homework 4 exterior angles of triangles

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4.18: Exterior Angles and Theorems

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Exterior angles equal the sum of the remote interiors.

Exterior Angles

An Exterior Angle is the angle formed by one side of a polygon and the extension of the adjacent side.

In all polygons, there are two sets of exterior angles, one that goes around clockwise and the other goes around counterclockwise.

Notice that the interior angle and its adjacent exterior angle form a linear pair and add up to \(180^{\circ}\).

\(m\angle 1+m\angle 2=180^{\circ} \)

There are two important theorems to know involving exterior angles: the Exterior Angle Sum Theorem and the Exterior Angle Theorem.

The Exterior Angle Sum Theorem states that the exterior angles of any polygon will always add up to \(360^{\circ}\).

\(m\angle 1+m\angle 2+m\angle 3=360^{\circ}\)

\(m\angle 4+m\angle 5+m\angle 6=360^{\circ}\).

The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of its remote interior angles . ( Remote Interior Angles are the two interior angles in a triangle that are not adjacent to the indicated exterior angle.)

\(m\angle A+m\angle B=m\angle ACD\)

What if you knew that two of the exterior angles of a triangle measured \(130^{\circ}\)? How could you find the measure of the third exterior angle?

Example \(\PageIndex{1}\)

Two interior angles of a triangle are \(40^{\circ}\) and \(73^{\circ}\). What are the measures of the three exterior angles of the triangle?

Remember that every interior angle forms a linear pair (adds up to \(180^{\circ}\)) with an exterior angle. So, since one of the interior angles is \(40^{\circ}\) that means that one of the exterior angles is \(140^{\circ}\) (because \(40+140=180\)). Similarly, since another one of the interior angles is \(73^{\circ}\), one of the exterior angles must be \(107^{\circ}\). The third interior angle is not given to us, but we could figure it out using the Triangle Sum Theorem . We can also use the Exterior Angle Sum Theorem. If two of the exterior angles are \(140^{\circ}\) and \(107^{\circ}\), then the third Exterior Angle must be \(113^{\circ}\) since \(140+107+113=360\).

So, the measures of the three exterior angles are 140, 107 and 113.

Example \(\PageIndex{2}\)

Find the value of \(x\) and the measure of each angle.

Set up an equation using the Exterior Angle Theorem.

\(\begin{align*} \underbrace{(4x+2)^{\circ}+(2x−9)^{\circ}}_\text{remote interior angles}&=\underbrace{(5x+13)^{\circ}}_\text{exterior angle} \\ (6x−7)^{\circ}&=(5x+13)^{\circ} \\ x&=20 \end{align*}\)

Substitute in 20 for \(x\) to find each angle.

\([4(20)+2]^{\circ}=82^{\circ}[2(20)−9]^{\circ}=31^{\circ} \qquad Exterior \:angle:\: [5(20)+13]^{\circ}=113^{\circ}\)

Example \(\PageIndex{3}\)

Find the measure of \(\angle RQS\).

Notice that \(112^{\circ}\) is an exterior angle of \(\Delta RQS\) and is supplementary to \(\angle RQS\).

Set up an equation to solve for the missing angle.

\(\begin{align*}112^{\circ}+m\angle RQS &=180^{\circ} \\ m\angle RQS&=68^{\circ}\end{align*}\)

Example \(\PageIndex{4}\)

Find the measures of the numbered interior and exterior angles in the triangle.

We know that \(m\angle 1+92^{\circ}=180^{\circ}\) because they form a linear pair. So, m\angle 1=88^{\circ}\).

Similarly, \(m\angle 2+123^{\circ}=180^{\circ}\) because they form a linear pair. So, m\angle 2=57^{\circ}\).

We also know that the three interior angles must add up to 180^{\circ}\) by the Triangle Sum Theorem.

\(\begin{align*} m\angle 1+m\angle 2+m\angle 3&=180^{\circ} \qquad by\: the \:Triangle \:Sum \:Theorem. \\ 88^{\circ}+57^{\circ}+m\angle 3&=180 \\ m\angle 3&=35^{\circ}\end{align*}\)

Lastly, \(m\angle 3+m\angle 4=180^{\circ} \qquad because\: they\: form \:a \:linear \:pair.\)

\(\begin{align*} 35^{\circ}+m\angle 4&=180^{\circ} \\ m\angle 4&=145^{\circ}\end{align*}\)

Example \(\PageIndex{5}\)

What is the value of \(p\) in the triangle below?

First, we need to find the missing exterior angle, which we will call \(x\). Set up an equation using the Exterior Angle Sum Theorem.

\(\begin{align*} 130^{\circ}+110^{\circ}+x&=360^{\circ} \\ x&=360^{\circ}−130^{\circ}−110^{\circ} \\ x&=120^{\circ}\end{align*} \)

\(x\) and \(p\) add up to \(180^{\circ}\) because they are a linear pair.

\(\begin{align*} x+p&=180^{\circ} \\ 120^{\circ}+p&=180^{\circ} \\ p&=60^{\circ}\end{align*}\)

Determine \(m\angle 1\).

Use the following picture for the next three problems:

• What is \(m\angle 1+m\angle 2+m\angle 3\)?
• What is \(m\angle 4+m\angle 5+m\angle 6\)?
• What is \(m\angle 7+m\angle 8+m\angle 9\)?

Solve for \(x\).

Additional Resources

Interactive Element

Video: Exterior Angles Theorems Examples - Basic

Activities: Exterior Angles Theorems Discussion Questions

Study Aids: Triangle Relationships Study Guide

Practice: Exterior Angles and Theorems

Real World: Exterior Angles Theorem

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Course: 8th grade   >   Unit 5

• Angles in a triangle sum to 180° proof
• Find angles in triangles
• Isosceles & equilateral triangles problems
• Find angles in isosceles triangles
• Triangle exterior angle example
• Worked example: Triangle angles (intersecting lines)
• Worked example: Triangle angles (diagram)
• Finding angle measures using triangles
• Triangle angle challenge problem
• Triangle angle challenge problem 2

Triangle angles review

Sum of interior angles in triangles.

