measures equals 180 degrees. Any two angles whose sum is 180 degrees are supplementary! 
For two angles X and Y to be supplementary, we must have m ∠ X + m ∠ Y = 180° You can also find the other counterpart of a given angle to find two supplementary angles. Example: What will be the supplementary angle of an angle of 70°? We know that the sum of two supplementary angles should be 180° Let us assume the missing angle to be x. 70° + x = 180° On solving for x, we get x = 110°. Supplementary Angles Theorem and ProofStatement: If two angles are supplementary to the same angle, they are congruent. Suppose ∠x and ∠y are two different angles supplementary to a third angle ∠a. ∠x + ∠a = 180° …….(1) ∠y + ∠a = 180° …….(2) From equations (1) and (2), we get ∠x + ∠a = ∠y + ∠a ∠x = ∠y Hence, proved. Facts about Supplementary Angles
In this article, we learned about the supplementary angles, definition, properties, and also the theorem.We also discussed how to find supplementary angles. Let us move ahead to the numerical section to have a better comprehension of the concepts through solved examples and practice problems. Example 1: Two angles are supplementary. Find the other angle if one angle is 80° . Let the missing angle be x. x + 80° = 180° …Angles are supplementary. Solving for x, we get x = 100° Therefore, the measure of the other supplementary angle is 100°. Example 2: Two angles that are supplementary. One angle is 35° greater than the other. Find the missing angle. Let us assume that the measure of one angle is x°. Then, other angle = x° + 35° So, x° + (x° + 35°) = 180° On simplifying: 2x° + 35° = 180° 2x°= 145° x° = 72.5° So, the other angle is: x° + 35° = 72.5° + 35° = 107.5° Therefore, 72.5° and 107.5° are the measures of the smaller and larger angles, respectively. Example 3: What will be the measures of two supplementary angles if the first angle is three times the second angle? Supplementary Angles - Definition, Types, Facts, Examples, FAQsAttend this quiz & Test your knowledge. What is the sum of two supplementary angles?What will be the other supplementary angle if one angle is 70 degrees, which of the following pairs of angles are supplementary, the supplementary angle of a right angle is always, a pair of supplementary angles are. Let the smaller angle be x°. Thus, the larger angle = 3x°. x° + 3x° = 180° …Angles are supplementary. On simplifying, we get 4x° = 180° x° = 45° So, the larger angle is: 3x° = 3(45°) x° = 135° Therefore, 45° and 135° are the values of the two supplementary angles. Example 4: If two supplementary angles are in the ratio 3 : 2, find the measures of angles. Solution: Let the measures of two angles be 3k° and 2k° respectively. 3k° + 2k° = 180° 5k° = 180° Now, we get the measures of the two angles. 3k° = (3 × 36)° = 108° 2k° = (2 × 36)° = 72° Therefore, 108° and 72° are the measures of two supplementary angles. What is the difference between complementary and supplementary angles? The complementary angles are a pair of angles that add up to 90°, while the supplementary angles add up to 180°. Can two acute angles be supplementary angles? No, two acute angles can never be supplementary because they can never add up to 180°. Do supplementary angles need to be adjacent? No, two supplementary angles may or may not be adjacent. What are angles in a linear pair? Are they supplementary? Angles in a linear pair are a pair of adjacent angles formed when two lines intersect each other. They are always supplementary. RELATED POSTS
 Math & ELA | PreK To Grade 5Kids see fun., you see real learning outcomes.. Make study-time fun with 14,000+ games & activities, 450+ lesson plans, and more—free forever. Parents, Try for Free Teachers, Use for Free SOLVING COMPLEMENTARY AND SUPPLEMENTARY WORD PROBLEMSComplementary angles : Two angles are complementary, if the sum of their measures is equal to 90. Supplementary angles : Two angles are supplementary angles if the sum of their measures is equal to 180 degrees. Problem 1 : Angles G and H are complementary. If m∠G = 3x + 6 and m∠H = 2x - 11. what is the measure of each angle? Complementary angles measure 90˚. ∠G + ∠H = 90˚ 3x + 6 + 2x - 11 = 90˚ 5x - 5 = 90˚ m∠G = 3x + 6 m∠G = 3(19) + 6 = 57 + 6 m∠H = 2x - 11 = 2(19) - 11 = 38 - 11 So, the measure of angles are m∠G = 63˚ and m∠H = 27˚. Problem 2 : The measures of angles A and B are supplementary. What is the measure of each angle? Supplementary angles measure 180˚. m∠A+ m∠B = 180˚ For example, Let m∠A = 60˚ 60 + m∠B = 180˚ m∠B = 180˚ - 60˚ m∠B = 120˚ So, the measure of angles are m∠A = 60˚ and m∠B = 120˚ Problem 3 : An angle is five its supplement. Find both angles. The two angles x and y are to be supplementary angles. x + y = 180˚ 5y + y = 180˚ Therefore, So, both angles are 150˚ and 30˚. Problem 4 : An angle is 74 degrees more than its complement. Find both angles. Let one angle = x Complementary angle = x + 74 x + x + 74 = 90˚ 2x + 74 = 90˚ 2x = 90 - 74 one angle = 8˚ Second angle = x + 74 So, both angles are 8˚ and 82˚. Problem 5 : The supplement of an angle exceeds the angle by 60 degrees. Find both angles. Solution: Let one angle is x and second angle is x + 60˚ x + x + 60 = 180˚ 2x + 60 = 180 2x = 180 - 60 Second angle = x + 60˚ So, both angles are 60˚ and 120˚. Problem 6 : Find the number of degrees in an angle which is 42 less than its complement. Let the angle be x. Complementary angle = 90˚ - x x = (90 - x) - 42 First angle = 24˚ Second angle = 90 - x Problem 7 : Find the number of degrees in an angle which is 120 less than its supplement. Supplementary angle = 180˚ - x x = (180 - x) - 120 First angle = 30˚ Second angle = 180 - x Problem 8 : The complement of an angle is 30 less than twice the angle. Find the larger angle. 