homework 2 central angles arc measures and arc lengths

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Lesson Explainer: Central Angles and Arcs Mathematics • Third Year of Preparatory School

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In this explainer, we will learn how to identify central angles, use their measures to find measures of arcs, identify adjacent arcs, find arc lengths, and identify congruent arcs in congruent circles.

We begin by defining exactly what is meant by an arc of a circle.

Definition: Arc of a Circle

• An arc of a circle is a section of the circumference of the circle between two radii.

We can see some examples of arcs of circles in the following diagrams.

To help us differentiate between different arcs, we introduce the idea of the central angle.

Definition: Central Angle

A central angle of a circle is an angle between two radii with the vertex at the center. In the following diagram, ∠ 𝐴 𝐵 𝐶 is an example of a central angle.

We can extend this idea to say that the central angle of an arc is the central angle subtended by the arc.

For example, the central angles of the two given arcs are shown in the following diagrams.

We can see that the larger the central angle, the larger the arc. Therefore, it would be useful to talk about the measure of the central angle of the arc with regards to the length of the arc. We do this by introducing the following definition.

Definition: Measure of an Arc

• The measure of an arc is the measure of its central angle.

For example, in the diagram below, the measure of the arc in red is 2 6 ∘ .

We can notice something interesting in this diagram: there are two possible arcs from 𝐴 to 𝐵 , the shorter one in red and the longer one in green. To help us differentiate between these two cases, we call the longer arc the major arc and the shorter arc the minor arc.

Definition: Major and Minor Arcs of a Circle

Given two radii, we denote the longer of the two arcs between the radii as the major arc and the shorter of the arcs as the minor arc . Equivalently, the arc with the smaller central angle is the minor arc and the arc with the larger central angle is the major arc.

To help differentiate between the major and minor arcs, we denote the minor arc as 𝐴 𝐵 and label the major arc using an extra point (for example, 𝐴 𝐶 𝐵 ).

We can also use the notation 𝑚 𝐴 𝐵 for the measure of the minor arc from 𝐴 to 𝐵 . In this case, we can use 𝑚 𝐴 𝐶 𝐵 for the measure of the major arc from 𝐴 to 𝐵 .

If the two arcs are the same length, then we call these semicircular arcs . These occur when the radii form a diameter or when their central angles have equal measures.

Since the size of the central angle of an arc determines its size, we define major and minor arcs in terms of their central angles. If the central angle is greater than 1 8 0 ∘ , then the arc is major. If the central angle is less than 1 8 0 ∘ , then the arc is minor. If the central angle is equal to 1 8 0 ∘ , then the arc is semicircular.

In our first example, we will determine the measure of an arc given its central angle.

Example 1: Finding the Measure of an Arc given Its Central Angle

Find 𝑚 𝐴 𝐷 .

We recall that the notation 𝑚 𝐴 𝐷 means the measure of the minor arc between 𝐴 and 𝐷 and that the measure of an arc is defined to be its central angle. We highlight this arc on the following diagram.

The central angle of an arc is the angle at the center of the circle between the two radii subtended by the arc. For the minor arc 𝐴 𝐷 , this is 3 3 ∘ . The measure of the arc is defined to be equal to this value. Hence, 𝑚 𝐴 𝐷 = 3 3 . ∘

Before we move on to more examples, there is one more definition we need to discuss, which is that of adjacent arcs.

Definition: Adjacent Arcs

• We say that two arcs are adjacent if they share a single point in common or if they share only their endpoints in common.

In the circle above, 𝐴 𝐵 and 𝐵 𝐶 are adjacent since they share a single point in common. Similarly, 𝐴 𝐶 and 𝐴 𝐷 𝐶 are adjacent since they only share both endpoints in common.

In fact, the major and minor arcs of a circle between two points will always be adjacent.

Since the measure of an arc is the measure of its central angle and adjacent arcs will have adjacent central angles, we can find the measure of adjacent arcs by adding their measures. For example, in the circle above, we have 𝑚 𝐴 𝐶 = 𝑚 𝐴 𝐵 + 𝑚 𝐵 𝐶 .

Let’s see an example of identifying adjacent arcs in a circle.

Example 2: Identifying Adjacent Arcs in a Circle

Which of the following arcs are adjacent in the given circle?