109 ∘ + 23 ∘ + 48 ∘ = 180 ∘ ‍  

Finding a missing angle

x ∘ + 42 ∘ + 106 ∘ = 180 ∘ ‍  

x ∘ = 180 ∘ − 106 ∘ − 42 ∘ ‍   x = 32 ‍  

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

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Exterior Angle of Triangle

The exterior angles of a triangle are those angles that are formed outside it. In other words, the exterior angle of a triangle is the angle that is formed between one of its sides and its adjacent extended side. Let us learn more about the exterior angle of triangle in this article.

What is the Exterior Angle of Triangle?

When any side of a triangle is extended, the angle that is formed with this side and its adjacent side is called the exterior angle of a triangle. There are three exterior angles in a triangle. It should be noted that each exterior angle forms a linear pair with its corresponding interior angle. We know that the interior angle of a triangle is formed inside it where the sides meet at a vertex. Observe the following figure to distinguish between the interior angles and the exterior angles of a triangle.

We can see that each interior angle forms a linear pair with its corresponding exterior angle. This means that the sum of each exterior angle and its respective interior angle is equal to 180°.

Exterior Angle of Triangle Properties

There are three basic properties of the exterior angles of a triangle.

• In a triangle, each exterior angle and its corresponding interior angle form a linear pair of angles . This means that the sum of the interior and exterior angle is equal to 180°.
• The exterior angle of a triangle is equal to the sum of the two opposite interior angles (remote interior angles). This is also known as the Exterior Angle theorem .
• The sum of all the exterior angles of a triangle is 360°.

Exterior Angle of Triangle Formula

Based on the properties of the exterior angles, the following formulas can be used to find the exterior angles of a triangle. Referring to the triangle given below, the formulas can be understood in a better way.

• Each Exterior angle = 180 - Interior angle. Here, ∠e = 180 - ∠b. Similarly, ∠d = 180 - ∠c, and ∠f = 180 - ∠a
• Exterior angle = Sum of Interior opposite angles. Here, ∠d = ∠a + ∠b, ∠f = ∠b + ∠c, and ∠e = ∠a + ∠c
• Sum of the Exterior angles of a triangle = 360°. Here, ∠d + ∠e + ∠f = 360°

Related Links

Check out the links given below related to the exterior angles of a triangle.

• Interior angles
• Exterior Angles of a Polygon
• Exterior Angle Theorem
• Linear Pair of Angles

Exterior Angle of Triangle Examples

Example 1: Find the value of the exterior angle in the triangle.

Solution: According to the property of the exterior angle of triangle, Exterior angle = Sum of Interior opposite angles. In this case, the exterior angle, ∠PRS = ∠RPQ + ∠PQR. Substituting the values in the formula, ∠PRS = 60° + 70° = 130°. Therefore, the unknown exterior angle, ∠PRS = 130°

Example 2: If one interior angle of a triangle is 56°, find the measure of its corresponding exterior angle.

Solution: According to the properties of the exterior angle of a triangle, each interior angle forms a linear pair with its respective exterior angle. This means, Exterior angle + Interior angle = 180°. In the question, one interior angle is given as 56°. Therefore, the corresponding exterior angle can be calculated using the formula: Each Exterior angle = 180° - Interior angle. Substituting the values in the formula, Exterior angle = 180 - 56 = 124°.

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Practice Questions on Exterior Angle of Triangle

Faqs on exterior angle of triangle.

The exterior angle of a triangle is the angle that is formed with one side and the adjacent extended side of a triangle. There are 3 exterior angles in a triangle and the sum of the exterior angles of a triangle is always equal to 360°.

How to Find the Exterior Angle of Triangle?

The value of the exterior angle of a triangle can be calculated using various formulas depending on the other known angles. The following formulas can be used according to the given parameters.

• Each Exterior angle = 180° - Interior angle. This formula can be used if the corresponding interior angle is given.
• Exterior angle = Sum of Interior opposite angles. This formula can be used to find the exterior angle when its remote interior opposite angles are given.
• The sum of all the exterior angles of a triangle is 360°. This formula can be used to find the unknown value of an exterior angle when the other two exterior angles are given.

What is the Sum of Exterior Angles of a Triangle?

The sum of the exterior angles of a triangle is always equal to 360°. This property applies to all polygons, which means that the sum of the exterior angles of all polygons is 360°.

How to find the Missing Exterior Angle of a Triangle?

A missing exterior angle of a triangle can be calculated using any of the following formulas. This depends on the angles that are given in the question.

• The first formula to find the exterior angle can be used if the corresponding interior angle is given. Each Exterior angle = 180 - Interior angle.
• The second formula can be used to find the exterior angle when its interior opposite angles are given. Exterior angle = Sum of Interior opposite angles.
• The third formula can be used to find the unknown value of an exterior angle when the other two exterior angles are given. The sum of all the exterior angles of a triangle is 360°.

Are the Exterior Angles of a Triangle Equal to 360°?

Yes, the sum of the exterior angles of a triangle is always equal to 360°.

Are the Exterior Angles of a Triangle Always Obtuse?

No, the exterior angles of a triangle may not always be obtuse (more than 90°). However, the sum of all the three exterior angles should always be 360°. For example, if two exterior angles of a triangle are 165° (obtuse) and 141° (obtuse), the third one is 54° (acute).

What is the Measure of Each Exterior Angle of an Equilateral Triangle?

The measure of each exterior angle of an equilateral triangle is 120°. An equilateral triangle is a triangle in which all the sides are equal in length and all the 3 interior angles are of equal measure. This means each interior angle of an equilateral triangle is 60° because the sum of the interior angles is 180°. Now, if each interior angle of an equilateral triangle is 60°, its corresponding exterior angle will be 120°. This is because in a triangle, the exterior angle and its corresponding interior angle form a linear pair of angles . This means that the sum of the interior and exterior angle is equal to 180°.

What is the Exterior Angle Theorem of a Triangle?

According to the exterior angle theorem , the measure of an exterior angle is equal to the sum of the interior opposite angles (remote interior angles). This means if we need to find the exterior angle of a triangle, and its remote interior angles are known, then the value of the exterior angle will be the sum of those two interior opposite angles.

Chapter 1: Triangles and Circles

Exercises: 1.1 Triangles and Angles

Practice each skill in the Homework Problems listed.