90 - x = 2x - 30 90 + 30 = 2x + x First angle = 40˚ Second angle = 90˚ - x Problem 9 : Angles A and B are complementary. If m∠A = 3x - 8 and m∠B = 5x + 10, what is the measure of each angle? ∠A + ∠B = 90˚ 3x - 8 + 5x + 10 = 90˚ 8x + 2 = 90˚ m∠A = 3x - 8 m∠A = 3(11) - 8 = 33 - 8 m∠B = 5x + 10 = 5(11) + 10 = 55 + 10 So, the measure of angles are m∠A = 25˚ and m∠B = 65˚. Problem 10 : Angles Q and R are supplementary. If m∠Q = 4x + 9 and m∠R = 8x + 3, what is the measure of each angle? ∠Q + ∠R = 180˚ 4x + 9 + 8x + 3 = 180˚ 12x + 12 = 180˚ m∠Q = 4x + 9 m∠Q = 4(14) + 9 = 56 + 9 m∠R = 8x + 3 = 8(14) + 3 = 112 + 3 So, the measure of angles are m∠Q = 65˚ and m∠R = 115˚.
Recent ArticlesFinding range of values inequality problems. May 21, 24 08:51 PM Solving Two Step Inequality Word ProblemsMay 21, 24 08:51 AM Exponential Function Context and Data ModelingMay 20, 24 10:45 PM © All rights reserved. intellectualmath.com You are using an outdated browser. Please upgrade your browser to improve your experience.  UPDATED 07:00 EDT / JUNE 17 2024  D-Wave announces new hybrid quantum solver for commercial applications supporting 2M variables  by Kyt Dotson Quantum computing systems maker D-Wave Quantum Inc. today announced the launch of a new commercial hybrid quantum computing system that can solve problems for practical commercial applications by supporting 2 million variables and constraints, double the number of its previous solutions. The announcement, which took place at the company’s Qubits 2024 user conference, revealed that its new system also demonstrated up to 10 times the capacity of D-Wave’s other solvers, according to the company’s benchmarking. The Burnaby, Canada-based D-Wave is an early trailblazer in the field of quantum computing and introduced the world’s first commercial-use quantum computer in 2011. Now it builds full-stack solutions for companies to tackle real-world practical commercial problems using quantum computers. It also provides companies solutions by combining classical and quantum computers for what it calls “ hybrid solvers. ” These are computing systems that take problems as far as they will go with traditional computing, determine which parts will benefit from quantum computing and spin those out. The new solution is now available on D-Wave’s Leap quantum cloud service, which provides real-time access to the company’s Advantage quantum computer and its full-stack platform for programming solutions. Murray Thom, vice president of quantum tech evangelism, told SiliconANGLE in an interview that this combination and D-Wave’s early entry into the quantum industry mean that it already has been rolling out practical applications for commercial business for years. Users don’t need Ph.D.s or science degrees to build solutions on its systems to get real applicable results. “T here are people talking about revolutionary applications for quantum computing, and certainly those folks who have very early-stage technology products, they’re talking about revolutionary applications because they’re a long way away from it, ” said Thom. “ So, they’re like, the application has to be big who weren’t the journey they need to take. Fortunately for our customers, we’ve passed that journey already .” Thom said prime examples include handling logistics and tracking for large companies such as grocery delivery, which have complex network effects taking place across teams in real time. “Pattison Food Group has 11 grocery brands in the western North American region and during the pandemic, at-home driver delivery was really important for their groceries, ” Thom explained. “ We worked with them to develop and optimize the scheduling of those at-home delivery drivers and that application saved them 80% of the time it took them to produce those schedules .” Though classical computers excel at optimizing predictable routes for deliveries , they can struggle when faced with real-world disruptions such as driver absences or accidents. These unforeseen events create branching possibilities that can overwhelm traditional optimization methods. That’s where quantum computers come into play with their ability to explore multiple solutions simultaneously using parallel processing and handle inherent uncertainties with probabilistic solving. This makes quantum computing ideal for tackling these complex scenarios. D-Wave doesn’t just