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**

The angles of a triangle have a relationship with each other. Draw three unique triangles. What do you notice about the measures of the interior angles of the triangles?

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

**Build Understanding**

1. What is the sum of the measures of the three interior angles of a triangle?

A. Find the sum of the measures of the angles in each of the three triangles.

Answer:

Triangle A:

90Â° + 25Â° + 65Â°

90Â° + 90Â° = 180Â°

Triangle B:

75Â° + 75Â° + 30Â° = 150Â° + 30Â° = 180Â°

Triangle C:

105Â° + 45Â° + 30Â° = 150Â° + 30Â° = 180Â°

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 180Â°

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 180Â°.

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 180Â°

2x + 3x + 4x = 180Â°

9x = 180Â°

x = 180/9

x = 20Â°

2x = 2 Ã— 20 = 40Â°

3x = 3 Ã— 20 = 60Â°

4x = 4 Ã— 20 = 80Â°

**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 triangleâ€™s interior angles.

2. The dashed line segment represents an extension of one side of the triangle. Together with the right side of the triangle, the segment forms an angle, âˆ 4.

A. What is the sum of the measures of âˆ 3 and âˆ 4?

__________________

Answer:

the sum of the measures of âˆ 3 and âˆ 4 is 180Â°

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: âˆ 4 is an exterior angle.

C. If the measure of âˆ 3 is 60Â°, what is the measure of âˆ 4?

_______________________

Answer:

âˆ 3 = 60Â°

âˆ 3 + âˆ 4 = 180Â°

60Â°+ âˆ 4 = 180Â°

âˆ 4 = 180Â° – 60Â°

âˆ 4 = 120Â°

D. If the measure of âˆ 3 is 60Â°, what is the sum of the measures of âˆ 1 and âˆ 2?

______________

Answer:

âˆ 3 = 60Â°

âˆ 1 + âˆ 2 + âˆ 3 = 180Â°

âˆ 1 + âˆ 2 + 60Â° = 180Â°

âˆ 1 + âˆ 2 = 180Â° – 60Â° = 120Â°

Thus the sum of the measures of âˆ 1 and âˆ 2 is 120Â°

E. Which angle has a measure equal to the sum of the measures of âˆ 1 and âˆ 2?

______________________________

Answer:

âˆ 4 = 120Â°

âˆ 1 + âˆ 2 = 180Â° – 60Â° = 120Â°

So, âˆ 4 has a measure equal to the sum of the measures of âˆ 1 and âˆ 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: âˆ 1 and âˆ 2 are the remote interior angles in relation to Angle 4.

G. If the sum of the measures of âˆ 1 and âˆ 2 is 115Â°, what is the measure of âˆ 4?

_______________________

Answer:

If the sum of the measures of âˆ 1 and âˆ 2 is 115Â° then the measure of âˆ 4 is 115Â°.

**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?

3. A machinist is drawing a triangular piece of an industrial machine.

A. Write an equation and solve to find the value of x. Show your work.

___x + ___ = x + ___

__x – x = 80 – ___

x = ___

Answer:

2x + 45Â° = x + 80Â°

2x – x = 80Â° – 45Â°

x = 35Â°

B. What is the measure of the unknown remote interior angle?

_____________________

Answer: the measure of the unknown remote interior angle is 35Â°

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

70Â° + 45Â° = 115Â°

**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 115Â°

**Check Understanding**

Question 1.

Two angles of a triangle have measures of 30Â° and 45Â°. What is the measure of the remaining angle?

Answer:

Given,

Two angles of a triangle have measures of 30Â° and 45Â°.

Sum of three angles of a triangle = 180Â°

30Â°+ 45Â° + xÂ° = 180Â°

75Â° + xÂ° = 180

xÂ° = 180Â° – 75Â°

xÂ° = 105Â°

Question 2.

Dana draws a triangle with one angle that has a measure of 40Â°.

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 40Â°.

180Â°- 40Â° = 140Â°

Thus the measure of the angle’s adjacent exterior angle is 140Â°

B. What is the sum of the measures of the remote interior angles for the exterior angle adjacent to the 40Â° angle?

______________

Answer:

140Â° + 40Â° = 180Â°

Question 3.

An exterior angle of a triangle has a measure of 80Â°, and one of the remote interior angles has a measure of 20Â°. 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 80Â°,

180Â° – 80Â° = 100Â°

and one of the remote interior angles has a measure of 20Â°.

180Â° – 20Â° – 100Â° = 60Â°

**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 30Â°, what is the measure of the third angle?

Answer:

x + 30Â° + 30Â° = 180

x + 60Â° = 180Â°

x = 180Â° – 60Â°

x = 120Â°

The triangle is an isosceles triangle and the measure of the third angle is 120Â°

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 = 180Â°

100 + 100 = 200Â° (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 = 180Â°

3xÂ° = 180Â°

x = 180/3

x = 60Â°

Question 7.

A triangle has one 30Â° 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 30Â° angle, an unknown angle, and an angle with a measure that is twice the measure of the unknown angle.

x + 2x + 30Â° = 180Â°

3x + 30Â° = 180Â°

3x = 180Â° – 30Â°

3x = 150Â°

x = 150/3

x = 50Â°

2x = 2 Ã— 50 = 100Â°

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

Question 8.