• 𝐴 𝐵 and 𝐶 𝐷
• 𝐴 𝐵 and 𝐵 𝐶
• 𝐴 𝐷 and 𝐵 𝐶
• 𝐴 𝐶 and 𝐷 𝐵

We recall that two arcs are adjacent if they share a single point in common and that the notation 𝐴 𝐵 means the minor (or smaller) arc from 𝐴 to 𝐵 . Therefore, we can answer this question by highlighting each pair of arcs. Let’s start with 𝐴 𝐵 and 𝐶 𝐷 .

We see that the arcs share no points in common, so they cannot be adjacent. Next, we highlight 𝐴 𝐵 and 𝐵 𝐶 .

We can see that 𝐴 𝐵 and 𝐵 𝐶 only share the point 𝐵 in common; it is the end point of both arcs, so these arcs are adjacent. For due diligence, we will also check the other options.

We have 𝐴 𝐷 and 𝐵 𝐶 .

We can see that these arcs share no points in common, so they are not adjacent.

Finally, we check 𝐴 𝐶 and 𝐷 𝐵 .

We can see that every point on arc 𝐵 𝐶 lies on both arcs, so this pair of arcs share more than one point in common. Hence, they are not adjacent.

The only pair of arcs to share a single point in common is the pair of arcs 𝐴 𝐵 and 𝐵 𝐶 , which is option B.

In our next example, we will determine the measure of an arc using a diagram and knowledge of the ratio of two other arc measures.

Example 3: Finding an Arc’s Measure in a Circle given the Other Arcs’ Measures by Solving Linear Equations

Given that 𝐴 𝐵 is a diameter in a circle of center 𝑀 and 𝑚 𝐴 𝐶 ∶ 𝑚 𝐷 𝐵 = 8 5 ∶ 6 7 , determine 𝑚 𝐴 𝐶 𝐷 .

We want to determine the value of 𝑚 𝐴 𝐶 𝐷 . We recall that this is the measure of the arc from 𝐴 to 𝐶 to 𝐷 , as shown in the following diagram.

We can see that this arc consists of two adjacent arcs: 𝐴 𝐶 and 𝐶 𝐷 . We can therefore find the measure of 𝐴 𝐶 𝐷 by finding the sum of the measures of 𝐴 𝐶 and 𝐶 𝐷 .

Since the measure of an arc is equal to its central angle, 𝑚 𝐶 𝐷 = 𝑚 ∠ 𝐶 𝑀 𝐷 . We are given 𝑚 ∠ 𝐶 𝑀 𝐷 = 2 8 ∘ , so we have 𝑚 𝐶 𝐷 = 2 8 . ∘

We know that the sum of the measures of all the arcs that make up the circle will be 3 6 0 ∘ . In particular, the sum of the measures of the arcs that make up 𝐴 𝐵 will be 1 8 0 ∘ since 𝐴 𝐵 is a diameter. This means

We are told that 𝑚 𝐴 𝐶 ∶ 𝑚 𝐷 𝐵 = 8 5 ∶ 6 7 .

Therefore, the quotients of each part of the ratio must be equal: 𝑚 𝐴 𝐶 𝑚 𝐷 𝐵 = 8 5 6 7 .

We can rearrange this equation to get 𝑚 𝐷 𝐵 = 6 7 𝑚 𝐴 𝐶 8 5 .

We can substitute our expression for 𝑚 𝐷 𝐵 into equation (1) and simplify to get 𝑚 𝐴 𝐶 + 6 7 𝑚 𝐴 𝐶 8 5 = 1 5 2 1 5 2 𝑚 𝐴 𝐶 8 5 = 1 5 2 𝑚 𝐴 𝐶 =  8 5 × 1 5 2 1 5 2  𝑚 𝐴 𝐶 = 8 5 . ∘ ∘ ∘ ∘

Finally, 𝑚 𝐴 𝐶 𝐷 = 𝑚 𝐴 𝐶 + 𝑚 𝐶 𝐷 = 8 5 + 2 8 = 1 1 3 . ∘ ∘ ∘

Since an arc of a circle is a portion of its circumference, we can use the circumference of the circle to determine the length of the arc. We can do this by using the measure of the arc or, equivalently, its central angle.

To help us determine the length of an arc, let’s start with an example. We want to determine the length of the minor arc in the following diagram.

First, recall that a circle of radius 𝑟 has circumference 2 𝜋 𝑟 . This means the circumference of this circle is 2 𝜋 𝑟 .