• Sketch a triangle with given properties #1–6
• Find an unknown angle in a triangle #7–12, 17–20
• Find angles formed by parallel lines and a transversal #13–16, 35–44
• Find exterior angles of a triangle #21–24
• Find angles in isosceles, equilateral, and right triangles #25–34
• State reasons for conclusions #45–48

Suggested Problems

Exercises for 1.1 Triangles and Angles

Exercise group, 1. an isosceles triangle with a vertex angle [latex]306^{\circ}[/latex], 2. a scalene triangle with one obtuse angle ( scalene means three unequal sides.), 3. a right triangle with legs [latex]4[/latex] and [latex]7[/latex], 4. an isosceles right triangle, 5. an isosceles triangle with one obtuse angle, 6. a right triangle with one angle [latex]20°[/latex].

In parts (a) and (b), find the exterior angle [latex]\phi[/latex].

• Use your answer to part (c) to write a rule for finding an exterior angle of a triangle.

In Problems 25 and 26, the figures inscribed are regular polygons , which means that all their sides are the same length, and all the angles have the same measure. Find the angles [latex]\theta[/latex] and [latex]\phi[/latex].

In problems 27 and 28, triangle ABC is equilateral. Find the unknown angles.

a. [latex]2\theta + 2\phi =[/latex]

b. [latex]\theta + \phi =[/latex]

c. [latex]\triangle ABC[/latex] is

Find [latex]\alpha[/latex] and [latex]\beta[/latex]

• Explain why [latex]\angle OAB[/latex] and [latex]\angle ABO[/latex] are equal in measure.
• Explain why [latex]\angle OBC[/latex] and [latex]\angle BCO[/latex] are equal in measure.
• Explain why [latex]\angle ABC[/latex] is a right angle. (Hint: Use Problem 29.)
• Compare [latex]\theta[/latex] with [latex]\alpha + \beta[/latex] (Hint: What do you know about supplementary angles and the sum of angles in a triangle?)
• Compare [latex]\alpha[/latex] and [latex]\beta[/latex]
• Explain why the inscribed angle [latex]\angle BAO[/latex] is half the size of the central angle [latex]\angle BOD[/latex]

Find [latex]\alpha[/latex] and [latex]\beta[/latex]

• [latex]\angle 4 + \angle 2 + \angle 5 =[/latex]
• Use parts (a) and (b) to explain why the sum of the angles of a triangle is [latex]180^{\circ}[/latex]

ABCD is a rectangle. The diagonals of a rectangle bisect each other. In the figure,  [latex]\angle AQD = 130^{\circ}[/latex]. Find the angles labeled 1 through 5 in order, and give a reason for each answer.

A tangent meets the radius of a circle at a right angle. In the figure, [latex]\angle AOB = 140^{\circ}[/latex]. Find the angles labeled 1 through 5 in order, and give a reason for each answer.

Trigonometry Copyright © 2024 by Bimal Kunwor; Donna Densmore; Jared Eusea; and Yi Zhen. All Rights Reserved.

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1.0: Angles and Triangles

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• Page ID 112400

• Katherine Yoshiwara
• Los Angeles Pierce College

Historically, trigonometry began as the study of triangles and their properties. Let’s review some definitions and facts from geometry.

• We measure angles in degrees.
• One full rotation is \(360^{\circ}\), as shown below.
• Half a full rotation is \(180^{\circ}\) and is called a straight angle .
• One quarter of a full rotation is \(90^{\circ}\) and is called a right angle .

If you tear off the corners of any triangle and line them up, as shown below, they will always form a straight angle.

Sum of angles in a triangle.

1. The sum of the angles in a triangle is \(180^{\circ}\).

Example 1.1

Two of the angles in the triangle at right are \(25^{\circ}\) and \(115^{\circ}\). Find the third angle.

To find the third angle, we write an equation.

\begin{aligned} x+25+115 &=180 \quad \quad &&\text{Simplify the left side.} \\ x+140 &=180 \quad \quad &&\text{Subtract 140 from both sides.}\\ x &=40 \end{aligned}

The third angle is \(40^{\circ}\).

Checkpoint 1.2

Find each of the angles in the triangle at right.

\(x = 39^{\circ}, 2x = 78^{\circ}, 2x-15 = 63^{\circ}\)

Some special categories of triangles are particularly useful. Most important of these are the right triangles .

Right triangle.

2. A right triangle has one angle of \(90^{\circ}\).

Example 1.3

One of the smaller angles of a right triangle is \(34^{\circ}\). What is the third angle?

The sum of the two smaller angles in a right triangle is \(90^{\circ}\). So

\begin{aligned} x+34 &=90 \quad \quad \text{Subtract 34 from both sides} \\ x &=56 \end{aligned}

The unknown angle must be \(56^{\circ}\).

Checkpoint 1.4

Two angles of a triangle are \(35^{\circ}\) and \(45^{\circ}\). Can it be a right triangle?

An equilateral triangle has all three sides the same length.

Angles of equilateral triangle.

3. All of the angles of an equilateral triangle are equal.

Example 1.5

All three sides of a triangle are 4 feet long. Find the angles.

The triangle is equilateral, so all of its angles are equal. Thus

\begin{aligned} 3 x &=180 \quad \quad \quad \text{Divide both sides by 3.}\\ x &=60 \end{aligned}

Each of the angles is \(60^{\circ}\).

Checkpoint 1.6

Find \(x, y\), and \(z\) in the triangle at right.

\(x=60^{\circ}, y=8, z=8\)

An isosceles triangle has two sides of equal length. The angle between the equal sides is the vertex angle . The other two angles are the base angles.

Base angles of an isoceles triangle.

4. The base angles of an isosceles triangle are equal.

Example 1.7

Find \(x\) and \(y\) in the triangle at right.

The triangle is isosceles, so the base angles are equal. Therefore, \(y=38^{\circ}\). To find the vertex angle, we solve

\begin{aligned} x+38+38 &=180 \\ x+76 &=180 \quad \quad \quad \text{Subtract 76 from both sides.}\\ x &=104 \end{aligned}

The vertex angle is \(104^{\circ}\).