_____________

Answer:

Sum of three angles of a triangle is 180Â°

31.5Â° + 90Â° + xÂ° = 180Â°

121.5Â° + xÂ° = 180Â°

xÂ° = 180Â° – 121.5Â°

xÂ° = 58.5Â°

Thus the unknown angle is 58.5Â°

Question 9.

_____________

Answer:

Sum of three angles of a triangle is 180Â°

25Â° + xÂ° + 30Â° = 180Â°

55Â° + xÂ° = 180Â°

xÂ° = 180Â° – 55Â°

xÂ° = 125Â°

Thus the unknown angle is 125Â°

Question 10.

Answer:

Sum of three angles of a triangle is 180Â°

45Â° + 80Â° + xÂ° = 180Â°

125Â° + xÂ° = 180Â°

xÂ° = 180Â° – 125Â°

xÂ° = 155Â°

Thus the unknown angle is 155Â°

Question 11.

**Open Ended** The measure of an exterior angle of a triangle is xÂ°. 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 xÂ°.

The measure of the adjacent interior angle is at least twice x.

xÂ° + Î¸ = 180Â°

Î¸ = 180Â° – x â‰¥ 2xÂ°

180Â° â‰¥ 3xÂ°

0Â° < x â‰¤ 60Â°

Any three numbers in (0, 60).

Question 12.

The measure of an exterior angle of a triangle is 40Â°. What is the sum of the measures of the corresponding remote interior angles?

Answer:

The measure of an exterior angle of a triangle is 40Â°.

2x + 40Â° = 180Â°

2x = 180 – 40

2x = 140

x = 140/2

x = 70Â°

Thus the sum of the measures of the corresponding remote interior angles is 140Â°

Question 13.

Steven is building a fin for his surfboard. In order to make the fin, he needs to know the value of x in the following diagram. Use your knowledge of triangle angle relationships to find the value of x.

Answer:

Sum of three angles of a triangle is 180Â°

30Â° + 15Â° + yÂ° = 180Â°

45Â° + yÂ° = 180Â°

yÂ° = 180Â° – 45Â°

y = 135Â°

135Â° + x = 180Â°

xÂ° = 180Â° – 135Â°

xÂ° = 45Â°

Question 14.

Find the value of x in the diagram. Explain how you found the answer.

Answer:

x = y

y + 60Â° + 55Â° = 180Â°

y + 115Â° = 180Â°

y = 180Â° – 115Â°

y = 65Â°

So, x = 65Â°

**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 1.

Find the value of x using your knowledge of the relationship between interior and exterior angles.

Answer:

Sum of interior and exterior angles is 180Â°

27.4Â° + xÂ° = 180Â°

xÂ° = 180Â° – 27.4Â°

x = 152.6Â°

Question 2.

**Math on the Spot** Find the unknown measure in the triangle.

Answer:

xÂ° + 78Â° + 27Â° = 180Â°

xÂ° + 105Â° = 180Â°

xÂ° = 180Â° – 105Â°

xÂ° = 75Â°

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 180Â° in adding all the three angles of a triangle.

Question 4.

**STEM** The measure of the angle formed at the center of an oxygen atom in a water molecule is about 105Â°. The angles formed at each hydrogen atom are congruent. What is the size of the angle at each hydrogen atom?

Answer:

Congruent angles are the ones that are the same in value, in the attached image we can see the atom, the O atom forms 105 with both H atoms, if we use trigonometry we have

x + x + 105Â° = 180Â°

2x + 105Â° = 180Â°

2x = 180 – 105

2x = 75

x = 75/2

x = 37.5Â°

Question 5.

**Open Ended** One of the angles in a triangle measures 90Â°. Name three possibilities for the measures of the remaining two angles.

Answer:

One of the angles in a triangle measures 90Â°

30Â° + 60Â° + 90Â° = 180Â°

90Â° + 45Â° + 45Â° = 180Â°

90Â° + 50Â° + 40Â° = 180Â°

Question 6.

Find the value of x in the following diagram.

Answer:

xÂ° + 60Â° = 180Â°

xÂ° = 180Â° – 60Â°

xÂ° = 120Â°

Thus the measure of angle x is 120.

**Test Prep**

Question 7.

Complete the tÃ¡ble by entering the measures of the unknown angles for the following two triangles.

Answer:

Sum of three angles = 180Â°

Triangle 1:

30Â° + 60Â° + x = 180

90Â° + xÂ° = 180Â°

xÂ° = 180Â° – 90Â°

xÂ° = 90Â°

Triangle 2:

45Â° + 20Â° + y = 180Â°

65Â° + y = 180Â°

y = 180Â° – 65Â°

y = 115Â°

Question 8.

If an exterior angle of a triangle has a measure of 35Â°, what is the measure of the adjacent interior angle?

Answer:

35Â° + x = 180Â°

x = 180 – 35

x = 145Â°

Thus the measure of the adjacent interior angle is 145Â°

Question 9.

Find the value of x.

x = ____

Answer:

120Â° + y = 180Â°

y = 180Â° – 120Â°

y = 60Â°

(x – 5)Â° + (x + 5)Â° + 60 = 180Â°

x – 5 + x + 5 + 60 = 180

2x + 60 = 180

2x = 180 – 60

2x = 120

x = 120/2 = 60

x = 60Â°

x – 5 = 60 – 5 = 55Â°

x + 5 = 60 + 5 = 65Â°

Question 10.

The measures of an exterior angle of a triangle and its adjacent interior angle add to what value?

A. 90Â°

B. 100Â°

C 180Â°

D. 360Â°

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