We can see that this arc represents one-quarter of the circle, but it is good practice to see why this is the case. A full turn is an angle of 3 6 0 ∘ , so a 9 0 ∘ angle is 9 0 3 6 0 = 1 4 of the circle.

Hence, the arc length is one-quarter of the circumference: a r c l e n g t h = 1 4 ( 2 𝜋 𝑟 ) = 𝜋 𝑟 2 .

In general, if the central angle (or arc measure) is 𝜃 ∘ , then the arc length is 𝜃 3 6 0 ( 2 𝜋 𝑟 ) ∘ ∘ . We can state this formally as follows.

Definition: Length of an Arc

If the central angle (or measure) of an arc in a circle of radius 𝑟 is 𝜃 ∘ , then the length of the arc, 𝑙 , is given by 𝑙 = 𝜃 3 6 0 ( 2 𝜋 𝑟 ) . ∘ ∘

In our next example, we will use the formula for the length of an arc to determine the measure of an arc that gives a specific proportion of the circumference of a circle.

Example 4: Finding the Measure of the Arc That Represents a Known Part of the Circumference of a Circle

Find the measure of the arc that represents 1 6 of the circumference of a circle.

To answer this question, we first recall that the length, 𝑙 , of an arc of measure 𝑥 ∘ in a circle of radius 𝑟 is given by 𝑙 = 𝑥 3 6 0 ( 2 𝜋 𝑟 ) . ∘ ∘

We want this value to be equal to 1 6 the circumference of the circle, and we know a circle of radius 𝑟 has circumference 2 𝜋 𝑟 . So, we want 𝑙 = 1 6 ( 2 𝜋 𝑟 ) = 1 3 ( 𝜋 𝑟 ) .

Setting these two expressions for 𝑙 to be equal gives us 𝑥 3 6 0 ( 2 𝜋 𝑟 ) = 1 3 ( 𝜋 𝑟 ) . ∘ ∘

We can then solve for 𝑥 ∘ . We divide through by 𝜋 𝑟 to get 𝑥 3 6 0 ( 2 ) = 1 3 . ∘ ∘

Finally, we multiply through by 1 8 0 ∘ and simplify: 𝑥 = 1 3 ( 1 8 0 ) = 6 0 . ∘ ∘ ∘

It is worth noting there is another method of answering this question. We can note that the proportion of the measure of an arc to 3 6 0 ∘ is exactly the same as the proportion of the arc length to the circumference. In other words, 𝑥 3 6 0 = . ∘ ∘ a r c l e n g t h c i r c u m f e r e n c e

We are told a r c l e n g t h c i r c u m f e r e n c e = 1 6 , so we have 𝑥 3 6 0 = 1 6 , ∘ ∘ which we solve and get 𝑥 = 6 0 ∘ ∘ .

There is an important corollary to the arc length formula involving congruent arcs in a circle, which we will discuss now.

Property: Congruent Arcs

Since the length of an arc is determined by its central angle (or measure) and the radius of the circle, we can conclude that if two arcs in circles with equal radii have the same length, then their central angles (and measures) will be equal. In other words, two arcs are congruent if and only if their central angles (or measures) are equal.

For example, we can use this formula to determine the arc length in the following diagram.

The radius of this circle is 2 and the central angle is 3 0 ∘ , so we have 𝑙 = 3 0 3 6 0 ( 2 𝜋 ( 2 ) ) = 𝜋 3 . ∘ ∘ l e n g t h u n i t s

This also tells us that any arc of length 𝜋 3 in this circle, or in any circle with a radius of 2 length units, will have a measure of 3 0 ∘ .

Let’s now see an example of how we can apply this congruence property of arcs to determine an arc length in a circle.

Example 5: Understanding the Relationship between Arcs with Equal Lengths

Consider a circle 𝑀 with two arcs, 𝐴 𝐵 and 𝐶 𝐷 , that have equal measures. 𝐴 𝐵 has a length of 5 cm . What is the length of 𝐶 𝐷 ?

We are told that 𝐴 𝐵 and 𝐶 𝐷 have the same measure and we can recall that if two arcs have the same measure, then they are congruent. So, their lengths are equal. Hence, 𝐶 𝐷 has a length of 5 cm .

Although not necessary in answering this question, it can be worth seeing why this result holds true from the formula for the length of an arc. We recall that the length, 𝑙 , of an arc between 𝑃 and 𝑄 in a circle of radius 𝑟 is given by the formula 𝑙 = 𝑚 𝑃 𝑄 3 6 0 ( 2 𝜋 𝑟 ) .