Checkpoint 1.8

Find \(x\) and \(y\) in the figure at right.

\(x=140^{\circ}, y=9\)

In addition to the facts about triangles reviewed above, there are several useful properties of angles.

• Two angles that add to \(180^{\circ}\) are called supplementary .
• Two angles that add to \(90^{\circ}\) are called complementary .
• Angles between \(0^{\circ}\) and \(90^{\circ}\) are called acute .
• Angles between \(90^{\circ}\) and \(180^{\circ}\) are called obtuse .

Example 1.9

In the figure at right,

• \(\angle A O C\) and \(\angle B O C\) are supplementary.
• \(\angle D O E\) and \(\angle B O E\) are complementary.
• \(\angle A O C\) is obtuse,
• and \(\angle B O C\) is acute.

In trigonometry we often use lower-case Greek letters to represent unknown angles (or, more specifically, the measure of the angle in degrees). In the next Exercise, we use the Greek letters \(\alpha\) (alpha), \(\beta\) (beta), and \(\gamma\) (gamma).

Checkpoint 1.10

In the figure, \(\alpha, \beta\), and \(\gamma\) denote the measures of the angles in degrees.

a. Find the measure of angle \(\alpha\). b. Find the measure of angle \(\beta\). c. Find the measure of angle \(\gamma\). d. What do you notice about the measures of the angles?

\(\quad \alpha=130^{\circ}, \beta=50^{\circ}, \gamma=130^{\circ}\). The non-adjacent angles are equal.

Non-adjacent angles formed by the intersection of two straight lines are called vertical angles . In the previous exercise, the angles labeled \(\alpha\) and \(\gamma\) are vertical angles, as are the angles labeled \(\beta\) and \(50^{\circ}\).

Vertical Angles.

5. Vertical angles are equal.

Example 1.11

Explain why \(\alpha=\beta\) in the triangle at right.

Because they are the base angles of an isosceles triangle, \(\theta\) (theta) and \(\phi(\mathrm{phi})\) are equal. Also, \(\alpha=\theta\) because they are vertical angles, and similarly \(\beta=\phi\). Therefore, \(\alpha=\beta\) because they are equal to equal quantities.

Checkpoint 1.12

Find all the unknown angles in the figure at right. (You will find a list of all the Greek letters and their names at the end of this section.)

\(\alpha=40^{\circ}, \beta=140^{\circ}, \gamma=75^{\circ}, \delta=65^{\circ}\)

A line that intersects two parallel lines forms eight angles, as shown in the figure below. There are four pairs of vertical angles, and four pairs of corresponding angles , or angles in the same position relative to the transversal on each of the parallel lines.

For example, the angles labeled 1 and 5 are corresponding angles, as are the angles labeled 4 and 8. Finally, angles 3 and 6 are called alternate interior angles , and so are angles 4 and 5.

Parallel lines cut by a transversal.

6. If parallel lines are intersected by a transversal, the alternate interior angles are equal. Corresponding angles are also equal.

Example 1.13

The parallelogram \(A B C D\) shown at right is formed by the intersection of two sets of parallel lines. Show that the opposite angles of the parallelogram are equal.

Angles 1 and 2 are equal because they are alternate interior angles, and angles 2 and 3 are equal because they are corresponding angles. Therefore angles 1 and 3 , the opposite angles of the parallelogram, are equal. Similarly, you can show that angles 4,5 , and 6 are equal.

Checkpoint 1.14

Show that the adjacent angles of a parallelogram are supplementary. (You can use angles 1 and 4 in the parallelogram of the previous example.)

Note that angles 2 and 6 are supplementary because they form a straight angle. Angle 1 equals angle 2 because they are alternate interior angles, and similarly angle 4 equals angle 5. Angle 5 equals angle 6 because they are corresponding angles. Thus, angle 4 equals angle 6, and angle 1 equals angle 2. So angles 4 and 1 are supplementary because 2 and 6 are.

Note 1.15 In the Section 1.1 Summary, you will find a list of vocabulary words and a summary of the facts from geometry that we reviewed in this section. You will also find a set of study questions to test your understanding, and a list of skills to practice in the homework problems.

Table 1.16 Lower Case Letters in the Greek Alphabet

\begin{aligned} &\quad \quad \quad \quad \quad \text { Greek Alphabet }\\ &\begin{array}{cc|cc|cc|} \hline \alpha & \text { alpha } & \beta & \text { beta } & \gamma & \text { gamma } \\ \hline \delta & \text { delta } & \epsilon & \text { epsilon } & \gamma & \text { gamma } \\ \hline \eta & \text { eta } & \theta & \text { theta } & \iota & \text { iota } \\ \hline \kappa & \text { kappa } & \lambda & \text { lambda } & \mu & \text { mu } \\ \hline \nu & \text { nu } & \xi & \text { xi } & o & \text { omicron } \\ \hline \pi & \text { pi } & \rho & \text { rho } & \sigma & \text { sigma } \\ \hline \tau & \text { tau } & v & \text { upsilon } & \phi & \text { phi } \\ \hline \chi & \text { chi } & \psi & \text { psi } & \omega & \text { omega } \\ \hline \end{array} \end{aligned}

Review the following skills you will need for this section.

Algebra Refresher 1.2

Solve the equation.

1. \(x-8=19-2 x\) 2. \(2 x-9=12-x\) 3. \(13 x+5=2 x-28\) 4. \(4+9 x=-7+x\)

Solve the system.

5. \(5x - 2y = -13\)

\(2x + 3y = -9\)

6. \(4x + 3y = 9\)

\(3x + 2y = 8\)

5. \(x=-3,y=-1\)

6. \(x=6,y=-5\)

Section 1.1 Summary

• Right angle

• Straight angle

• Right triangle

• Equilateral triangle

• Isosceles triangle

• Vertex angle

• Base angle

• Supplementary

• Complementary

• Acute

• Obtuse

• Vertical angles

• Transversal

• Corresponding angles

• Alternate interior angles

Facts from Geometry.

1. The sum of the angles in a triangle is \(180^{\circ}\). 2. A right triangle has one angle of \(90^{\circ}\). 3. All of the angles of an equilateral triangle are equal. 4. The base angles of an isosceles triangle are equal. 5. Vertical angles are equal. 6. If parallel lines are intersected by a transversal, the alternate interior angles are equal.