want to provide a great solution for companies to be able to tackle complex problems. I t also wants to make it simple for them to describe their problems and invent new ways to tackle them using the solver. “ W e’re focused on making it easier for people to build applications with this technology, ” said Thom. “ And that’s the focus of this new nonlinear programming solver that we’re releasing. So, it’s got a more powerful way to kind of represent problems .” As a result, developers can quickly represent the problem in a way that can be resolved by a quantum solution. Especially with the larger number of variables and constraints, the new quantum solver is capable of handling larger versions of real-world problems. D-Wave also provides access to expert guidance to help developers better describe their problems in a way that be represented in quantum instructions. Image: SiliconANGLE/Microsoft DesignerA message from john furrier, co-founder of siliconangle:, your vote of support is important to us and it helps us keep the content free., one click below supports our mission to provide free, deep, and relevant content. , join our community on youtube, join the community that includes more than 15,000 #cubealumni experts, including amazon.com ceo andy jassy, dell technologies founder and ceo michael dell, intel ceo pat gelsinger, and many more luminaries and experts.. Like Free Content? Subscribe to follow. LATEST STORIES Runway debuts new Gen-3 Alpha model for generating videos  NICE customers provide best practices for adopting AI in the contact center  Adobe integrates more generative AI features into Acrobat  Oracle's APEX now supports natural language prompts to ease app development  Israeli startup Aim Security raises $18M to bolster generative AI security  AI - BY MARIA DEUTSCHER . 52 MINS AGO AI - BY ZEUS KERRAVALA . 2 HOURS AGO AI - BY MARIA DEUTSCHER . 4 HOURS AGO AI - BY MIKE WHEATLEY . 5 HOURS AGO SECURITY - BY DUNCAN RILEY . 8 HOURS AGO EMERGING TECH - BY KYT DOTSON . 9 HOURS AGO A cloud-storage company is using AI to save employees weeks of cybersecurity-threat work
 For "CXO AI Playbook," Business Insider takes a look at mini case studies about AI adoption across industries, company sizes, and technology DNA. We've asked each of the featured companies to tell us about the problems they're trying to solve with AI, who's making these decisions internally, and their vision for using AI in the future. Pure Storage has been providing cloud-storage systems for more than a decade and is trusted by some of the world's largest organizations, such as ServiceNow and Domino's Pizza. It uses generative AI to help make its 2,000 engineers even more efficient. Situation analysis: What problem was the company trying to solve?Pure Storage was founded in 2009 and, according to its website, serves over 11,000 global customers. As a result, it has extensive institutional knowledge in providing data storage to companies. However, making use of that knowledge across complex business processes was a challenge, Ratinder Paul Singh Ahuja, the company's chief technology officer for security and networking, told Business Insider. That changed with the popularization of generative-AI platforms in 2022. "I could see how this could be used in a number of business processes," Ahuja said. "We put together a company-level initiative to do what we call a generative-AI-powered enterprise." Ahuja said the company considered several options for deploying generative artificial intelligence, such as using it to sift through queries, support its internal help desk, or help the company's finance arm. But the most pressing improvement Ahuja wanted to make was speeding up and bolstering the checks his security team carried out. He quickly hit upon two key areas where AI could help. Normally, his development, security, and operations program would have to sit through design discussions with company teams and try to find security issues to fix before rolling out products. It was laborious and time-consuming. The security team at Pure Storage would also be overwhelmed by threat announcements — when providers of hardware or software used by companies, including Pure Storage, announce they've found vulnerabilities in their code that need fixing. But these reports are often for products not used by Pure Storage or don't affect them, so filtering through the announcements is hard. Key staff and partnersThe process of implementing these uses of AI was led by Ahuja, who demonstrated the early examples to the executive team at Pure Storage. He said that at Pure Storage, the office of the technology officer had more freedom and ability to explore new technologies than the IT department: "They kept it under the office of CTO, as opposed to IT services, just because the field is changing rapidly, and we wanted to have the ability to not be rigidly covered by an IT process." AI in actionIn its security department, Pure Storage now uses a generative-AI tool that was trained with Ahuja's presentation slides and knowledge