So, the length of 𝐴 𝐵 is given by 𝑙 = 𝑚 𝐴 𝐵 3 6 0 ( 2 𝜋 𝑟 ) .

Since 𝑚 𝐴 𝐵 = 𝑚 𝐶 𝐷 , we have 𝑙 = 𝑚 𝐴 𝐵 3 6 0 ( 2 𝜋 𝑟 ) = 𝑚 𝐶 𝐷 3 6 0 ( 2 𝜋 𝑟 ) .

However, this expression is the length of 𝐶 𝐷 , so their lengths are equal.

Hence, 𝐶 𝐷 has a length of 5 cm .

Another similar property to the one in the example above is that if the lengths of the chords between the endpoints of two arcs in a circle are equal, then the arcs have equal measures. In fact, the same is true in reverse; if the arcs have the same measure, then the chords between their respective endpoints will have equal lengths.

To see why this is true, consider the following circle.

Let’s suppose that 𝐴 𝐵 and 𝐶 𝐷 have equal measures. Then, the measures of the central angles are equal: 𝑚 ∠ 𝐷 𝑀 𝐶 = 𝑚 ∠ 𝐴 𝑀 𝐵 .

We also know that 𝐴 𝑀 , 𝐵 𝑀 , 𝐶 𝑀 , and 𝐷 𝑀 are radii, so they have the same length. Hence, triangles 𝐴 𝑀 𝐵 and 𝐷 𝑀 𝐶 are congruent by the SAS rule, so 𝐴 𝐵 and 𝐷 𝐶 must have the same length.

Similarly, if 𝐴 𝐵 and 𝐶 𝐷 have the same length, then, by using the radii of the circle, we have that triangles 𝐴 𝑀 𝐵 and 𝐷 𝑀 𝐶 are congruent by the SSS rule. So, the measures of the internal angles are equal. In particular, 𝑚 ∠ 𝐷 𝑀 𝐶 = 𝑚 ∠ 𝐴 𝑀 𝐵 .

Then, since the measures of the central angles are equal, we know that their measures (and arc lengths) are equal.

Property: Congruent Chords of Congruent Arcs

In the same circle or in congruent circles, if two arcs have the same measure, then the chords between their respective endpoints will have equal lengths. In fact, the same is true in reverse; in the same circle or in congruent circles, if two chords between the endpoints of two arcs are congruent, then the two arcs have the same measure. We can see this in the following diagram.

• If 𝑚 𝐴 𝐵 = 𝑚 𝐷 𝐶 , then 𝐴 𝐵 = 𝐷 𝐶 .
• If 𝐴 𝐵 = 𝐷 𝐶 , then 𝑚 𝐴 𝐵 = 𝑚 𝐷 𝐶 .

Let’s see an example of how we can apply this property.

Example 6: Understanding the Relationship between Arcs and Chords

Consider circle 𝑀 with two chords of equal lengths, 𝐴 𝐷 and 𝐵 𝐶 . If 𝐴 𝐷 has a length of 5 cm , what is the length of 𝐵 𝐶 ?

We see that 𝐴 𝐷 and 𝐵 𝐶 are the chords between the endpoints of arcs 𝐴 𝐷 and 𝐵 𝐶 as shown.

We then recall that if the lengths of the chords between the endpoints of two arcs in a circle are equal, then the arcs have equal lengths and measures. Therefore, since 𝐴 𝐷 and 𝐵 𝐶 have the same length, 𝐴 𝐷 and 𝐵 𝐶 will also have the same length.

Hence, since 𝐴 𝐷 has a length of 5 cm , 𝐵 𝐶 also has a length of 5 cm .

In our next example, we will use a diagram and the properties of central angles to determine the measure of a given arc.

Example 7: Finding the Measure of an Arc in a Circle given a Diameter and the Measures of Two Central Angles in the Form of Algebraic Expressions

Given that 𝐴 𝐵 is a diameter in circle 𝑀 and 𝑚 ∠ 𝐷 𝑀 𝐵 = ( 5 𝑥 + 1 2 ) ∘ , determine 𝑚 𝐴 𝐶 .

We are asked to find 𝑚 𝐴 𝐶 , which is the measure of the minor arc from 𝐴 to 𝐶 , the arc shown in the following diagram.