Corresponding angles are also equal.

Study Questions

1. Is it possible to have more than one obtuse angle in a triangle? Why or why not?

2. Draw any quadrilateral (a four-sided polygon) and divide it into two triangles by connecting two opposite vertices by a diagonal. What is the sum of the angles in your quadrilateral?

3. What is the difference between a vertex angle and vertical angles?

4. Can two acute angles be supplementary?

5. Choose any two of the eight angles formed by a pair of parallel lines cut by a transversal. Those two angles are either equal or _______ .

Practice each skill in the Homework Problems listed.

1. Sketch a triangle with given properties #1–6

2. Find an unknown angle in a triangle #7–12, 17–20

3. Find angles formed by parallel lines and a transversal #13–16, 35–44

4. Find exterior angles of a triangle #21–24

5. Find angles in isosceles, equilateral, and right triangles #25–34

6. State reasons for conclusions #45–48

Homework 1.1

For Problems 1–6, sketch and label a triangle with the given properties.

1. An isosceles triangle with vertex angle \(30^{\circ}\) 2. A scalene triangle with one obtuse angle ( Scalene means three unequal sides.) 3. A right triangle with \(\operatorname{legs} 4\) and 7 4. An isosceles right triangle 5. An isosceles triangle with one obtuse angle 6. A right triangle with one angle \(20^{\circ}\)

For Problems 7–20, find each unknown angle.

In Problems 21 and 22, the angle labeled \(\phi\) is called an exterior angle of the triangle, formed by one side and the extension of an adjacent side. Find \(\phi\).

23. In parts (a) and (b), find the exterior angle \(\phi\).

c. Find an algebraic expression for \(\phi\).

d Use your answer to part (c) to write a rule for finding an exterior angle of a triangle.

a Find the three exterior angles of the triangle. What is the sum of the exterior angles?

b Write an algebraic expression for each exterior angle in terms of one of the angles of the triangle. What is the sum of the exterior angles?

In Problems 25 and 26, the figures inscribed are regular polygons , which means that all their sides are the same length, and all the angles have the same measure. Find the angles \(\theta\) and \(\phi\).

In problems 27 and 28, \(\Delta ABC\) is equilateral. Find the unknown angles

a \(2\theta + 2\phi = ________\)

b \(\theta + \phi = ________\)

c \(\Delta ABC\) is ________

30. Find \(\alpha\) and \(\beta\).

a Explain why \(\angle O A B\) and \(\angle A B O\) are equal in measure.

b Explain why \(\angle O B C\) and \(\angle B C O\) are equal in measure.

c Explain why \(\angle A B C\) is a right angle. (Hint: Use Problem 29.)

a Compare \(\theta\) with \(\alpha+\beta\). (Hint: What do you know about supplementary angles and the sum of angles in a triangle?

b Compare \(\alpha\) and \(\beta\).

c Explain why the inscribed angle \(\angle B A O\) is half the size of the central angle \(\angle B O D\).

33. Find \(\alpha\) and \(\beta\).

34. Find \(\alpha\) and \(\beta\).

In Problems 35–44, arrows on a pair of lines indicate that they are parallel. Find \(x\) and \(y\).

a Among the angles labeled 1 through 5 in the figure at right, find two pairs of equal angles.

b \(\angle 4+\angle 2+\angle 5= _________\)

c Use parts (a) and (b) to explain why the sum of the angles of a triangle is \(180^{\circ}\)

a In the figure below, find \(\theta\), and justify your answer.

b Write an algebraic expression for \(\theta\) in the figure below.

47. \(A B C D\) is a rectangle. The diagonals of a rectangle bisect each other. In the figure, \(\angle A Q D=130^{\circ}\). Find the angles labeled 1 through 5 in order, and give a reason for each answer.

48. A tangent meets the radius of a circle at a right angle. In the figure, \(\angle AOB = 140^{\circ}\). Find the angles labeled 1 through 5 in order, and give a reason for each answer.

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Angle Sums and Exterior Angles of Triangles Worksheets

Triangles are one the most interesting geometric shapes there are. They have some unique properties that can be used to help us much better understand exact measures within them and around. This may be why a complete branch of mathematics (trigonometry) is entirely devoted to them. Triangles have three sides and three internal angles. The angle sum theorem states that when you add all those interior (inside) angles of a triangle together, the sum will always measure 180 degrees. You will find that when you advance to writing geometric proofs, this is a commonly used theorem. Exterior angles of a triangle are created by extending the line of an adjacent side. Since this straight line forms a 180 degree angle, to determine the measure of the exterior angle we would subtract 180 by the measure of the interior angle. What is interesting, if you are keeping up with the math, the measure of that exterior angle is also equal to the sum of the two opposite interior angles. This gives us two methods for determining the value of the exterior angle. These worksheets and lessons will help you learn how to determine all these measures.

Aligned Standard: Grade 8 Geometry - 8.G.A.5

• Using References of a Triangle Step-by-Step Lesson - Use the measures of the triangles to determine the missing supplementary angles. Just line them up and find the difference between them.
• Guided Lesson - Use parallel lines, a transversal, and one angle to figure out the measure of a whole bunch of unknowns.
• Guided Lesson Explanation - Once you get the hang of these types of problems, geometry just comes a lot easier to you.
• Independent Practice - These are some really great practice problems for you to tackle.
• Matching Worksheet - One scenario and a whole bunch of missing measures to figure out.
• Transversals Five Pack - Which one is the transversal? There are three lines flapping all about that you need to make sense of.
• Exterior Angle Worksheet Five Pack - A mixed bag of questions and difficulty levels.
• Sum of Interior Worksheet Five Pack - Find the sum of the outside measures in most cases. There are sum problems as well.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

A unique set of problems that are all about finding that missing angle.

• Homework 1 - Solve by using the Triangle Angle Sum Theorem.
• Homework 2 - We will first find ∠c. This will help us find d since they should be the same measure. 102 degrees and angle c makes linear pair so their sum is 180 degrees.
• Homework 3 - The line de is drawn parallel to the base bc in the triangle abc. If the ∠a = 15 degrees, find the

Practice Worksheets

This skill has made its way into every single assessment we have seen on this grade level, since the core curriculum was adopted.

