# Texas Go Math Grade 8 Lesson 7.1 Answer Key Parallel Lines Cut by a Transversal

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## Texas Go Math Grade 8 Lesson 7.1 Answer Key Parallel Lines Cut by a Transversal

Explore Activity 1

Parallel Lines and Transversals

A transversal is a line that intersects two lines in the same plane at two different points. Transversal f and lines a and b form eight angles.

Use geometry software to explore the angles formed when a transversal intersects parallel lines.

A. Construct a line and label two points on the line A and B.
B. Create point C not on $$\overleftrightarrow{A B}$$. Then construct a line parallel to $$\overleftrightarrow{A B}$$ through point C. Create another point on this line and label it D.
C. Create two points outside the two parallel lines and label them E and F. Construct transversal $$\overleftrightarrow{E F}$$. Label the points of intersection G and H.

D. Measure the angles formed by the parallel lines and the transversal. Write the angle measures in the table below.
E. Drag point E or point F to a different position. Record the new angle measures in the table.

Reflect

Make a Conjecture Identify the pairs of angles in the diagram. Then make a conjecture about their angle measures. Drag a point in the diagram to confirm your conjecture.

Question 1.
corresponding angles
Corresponding angles âˆ CGE = âˆ AHG and âˆ DGE = âˆ BHG

Two Parallel Lines Cut by a Transversal Worksheet Answer Key Question 2.
alternate interior angles
Alternate interior angles âˆ CGH and âˆ GHB, and the second pair âˆ DGH and âˆ AHG

Question 3.
alternate exterior angles
Alternate exterior angles âˆ CGE = âˆ BHF and second pair âˆ EGD = âˆ AHF

Question 4.
same-side interior angles
Same-side interior angles âˆ CGH and angle âˆ AHG, and second pair âˆ DGH and âˆ GHB

Explore Activity 2

Justifying Angle Relationships

You can use tracing paper to informally justify your conclusions from the first Explore Activity.

Lines a and b are parallel. (The black arrows on the diagram indicate parallel lines.)
A. Trace the diagram onto tracing paper.
B. Position the tracing paper over the original diagram so that âˆ 1 on the tracing is over âˆ 5 on the original diagram. Compare the two angles. Do they appear to be congruent?
C. Use the tracing paper to compare all eight angles in the diagram to each other. List all of the congruent angle pairs.

Find each angle measure.

Parallel Lines Cut by a Transversal Worksheet 8th Grade Pdf Question 5.
mâˆ GDE = __________
Find mâˆ GDE
Angles mâˆ GDE and mâˆ DEH are vertical angles so
mâˆ GDE = mâˆ DEH = 4xÂ°
Angle mâˆ GDE is supplementary to mâˆ DEH because they are same-side interior angles
mâˆ GDE + mâˆ DEH = 180Â° …………………… (1)
4xÂ° + 6xÂ° = 180Â° (Replace mâˆ GDE with 4xÂ°) and mâˆ DEH with 6xÂ° ………….. (2) & (3)
10xÂ° = 180Â° (Combine like terms.) ……………. (4)
xÂ° = $$\frac{1}{2}$$ (Divide both sides with 10) …………… (5)
x = 18Â° (Simplify) ………….. (6)
mâˆ GDE = 4xÂ° = (4 âˆ™ 18)Â° = 72Â°

Question 6.
mâˆ BEF = ___________
Find mâˆ BEF.
Angles mâˆ GDE and mâˆ DEH are vertical angles so
mâˆ GDE = mâˆ DEH = 4xÂ°
Angle mâˆ GDE is supplementary to mâˆ DEH because they are same-side interior angles.
mâˆ GDE + mâˆ DEH = 1800 …………. (1)
4xÂ° + 6xÂ° = 180Â° (RepLace mâˆ GDE with 4xÂ°) and mâˆ DEH with 6xÂ° …………… (2) & (3)
10xÂ° = 180Â° (Combine like terms.) …………….. (4)
xÂ° = $$\frac{180^{\circ}}{10}$$ (Divide both sides with 10) ……………… (5)
x = 18Â° (Simplify) ……………. (6)
mâˆ BEF = 6xÂ° = (6 âˆ™ 18)Â° = 108Â°