We recall that the measure of an arc is equal to its central angle, and we can see in the diagram the central angle for this arc is 4 𝑥 ∘ . So, 𝑚 𝐴 𝐶 = 4 𝑥 ∘ . Therefore, we need to determine the value of 𝑥 . To find the value of 𝑥 , we will start by adding the angle we are given in the question to the diagram.

We then note that 𝐴 𝐵 is a diameter of the circle, which means it is a straight line. So, we must have 𝑚 ∠ 𝐷 𝑀 𝐵 + 𝑚 ∠ 𝐷 𝑀 𝐴 = 1 8 0 ( 5 𝑥 + 1 2 ) + 2 𝑥 = 1 8 0 . ∘ ∘ ∘ ∘

We can then solve this for 𝑥 7 𝑥 + 1 2 = 1 8 0 7 𝑥 = 1 6 8 𝑥 = 2 4 . ∘ ∘ ∘ ∘ ∘ ∘ ∘

Finally, we know that 𝑚 𝐴 𝐶 = 4 𝑥 . ∘

Substituting in the value for 𝑥 , we get that 𝑚 𝐴 𝐶 = 4 ( 2 4 ) = 9 6 . ∘ ∘

Let’s finish by recapping some of the important points of this explainer.

• A central angle of a circle is an angle between two radii with the vertex at the center.
• The central angle of an arc is the central angle subtended by the arc.
• Given two radii, we denote the longer of the two arcs between the radii as the major arc and the shorter of the arcs as the minor arc . Arcs of equal lengths are called semicircular arcs; these occur when the radii form a diameter.
• An arc is major if its measure (or the measure of its central angle) is greater than 1 8 0 ∘ , an arc is minor if its measure (or the measure of its central angle) is less than 1 8 0 ∘ , and an arc is semicircular if its measure (or the measure of its central angle) is equal to 1 8 0 ∘ .
• We denote the minor arc from 𝐴 to 𝐵 as 𝐴 𝐵 , and major arcs can be labeled by using an extra point (for example, 𝐴 𝐶 𝐵 ).
• We use the notation 𝑚 𝐴 𝐵 for the measure of the minor arc from 𝐴 to 𝐵 and 𝑚 𝐴 𝐶 𝐵 for the measure of the major arc from 𝐴 to 𝐵 that passes through 𝐶 .
• If the central angle (or measure) of an arc in a circle of radius 𝑟 is 𝜃 ∘ , then its length, 𝑙 , is given by 𝑙 = 𝜃 3 6 0 ( 2 𝜋 𝑟 ) . ∘ ∘
• If two arcs in the same circle have equal lengths, then their central angles and measures are equal. The same is true in reverse; if two arcs have equal lengths, then their central angles and measures are equal.
• If two arcs in the same circle have equal measures, then the chords between their respective endpoints have the same length. The same is true in reverse; if the chords have the same length, then the arcs between the endpoints of the chords must have equal measures.

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CENTRAL ANGLES AND ARC MEASURES

1. A central angle is an angle with its vertex at the center of the circle and its two sides are radii. 

2. For example : m ∠POQ is a central angle in circle P shown below. 

3. The sum of all central angle is 360 °.

4. The measure of the arc formed by the endpoints of a central angle is equal to the degree of the central angle.

In the above diagram, 

m ∠arc PQ = 85 °

m ∠arc PRQ = 360 ° - 85 ° = 275 °

5. The measure of the arc formed by the endpoints of the diameter is equal to 180 ° .

m∠arc PRQ = 180 °

Example 1 : 

From the diagram shown above, find the following arc measures. 

(i)  m ∠arc BC

(ii) m∠arc ABC

(i)  m ∠arc BC :

AB is the diameter of the above circle. 

m∠arc AB = 180 °

m∠arc BC +  m∠arc CA = 180 °

m∠arc BC + 123 °  = 180 °

m∠arc BC  = 57 °

(ii) m∠arc ABC :

m∠arc ABC = m ∠arc AB + m ∠arc BC

= 180 °  + 57 °

Example 2 :

From the diagram shown above, find the following measures. 

(i)  m ∠arc CD

(iii) m∠arc BD

(iv) m∠arc ABC

(v) m∠arc CBD

(i)  m ∠arc CD :

m∠AOB and m ∠COD are vertical angles. 

m ∠COD = m ∠AOB

m ∠arc CD = m ∠arc AB

m∠arc CD = 55 °

(ii) m∠AOC :

BC is the diameter of the above circle. 

m∠arc BAC = 180 °

m∠arc BA +  m∠arc AC = 180 °.