• Practice 1 - a and b are two parallel lines with eg and ts as the transversal. Find the value of x.
• Practice 2 - Given: Line f is parallel to line g. ∠1= 111 degrees and ∠4= 43 degrees. Find the measures of ∠2 and ∠3 in the figure.
• Practice 3 - Based on the information provided, determine the values of the angles.

Math Skill Quizzes

Special note: Make sure that students realize that these angles are not drawn to scale.

• Quiz 1 - The lines b and c are parallel. Find a. These are the types of problems that you can expect.
• Quiz 2 - You will find missing measures when given some statements to work off of.
• Quiz 3 - Locate the transversal in the pack of wild lines.

What Is the Angle Sum Property of a Triangle?

This property simply states that the sum of all three angles within a triangle add up to a total of 180 degrees. Regardless of how a triangle is classified (acute, obtuse, or right) all of those interior angles always equal 180 degrees. This can be very helpful to find the measures of unknown interior angles within a triangle. All you need is the measure of two of the angles and subtract their sum from 180 degrees to determine the missing angle. You will find that this property will be used very often when writing geometric proofs.

How to Find the Exterior Angle of a Triangle

Since triangles have three sides and are closed structures the also have three angles found inside them. These are referred to as interior angles because they are found on the interior of the triangle. The angles that are located outside of the triangle are referred to as exterior angles. These exterior angles have more specific characteristics than just lying on the outside of a triangle. An exterior must linearly pair with an interior angle. It means that an exterior angle must be right next to an interior angle (they share a side), and the exterior and interior angles together form a straight line. You can use this to your advantage by easily forming an exterior angle by extending any one side of the triangle. There can be multiple ways to form an exterior angle. It depends on which side of a triangle you choose to extend. For a triangle, the measure of an exterior angle (d) is equal to the sum of two interiors (a, b) where d = a + b. The exterior angle (d) is greater than two interior angles (a, b). If you know the values of two interior angles, you can easily find the measure of an exterior angle. These worksheets and lessons help students understand how to find the missing measures of angles by using other angles within or outside of the triangle itself.

Why Is It Important to Master These Skills?

Geometry skills like these do not suggest to many students that they are that vital to everyday life and for the most part that is true. There are many situations in real life where these skills are not only vital, but paramount to be able to complete the project and life and death, when you consider safety. The Angle Sum Theorem allows architects to design structures that are free floating. Any building that is not a rectangle or square from top to bottom has this theorem to thank, not only for the construction, but also for the ability to create and fulfillment of a safe structure or dwelling that people can inhabit. This is not only true of building and homes, but just about any physical design you are making. Almost all electronics today run off of a chip set of some kind. The housing that encases them needs to sturdy and secure. Otherwise, every time you slightly rattled this device it would stop working. These housing that encase the chipsets are all designed with this Theorem in mind. The next time you drop your phone and it still works, you know who to thank.

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Exterior Angle Theorem Lesson Plan

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Exterior Angle Theorem Guided Notes w Doodles | Graphic & Sketch Notes 8th Grade

Ever wondered how to teach the Exterior Angle Theorem in an engaging way to your 8th-grade students?

In this lesson plan, students will learn about exterior angles of triangles and the Exterior Angle Theorem. Through artistic guided notes, interactive practice worksheets, and a real-life application, students will gain a comprehensive understanding of the topic.

The lesson ends with a real-life example that explores how the Exterior Angle Theorem can be applied to solve practical problems involving angles in the real world.

• Standard : CCSS 8.G.A.5
• Topics : Shapes & Angles , Triangles & Pythagorean Theorem
• Grade : 8th Grade
• Type : Lesson Plans

Learning Objectives

After this lesson, students will be able to:

Define exterior angle of a triangle and the Exterior Angle Theorem

Calculate missing exterior angles of triangles

Apply the Exterior Angle Theorem to solve problems involving angles of triangles and missing variables

Identify the real-life applications of the Exterior Angle Theorem

Prerequisites

Before this lesson on the Exterior Angle Theorem, students should be familiar with:

Understanding of basic geometry concepts, such as angles, triangles, and polygons

How to solve one step or two step equations

Colored pencils or markers

Exterior Angle Theorem Guided Notes with Doodles

Key Vocabulary

Exterior Angle Theorem

Introduction

As a hook, ask students why understanding the exterior angle of triangles is important in real-life situations. Refer to the last page of the guided notes as well as the FAQs below for ideas.

Use the first page of the guided notes to introduce the concept of exterior angles of triangles. Walk through the definition of exterior angles and how they relate to the angles of a triangle. Emphasize the relationship between the exterior angle and the two remote interior angles. Refer to the FAQ below for a walk through on this, as well as ideas on how to respond to common student questions.

Continue with the second page of the guided notes to introduce the Exterior Angle Theorem. Teach students how to apply the theorem to find unknown exterior angles of triangles. Students will set up equations with Exterior Angle Theorem to solve for the missing variables. Highlight the significance of the theorem in solving geometric problems involving triangles.

Based on student responses, reteach concepts that students need extra help with. If your class has a wide range of proficiency levels, you can pull out students for reteaching, and have more advanced students begin work on the practice exercises.

Have students practice solving problems involving exterior angles of triangles and the Exterior Angle Theorem using the practice worksheet (pg. 2), maze activity (pg. 3), color by number activity (pg. 4) provided in the guided notes resource. Walk around to answer student questions. Yo u can also assign it as homework for independent practice.

Real-Life Application

Use the last page of the guided notes resource to bring the class back together, and introduce the concept of how the Exterior Angle Theorem can be used in real life situations, such as in architecture and engineering when designing structures with angles, in navigation when determining directions and angles, or in art and design when creating geometric patterns and shapes. Students will read about some real life scenarios of how the Exterior Angle Theorem is used and then reflect on their reading with short sentences (pg. 5 of guided notes). Then, they will rate their mastery of the topic in a mini self assessment box on the bottom of the page.

Refer to the FAQ for more ideas on how to teach real-life applications of the Exterior Angle Theorem!