about modeling threats — similar to the way a human staff member would be trained on best practices. "This GPT can now be cut and pasted as a picture of a design, any documentation or code you have written, and it will walk through the STRIDE methodology," Ahuja said, referring to a standard threat-modeling methodology that stands for spoofing, tampering, repudiation, information disclosure, denial of service, and elevation of privilege. Other Pure Storage teams, not just security, can use the program, meaning they don't have to wait for a security specialist to become free to check their plans. The generative-AI tool scanning the vast volumes of threat announcements and warnings can quickly triage what human security professionals need to pay attention to and what they can ignore. The tech analyzes the threat feed and asks what class of system is affected and which signs to look for to detect the issue. "Then it queries our asset database, and says, 'Do we have this class of systems? Should I even worry about it?" Ahuja said. If the answer is yes, it'll continue to analyze until it's convinced it needs to flag a human, he said. Did it work, and how did leaders know?Pure Storage's AI model is designed to poke holes in new features, products, or services, probing for weaknesses that cybercriminals could exploit. "What used to take a couple of weeks is now an hour's job, with the bot guiding the different teams through the STRIDE methodology," he said. "This is really popular with our engineering teams because they don't have to wait for a security expert." Meanwhile, Ahuja said, the triaging tool is so helpful it's as if the security-operations team has added another worker: "This is really powerful. You could not keep up. They were constantly underresourced." What's next?Ahuja wants to layer AI on top of Pure Storage's products. "Gen AI is really good at analyzing configurations — it's very good at generating code," he said. "If you look at Pure Storage and many other vendors, we put out complex systems, and you have to configure them." He believes generative AI can help automate large parts of that process. We want to hear from you. If you are interested in sharing your company's AI journey, email [email protected] .
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Supplementary angles, formula, examples, lessons, definition, rule and practice problems. Math Gifs; Algebra; ... since this is a ratio problem, we will let the larger angle be 2x and the smaller angle x. We know that $$ 2x + 1x = 180$$ , so now, let's first solve for x: $$ 3x = 180° \\ x = \frac{180°}{3} = 60° $$ Now, the larger angle ...
Two angles are called complementary if their measures add to 90 degrees, and called supplementary if their measures add to 180 degrees. Note that in these definitions, it does not matter whether or not the angles are adjacent; only their measures matter. For example, a 50-degree angle and a 40-degree angle are complementary; a 60-degree angle ...
Complementary and supplementary angles (visual) Google Classroom. What is the measure of ∠ x ? Angles are not necessarily drawn to scale. x = ∘. Loading... Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of ...
Example 6: Two supplementary angles are such that the measure of one angle is 3 times the measure of the other. Determine the measure of each angle. Let [latex]x^\circ[/latex] be the measure of the first angle. Since the second angle measures 3 times than the first, then it will be [latex]3x ^\circ[/latex].
Free supplementary angles math topic guide, including step-by-step examples, free practice questions, teaching tips, and more! Math Tutoring for Schools. How it Works ... Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, for example, by using an equation with a symbol for the ...
Supplementary angles add up to 180°. - example: 50° & 130° are supplementary. (added together, they form a straight line) Two facts: (1) 90° comes before 180° on the number line. (2) "C" comes before "S" in the alphabet. You can use this to help you remember! 90° goes with "C" for complementary. so complementary angles add up to 90°.
In a right triangle, the two acute angles are complementary. This is because the sum of angles in a triangle is 180˚ and the right angle is 90˚. Therefore, the other two angles must add up to 90˚. Example: x and y are complementary angles. Given x = 35˚, find the value y. Solution: x + y = 90˚. 35˚ + y = 90˚.
The angles with measures \(a\)° and \(b\)° lie along a straight line. Since straight angles have measures of 180°, the angles are supplementary. Example problems with supplementary angles. Let's look at a few examples of how you would work with the concept of supplementary angles. Example. In the figure, the angles lie along line \(m\).
Two Angles are Supplementary when they add up to 180 degrees. These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°: Notice that together they make a straight angle. But the angles don't have to be together. These two are supplementary because. 60° + 120° = 180°.