55 °  +  m∠arc AC = 180 °.

m∠arc AC = 125 °.

m∠AOC = 125 °.

(iii) m∠arc BD : 

m∠BOD and m ∠AOC are vertical angles. 

m ∠BOD = m ∠AOC

m ∠BOD = 125 °

m∠arc BD = 125°

(iv) m∠arc ABC : 

m∠arc ABC =  m∠arc ABD +  m∠arc DC

= 180 °  + 55 °

(v) m∠arc CBD : 

m∠arc CBD =  m∠arc CAB +  m∠arc BD

= 180 °  + 125 °

Example 3 :

Find the value of x in the diagram shown below. 

From the diagram shown above, find the  m ∠arc QTR.

Find m ∠arc QP :

PS is the diameter of the above circle.

m ∠arc PTS = 180 °

m∠arc PT +  m∠arc TS  = 180°

135 ° +  m∠arc TS  = 180°

m∠arc TS = 45°

Find m ∠arc QTR :

m∠QTR = m ∠arc QT + m ∠arc TS + m ∠arc SR

= 180 ° + 45 ° + 81 °

Example 4 :

m ∠BOD,   m ∠BOE and  m ∠BOC

Find  m ∠BOD :

In the circle above,

m ∠arc AB +  m ∠arc BCD +  m ∠arc DE +  m ∠arc EA = 360 °

60 °  +  m ∠arc BCD + 86 °  + 154 °  = 360 °

m ∠arc BCD + 300 °  = 360 °

m ∠arc BCD  = 60 °

m ∠BOD  = 60 °

Find  m ∠BOE :

m ∠BOE = m ∠arc BCD + m∠arc DE

= 60 ° + 86 °

Find m ∠BOC :

In the above diagram,  m∠BOC =  m ∠COD.

m∠BOC + m∠COD =  m∠BOD

m∠BOC + m∠BOC = m∠BOD

2m∠BOC = 60 °

m∠BOC = 30 °

Example 5 :

m ∠ KOL and  m∠arc MNK

In the diagram above,  m∠JON and  ∠KOM are vertical angles.

m∠KOM  = m ∠KOM

m∠KOM = 126 °

m∠KOL + m ∠LOM  = 126 °

In the above diagram,  m∠KOL =  m ∠LOM.

m∠KOL + m∠KOL = 126°

2m∠KOL = 126°

m ∠ KOL = 63°

Find m ∠arc MNK :

m∠arc MNK = 360 ° - m ∠arc KLM

m∠arc MNK = 360° - m∠KOM

m∠arc MNK = 360° - 126 °

m∠arc MNK = 234 °

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This Circles Unit Bundle contains guided notes, homework assignments, three quizzes, a study guide and a unit test that cover the following topics:

• Identifying Parts of Circles: Center, Radius, Chord, Diameter, Secant, Tangent, Central Angle, Inscribed Angle, Minor Arc, Major Arc, Semicircle

• Area and Circumference

• Central Angles

• Arc Lengths

• Congruent Chords and Arcs

• Inscribed Angles and Polygons

• Properties of Tangent Lines

• Arc and Angle Measures

• Segment Lengths

• Equation of a Circle (Graphing, identifying center, radius, circumference, area)

• Writing the Equation of a Circle in Standard Form (by completing the square)

ADDITIONAL COMPONENTS INCLUDED:

(1) Links to Instructional Videos: Links to videos of each lesson in the unit are included. Videos were created by fellow teachers for their students using the guided notes and shared in March 2020 when schools closed with no notice.  Please watch through first before sharing with your students. Many teachers still use these in emergency substitute situations. (2) Editable Assessments: Editable versions of each quiz and the unit test are included. PowerPoint is required to edit these files. Individual problems can be changed to create multiple versions of the assessment. The layout of the assessment itself is not editable. If your Equation Editor is incompatible with mine (I use MathType), simply delete my equation and insert your own.

(3) Google Slides Version of the PDF: The second page of the Video links document contains a link to a Google Slides version of the PDF. Each page is set to the background in Google Slides. There are no text boxes;  this is the PDF in Google Slides.  I am unable to do text boxes at this time but hope this saves you a step if you wish to use it in Slides instead! 