Additional Self-Checking Digital Practice

If you’re looking for digital practice for Exterior Angle Theorem, try my Pixel Art activities in Google Sheets. Every answer is automatically checked, and correct answers unlock parts of a mystery picture. It’s incredibly fun, and a powerful tool for differentiation.

Here’s 1 activity to explore:

Exterior Angle Theorem Digital Pixel Art

What is the Exterior Angle Theorem in geometry? Open

The Exterior Angle Theorem states that the exterior angle of a triangle is equal to the sum of the two interior opposite angles.

How do you find the exterior angle of a triangle? Open

To find the exterior angle of a triangle, look for the angle formed when one side of the triangle is extended.

The exterior angle of a triangle is equal to the sum of the two interior opposite angles.

Why is the Exterior Angle Theorem important in geometry? Open

The Exterior Angle Theorem is important because it helps us understand the relationship between the exterior and interior angles of a triangle, which is crucial in solving various geometry problems involving triangles.

What is the relationship between exterior angles and remote interior angles in a triangle? Open

In a triangle, each exterior angle is equal to the sum of the two remote interior angles ∠A + ∠B = ∠C.

This relationship helps in solving problems related to angles in triangles.

How can I apply the Exterior Angle Theorem in real-life situations? Open

Real-life applications of the Exterior Angle Theorem include architectural designs, construction projects, and navigation calculations.

Understanding this theorem can help in solving practical problems involving angles.

Can you explain how to use guided notes and doodles to teach the Exterior Angle Theorem effectively? Open

Guided notes provide structured information on the Exterior Angle Theorem, ensuring students stay on track with the lesson.

Doodles and sketches engage visual learners and help students remember key concepts through visual representations.

How can I incorporate color by code and mazes in teaching the Exterior Angle Theorem? Open

Color by code worksheets and mazes make learning fun and interactive for students.

These activities can be used for practice and reinforcement of the Exterior Angle Theorem concepts in an engaging way.

What makes the Exterior Angle Theorem Guided Notes with Doodles resource suitable for 8th-grade students? Open

The resource is designed to align with 8th-grade Common Core Standards.

It includes structured notes, practice worksheets, and real-life applications, catering to the specific learning needs of 8th-grade students.

Want more ideas and freebies?

Get my free resource library with digital & print activities—plus tips over email.

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Angles in a Triangle Worksheets

The free printables in this post deal with finding the unknown angles of triangles. Your concepts of interior and exterior angles of triangles should be sound if you want to solve the problems without hiccups.

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Into Math Grade 8 Module 4 Lesson 1 Answer Key Develop Angle Relationships for Triangles

We included H MH Into Math Grade 8 Answer Key PDF Module 4 Lesson 1 Develop Angle Relationships for Triangles to make students experts in learning maths.

HMH Into Math Grade 8 Module 4 Lesson 1 Answer Key Develop Angle Relationships for Triangles

I Can find an unknown angle measure in a triangle.

Spark Your Learning

Turn and Talk What conjecture can you make about the sum of the measures of the angles of a triangle?

Build Understanding

B. What do you notice about the sum of the measures of the three triangles? ____________________ Answer: The sum of the measures of the three triangles is 180°

C. Do you think this is true for all triangles? Explain. ____________________

The Triangle Sum Theorem states that the measures of the three interior angles of a triangle sum to 180°.

D. The angles in a triangle measure 2x, 3x, and 4x degrees. Write and solve an equation to determine the angle measures. ____________________ ____________________ Answer: The angles in a triangle measure 2x, 3x, and 4x degrees. The sum of the measures of the three triangles is 180° 2x + 3x + 4x = 180° 9x = 180° x = 180/9 x = 20° 2x = 2 × 20 = 40° 3x = 3 × 20 = 60° 4x = 4 × 20 = 80°

Turn and Talk Discuss how to find a missing measure of an angle in a triangle when the other two angle measures are given.

Step It Out

The Triangle Sum Theorem can be used to draw conclusions about a triangle’s interior angles.

A. What is the sum of the measures of ∠3 and ∠4? __________________ Answer: the sum of the measures of ∠3 and ∠4 is 180°

B. An exterior angle of a polygon is an angle formed by one side of the polygon and the extension of an adjacent side. Which angle in the diagram is an exterior angle? _____________ Answer: ∠4 is an exterior angle.

C. If the measure of ∠3 is 60°, what is the measure of ∠4? _______________________ Answer: ∠3 = 60° ∠3 + ∠4 = 180° 60°+ ∠4 = 180° ∠4 = 180° – 60° ∠4 = 120°

D. If the measure of ∠3 is 60°, what is the sum of the measures of ∠1 and ∠2? ______________ Answer: ∠3 = 60° ∠1 + ∠2 + ∠3 = 180° ∠1 + ∠2 + 60° = 180° ∠1 + ∠2 = 180° – 60° = 120° Thus the sum of the measures of ∠1 and ∠2 is 120°

E. Which angle has a measure equal to the sum of the measures of ∠1 and ∠2? ______________________________ Answer: ∠4 = 120° ∠1 + ∠2 = 180° – 60° = 120° So, ∠4 has a measure equal to the sum of the measures of ∠1 and ∠2.

F. A remote interior angle of an exterior angle of a polygon is an angle that is inside the polygon and is not adjacent to the exterior angle. Which two angles in the diagram are remote interior angles in relation to Angle 4? _____________________________ Answer: ∠1 and ∠2 are the remote interior angles in relation to Angle 4.

G. If the sum of the measures of ∠1 and ∠2 is 115°, what is the measure of ∠4? _______________________ Answer: If the sum of the measures of ∠1 and ∠2 is 115° then the measure of ∠4 is 115°.

Turn and Talk A triangle has exterior Angle P with remote interior Angles Q and R. Can you determine which angle has the greatest measure? Why or why not?

A. Write an equation and solve to find the value of x. Show your work. ___x + ___ = x + ___ __x – x = 80 – ___ x = ___ Answer: 2x + 45° = x + 80° 2x – x = 80° – 45° x = 35°

B. What is the measure of the unknown remote interior angle? _____________________ Answer: the measure of the unknown remote interior angle is 35°

C. Use the value of x from Part A to find the measure of the exterior angle. 2x + 45 = 2(___) + 45 = ___ + 45 = ___ Answer: 2x + 45 2(35) + 45 70° + 45° = 115°

Connect to Vocabulary The measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is the Exterior Angle Theorem.