Thus, the supplement of angle 105° = 180° - 105° = 75°. Solved problems on complementary and supplementary angles: 1. Find the complement of the angle 2/3 of 90°. Solution: Convert 2/3 of 90°. 2/3 × 90° = 60°. Complement of 60° = 90° - 60° = 30°. Therefore, complement of the angle 2/3 of 90° = 30°.
This geometry video tutorial explains how to solve algebra problems associated with complementary angles and supplementary angles.Geometry Introduction: ...
Step 1: Convert 1/3 of 210°. That is, (1/3) x 210° = 70°. Step 2: Supplement of 70° = 180° - 70° = 110°. Therefore, the supplement of the angle 1/3 of 210° is 110°. Example 3: The measures of the two angles are (x + 25)° and (3x + 15)°. Find the value of x if angles are supplementary angles. Solution:
So, the measure of the complementary angle is 45°. Problem 4 : Two angles are supplementary. If one angle is 36° less than twice of the other angle, find the two angles. Solution : Let x and y be the two angles which are supplementary. x + y = 180 ---->(1) Given : One angle is 36° less than twice of the other angle. x = 2y - 36 ---->(2)
Let's work on the following examples. Example 1. Check whether the angles 127° and 53° are a pair of supplementary angles. Solution. 127° + 53° = 180°. Hence, 127° and 53° are pairs of supplementary angles. Example 2. Check if the two angles, 170°, and 19° are supplementary angles. Solution.
If the angle is supplementary, the equation for x would be 2x+20+3x+60=180. If the angle is complementary, the equation would be 2x+20+3x+60=90. The answer for supplementary would be 60. The answer for complementary would be 24. Hope this helped! ( 0 votes) Upvote. Downvote.
Two angles are supplementary angles if the sum of their measures is equal to 180 degrees. Problem 1 : An angle measures 38° less than its complement. Solution : The required angle be x. If two angles are complementary to each other, then its sum will be 90 degree. Required angle = 90 - 38.
The angle formed by the hands of a clock at 3:00 is supplementary to the angle formed at 9:00. The angle of a ramp leading up to a building is supplementary to the angle of the ramp leading down. Importance of Supplementary Angles. Understanding supplementary angles is essential for: Solving geometric problems involving triangles ...
If the sum of the measures of two angles is 90∘ 90 ∘, then the angles are complementary. If angle A A and angle B B are complementary, then m∠A+m∠B =90∘ m ∠ A + m ∠ B = 90 ∘. In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications.
Supplementary angles : Two angles are supplementary angles if the sum of their measures is equal to 180 degrees. Problem 1 : Angles A and B are complementary. If m∠A = 3x - 8 and m∠B = 5x + 10, what is the measure of each angle ? Solution : Since angles A and B are complementary, m∠A + m∠B = 90. 3x - 8 + 5x + 10 = 90.
Step 1: Identify the two supplementary angles given as algebraic expressions in the given word problem. Step 2: Set an equation to show the sum of two algebraic expressions equals 180 ∘ . Step 3 ...
PROBLEMS ON COMPLEMENTARY AND SUPPLEMENTARY ANGLES. In each case. find the missing angle : Example 1 : Solution : In the diagram shown above, ∠ ABD and∠DBC are complementary angles. ∠ABD + ∠DBC = 90°. Substitute ∠ABD = a° and ∠DBC = 25°. a° + 25° = 90°. Subtract 25° from both sides.
Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. ... Complementary and supplementary angles (no visual) Complementary and supplementary angles review. ... Report a problem
So, the larger angle is: 3x° = 3 (45°) x° = 135°. Therefore, 45° and 135° are the values of the two supplementary angles. Example 4: If two supplementary angles are in the ratio 3 : 2, find the measures of angles. Solution: Let the measures of two angles be 3k° and 2k° respectively. 3k° + 2k° = 180°. 5k° = 180°.
Problem 3 : An angle is five its supplement. Find both angles. Solution: The two angles x and y are to be supplementary angles. x + y = 180˚ Then, x = 5y. 5y + y = 180˚ 6y = 180˚ y = 30˚ Therefore, x = 5(30˚) x = 150˚ So, both angles are 150˚ and 30˚. Problem 4 : An angle is 74 degrees more than its complement. Find both angles. Solution:
Quantum computing systems maker D-Wave Quantum Inc. today announced the launch of a new commercial hybrid quantum computing system that can solve problems for practical commercial applications by sup
An icon in the shape of an angle pointing down. ... Situation analysis: What problem was the company trying to solve? Pure Storage was founded in 2009 and, according to its website, serves over ...