This resource is included in the following bundle(s):

Geometry Second Semester Notes Bundle

Geometry Curriculum

Geometry Curriculum (with Activities)

More Geometry Units:

Unit 1 – Geometry Basics

Unit 2 – Logic and Proof

Unit 3 – Parallel and Perpendicular Lines

Unit 4 – Congruent Triangles

Unit 5 – Relationships in Triangles

Unit 6 – Similar Triangles

Unit 7 – Right Triangles and Trigonometry Unit 8 – Polygons and Quadrilaterals

Unit 9 – Transformations

Unit 11 – Volume and Surface Area Unit 12 – Probability

LICENSING TERMS: This purchase includes a license for one teacher only for personal use in their classroom. Licenses are non-transferable , meaning they can not be passed from one teacher to another. No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses. If you are a coach, principal, or district interested in transferable licenses to accommodate yearly staff changes, please contact me for a quote at [email protected].

COPYRIGHT TERMS: This resource may not be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.

© All Things Algebra (Gina Wilson), 2012-present

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Kemerovo Oblast, Russia

The capital city of Kemerovo oblast: Kemerovo .

Kemerovo Oblast - Overview

Kemerovo Oblast is a federal subject of Russia located in the south-east of Western Siberia, part of the Siberian Federal District. This region is also known as Kuzbass . In 2019, “Kuzbass” officially became the second name of Kemerovo Oblast. Kemerovo is the capital city of the region.

The population of Kemerovo Oblast is about 2,604,300 (2022), the area - 95,725 sq. km.

Kemerovo oblast flag

Kemerovo oblast coat of arms.

Kemerovo oblast map, Russia

Kemerovo oblast latest news and posts from our blog:.

11 February, 2019 / Kemerovo - the view from above .

21 April, 2016 / The carnival-parade at the festival GrelkaFest in Sheregesh .

2 June, 2013 / Summer snowfall in Kemerovo .

9 February, 2012 / "BelAZ 75600" - the biggest truck in the former USSR .

31 October, 2010 / The ship-house in Kemerovo oblast .

History of Kemerovo Oblast

People began to settle in what is now the Kemerovo region several thousand years ago. The indigenous peoples of the region were Shortsy and Teleuts. In 1618, Russians founded Kuznetsky stockaded town in the south of the present region to protect Russian lands from the raids of the Mongols and Jungars. In 1698, Mariinsk was founded.

In 1721, Mikhailo Volkov found “burning mountain” (a burning coal seam) on the banks of the Tom River and thus became the person who discovered the Kuzbass coal deposits. Industrial development of this land started at the end of the 18th century.

During the 19th century, the territory of the Kemerovo region was part of the Tomsk province. During this period, the first industrial enterprises appeared here: Tomsk ironworks, Gavrilovsky and Gurievsky silver plants, Suharinsky and Salairsky mines. Construction of the Trans-Siberian Railway was one of the main reasons for the rapid development of the local industry.

More historical facts…

In Soviet times, the region became part of the West Siberian krai, and then - Novosibirsk oblast. The development of the coal, metallurgical and chemical industries continued: Kemerovo Coke Plant, Kuznetsk Metallurgical Plant, a lot of new mines. The workers’ settlements built near the industrial enterprises quickly obtained the status of towns: Kiselyovsk, Osinniki, Krasnobrodsky, Tashtagol, Kaltan, Mezhdurechensk and others.

During the Second World War, this region was a major supplier of coal and metal. More than 50 thousand tanks and 45 thousand aircraft were produced using steel from Novokuznetsk. 71 industrial enterprises were evacuated to Kuzbass from the occupied regions, most of them remained in the region after the war.

In 1943, Kemerovo Oblast became a separate region that included 17.5% of the territory and 42% of the total population of Novosibirsk Oblast. After the war, the region continued to grow rapidly. On September 18, 1984, about 100 km from Kemerovo, a peaceful underground nuclear explosion was carried out, the power of the explosive device was 10 kilotons.

In the 1990s, the region’s economy declined. However, by the end of the 20th century, there were some positive developments - the development of the coal industry in the first place. Special attention was paid to the development of open-pit coal mining, as a more effective and safe way.