D. What is the measure of the exterior angle? __________________ Answer: the measure of the exterior angle is 115°

Check Understanding

Question 1. Two angles of a triangle have measures of 30° and 45°. What is the measure of the remaining angle? Answer: Given, Two angles of a triangle have measures of 30° and 45°. Sum of three angles of a triangle = 180° 30°+ 45° + x° = 180° 75° + x° = 180 x° = 180° – 75° x° = 105°

Question 2. Dana draws a triangle with one angle that has a measure of 40°. A. What is the measure of the angle’s adjacent exterior angle? ______________ Answer: Dana draws a triangle with one angle that has a measure of 40°. 180°- 40° = 140° Thus the measure of the angle’s adjacent exterior angle is 140°

B. What is the sum of the measures of the remote interior angles for the exterior angle adjacent to the 40° angle? ______________ Answer: 140° + 40° = 180°

Question 3. An exterior angle of a triangle has a measure of 80°, and one of the remote interior angles has a measure of 20°. Write and solve an equation to find the measure of the other remote interior angle. Answer: Given, An exterior angle of a triangle has a measure of 80°, 180° – 80° = 100° and one of the remote interior angles has a measure of 20°. 180° – 20° – 100° = 60°

On Your Own

Question 4. A puppeteer is making a triangular hat for a puppet. If two of the three angles of the hat both measure 30°, what is the measure of the third angle? Answer: x + 30° + 30° = 180 x + 60° = 180° x = 180° – 60° x = 120° The triangle is an isosceles triangle and the measure of the third angle is 120°

Question 5. Construct Arguments Can a triangle have two obtuse angles? Explain your answer. Answer: No, a triangle does not have two obtuse angles Sum of three angles of a triangle = 180° 100 + 100 = 200° (Not possible)

Question 6. STEM In engineering, equilateral triangles can support the most weight and so are commonly found in the design of bridges and buildings. Equilateral triangles are triangles with three congruent sides and three congruent angles. What are the measures of the angles of an equilateral triangle? Answer: x + x + x = 180° 3x° = 180° x = 180/3 x = 60°

Question 7. A triangle has one 30° angle, an unknown angle, and an angle with a measure that is twice the measure of the unknown angle. Find the measures of the triangle’s unknown angles and explain how you found the answer. Answer: Given, A triangle has one 30° angle, an unknown angle, and an angle with a measure that is twice the measure of the unknown angle. x + 2x + 30° = 180° 3x + 30° = 180° 3x = 180° – 30° 3x = 150° x = 150/3 x = 50° 2x = 2 × 50 = 100°

For Problems 8-10, find the measures of the unknown third angles.

Question 11. Open Ended The measure of an exterior angle of a triangle is x°. The measure of the adjacent interior angle is at least twice x. List three possible solutions for x. Answer: The measure of an exterior angle of a triangle is x°. The measure of the adjacent interior angle is at least twice x. x° + θ = 180° θ = 180° – x ≥ 2x° 180° ≥ 3x° 0° < x ≤ 60° Any three numbers in (0, 60).

Question 12. The measure of an exterior angle of a triangle is 40°. What is the sum of the measures of the corresponding remote interior angles? Answer: The measure of an exterior angle of a triangle is 40°. 2x + 40° = 180° 2x = 180 – 40 2x = 140 x = 140/2 x = 70° Thus the sum of the measures of the corresponding remote interior angles is 140°

I’m in a Learning Mindset!

What did I learn from applying my knowledge of interior angles of a triangle to find the missing exterior angle in Problem 13 that I can explain clearly to a classmate?

Lesson 4.1 More Practice/Homework

Question 3. Construct Arguments Can the measure of an exterior angle of a triangle ever exceed 180? Explain your reasoning. Answer: An exterior angle of a triangle cannot be a straight line because a triangle has 180° in adding all the three angles of a triangle.

Question 5. Open Ended One of the angles in a triangle measures 90°. Name three possibilities for the measures of the remaining two angles. Answer: One of the angles in a triangle measures 90° 30° + 60° + 90° = 180° 90° + 45° + 45° = 180° 90° + 50° + 40° = 180°

Question 8. If an exterior angle of a triangle has a measure of 35°, what is the measure of the adjacent interior angle? Answer: 35° + x = 180° x = 180 – 35 x = 145° Thus the measure of the adjacent interior angle is 145°

Question 10. The measures of an exterior angle of a triangle and its adjacent interior angle add to what value? A. 90° B. 100° C 180° D. 360° Answer: The measures of an exterior angle of a triangle and its adjacent interior angle is equal to 180 degrees. So, option C is the correct answer.

Question 11. The measure of an exterior angle of a triangle and the sum of the measures of the two remote interior angles are _____________ Answer: The measure of an exterior angle of a triangle and the sum of the measures of the two remote interior angles are 180 degrees.

Spiral Review

Question 12. Hayden and Jamie completed 20 math problems together. Jamie completed 2 more than twice the number that Hayden completed. Let p represent the number of math problems Hayden completed. Write an equation that can be used to find the number of math problems that Jamie completed. Answer: Let p represent the number of Math problems Hayden completed. Let 2p+2 represent the number of Math problems Jamie completed. 2p + 2 + p = 20 3p + 2 = 20 3p = 20 – 2 3p = 18 p = 18/3 = 6 p = 6 Thus Hayden completed 6 math problems. 2p + 2 = 2(6) + 2 = 12 + 2 = 14 Thus Jamie completed 14 Math problems.

Question 13. Does the equation 5(x – 3) = 10x – 15 have one solution, infinitely many solutions, or no solution? Answer: 5(x – 3) = 10x – 15 5x – 15 = 10x – 15 5x – 10x = 15 – 15 -5x = 0 x = 0 Thus x = 0 has infinite number of solutions.

Question 14. Find the value of x, given that 4(3x + 2) = 44. Answer: Given, 4(3x + 2) = 44 12x + 8 = 44 12x = 44 – 8 12x = 36 x = 36/12 x = 3

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