Beautiful nature of Kemerovo Oblast

Mountain stream in the Kemerovo region

Author: Sergey Timofeev

On the shore of a small lake in Kemerovo Oblast

Kemerovo Oblast landscape

Kemerovo Oblast - Features

Kemerovo Oblast is one of the few Russian regions that has a recognized and well-known alternative name “Kuzbass” - the abbreviation of “Kuznetsk coal basin” occupying a large part of the territory of the region.

It is the most densely populated part of Siberia. The length of the region from north to south is about 500 km, from west to east - 300 km. Russians make up more than 90% of the population. There are small nations of Shortsy, Teleuts, Siberian Tatars who have preserved their cultural traditions.

The climate is sharply continental with long cold winters and warm short summers. The average temperature in January is minus 17-20 degrees Celsius, in July - plus 17-18 degrees Celsius.

Today, about 86% of the population of Kemerovo Oblast lives in cities and towns making it one of the most urbanized regions of Russia. The largest cities are Kemerovo (548,000), Novokuznetsk (540,000), Prokopievsk (185,000), Mezhdurechensk (95,400), Leninsk-Kuznetsky (91,600), Kisilyovsk (83,700), Yurga (79,700), Belovo (70,100), Anzhero-Sudzhensk (65,700).

Sheregesh, a village located at the foot of Zelenaya Mountain in Tashtagol district, is one of Russia’s most popular ski resorts. The ski season lasts from November to May. Kuznetsky Alatau Reserve and Shorsky National Park are the main natural attractions.

Kemerovo Oblast plays a significant role in Russian industry. The following mineral resources are mined here: coal, gold, silver, iron ore, manganese ore, aluminum, nepheline ore, lead, zinc, barite, quartz, limestone, clay, dolomite, sand.

Kuznetsk coal basin is one of the largest coal basins in the world. The most important centers of the local coal industry are Prokopyevsk, Mezhdurechensk, Belovo, Kemerovo, Novokuznetsk, Osinniki, Leninsk-Kuznetsky. Coal mines can be found almost everywhere in the Kemerovo region. About 180 million tons of coal is mined annually.

Rail transport is well developed in the region. The Trans-Siberian Railway, the South Kuzbas branch of West Siberian Railway cross its territory. There are large airports in Kemerovo (Kemerovo International Airport) and Novokuznetsk (Spichenkovo Airport).

Kemerovo oblast of Russia photos

Pictures of kemerovo oblast.

Kemerovo Oblast scenery

Author: Sergey Ustuzhanin

Autumn in Kemerovo Oblast

Winter in Kemerovo Oblast

Author: Max Palchevsky

Landscapes of Kemerovo Oblast

Churches in Kemerovo Oblast

Orthodox church in Kemerovo Oblast

Author: Yury Marchenko

Church in the Kemerovo region

Author: Ludmila Boriskina

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• 52.920514 87.995369 1 Sheregesh — the all-Russia ski resort in Gornaya Shoriya.

By Siberian standards, Kemerovo Oblast is populous, urban, and industrialized. It lies in the heart of the "Kuzbass" (Kuznets Basin) region, home to the world's largest deposits of coal. Accordingly, most of Kemerovo Oblast's cities developed because of economic opportunities related to the coal industry. As a result, the region's cities are often quite polluted, although the situation has improved since the fall of the USSR. Visitors to the region's cities will find them busy, but not significant tourist attractions in and of themselves. But this is all the more reason to get out into the beautiful and unspoilt (and uninhabited) Siberian countryside!

Russian is the only dish on the menu.

Most visitors will pass through Kemerovo Oblast on the Trans-Siberian Railway , which makes stops at (from west to east) Yurga, Taiga, Anzhero-Sudzhensk , Yaya , and Mariinsk.

There is an airport at Kemerovo ( KEJ   IATA ), with flights to/from Moscow , Krasnoyarsk , and in the summer: Anapa , Sochi and Khabarovsk .

Novokuznetsk also has an airport serving flights from Moscow (Domodedovo and Vnukovo Airports), Saint Petersburg , Tomsk and in the summer: Sochi , Krasnodar , and Anapa .

Rail is the most important means of transport in this industrialized region on the Trans-Siberian Railway . The most important junction town is Yurga, junction for the branch going south to Kemerovo .

• Horseback Riding
• Alpine Sports
• Looking for Yeti .  

The next major stops on the Trans-Siberian Railway are Novosibirsk to the west and Achinsk and Krasnoyarsk to the east.

Travelers heading to Tomsk should take the branch from the Trans-Siberian Railway junction at Taiga.

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