Question 7.
mâˆ CDG = ____________
âˆ BEF is congruent to âˆ DEH because they are vertical angles. Therefore,
mâˆ BEF = mâˆ DEH = 6xÂ°
âˆ DEH is supplementary to âˆ GDE because they are same-side interior angles. Therefore,
mâˆ DEH + mâˆ GDE = 180Â°
6xÂ° + 4xÂ° = 180Â°
10x = 180
$$\frac{10 x}{10}=\frac{180}{10}$$
x = 18
âˆ CDG is congruent to âˆ DEH because they are corresponding angles. Therefore,
mâˆ CDG = mâˆ DEH = 6xÂ° = (6 âˆ™ 18)Â° = 108Â°

Use the figure for Exercises 1-4. (Explore Activity 1 and Example 1)

Lesson 7.1 Interior and Exterior Angles Answer Key Question 1.
âˆ UVY and ___________ are a pair of corresponding angles.
âˆ UVY and âˆ VWZ are a pair of corresponding angles.

Question 2.
âˆ WVYand âˆ VWT are ____________ angles.
âˆ WVYand âˆ VWT are alternate interior angles.

Question 3.
Find mâˆ SVW ___________.
âˆ SVW is supplementary to âˆ VWT because they are same-side interior angles. Therefore,
mâˆ SVW + mâˆ VWT = 180Â°
4xÂ° + 5xÂ° = 180Â°
9x = 180
$$\frac{9 x}{9}=\frac{180}{9}$$
x = 20
mâˆ SVW = 4xÂ° = (4 âˆ™ 20)Â° = 80Â°

Question 4.
Find mâˆ VWT. ______________
âˆ SVW is supplementary to âˆ VWT because they are same-side interior angles Therefore.
mâˆ SVW + mâˆ VWT = 180Â°
4xÂ° + 5xÂ° = 180Â°
9x = 180
$$\frac{9 x}{9}=\frac{180}{9}$$
x = 20
mâˆ VWT = 5xÂ° = (5 âˆ™ 20)Â° = 100Â°

Question 5.
Vocabulary When two parallel lines are cut by a transversal, ___________ angles are supplementary. (Explore Activity 1)
If two parallel lines are cut by a transversal line, interior angles are formed. The pairs of consecutive interior angles are called same-side interior angles which are supplementary Another thing is when two parallel lines are cut by a transversal, the pairs of angles on either side of the transversal and inside the two lines are called alternate interior angles which are congruent.

By the description, when two parallel lines are cut by a transversal, same-side interior angleÂ are supplementary.

Essential Question Check-In

Practice and Homework Lesson 7.1 Go Math 8th Grade Answer Key Question 6.
What can you conclude about the interior angles formed when two parallel lines are cut by a transversal?
When two parallel lines are cut by a transversal, then four interior angles are formed. These four interior angles could be the alternate interior angles or the same-side interior angles.
Alternate interior angles lie on the alternate side of the transversal and they are congruent.
Same-side interior angles lie on the same side of the transversal which are not adjacent. Same-side interior angles are said to be supplementary.

Vocabulary Use the figure for Exercises 7-10.

Question 7.
Name all pairs of corresponding angles.
The corresponding angles identified in the given diagram are:

• âˆ 1 and âˆ 5
• âˆ 3 and âˆ 7
• âˆ 2 and âˆ 6
• âˆ 4 and âˆ 8

Question 8.
Name both pairs of alternate exterior angles.
Alternate exterior angles identified in the given diagram are:

• âˆ 1 and âˆ 8
• âˆ 2 and âˆ 7

Question 9.
Name the relationship between âˆ 3 and âˆ 6.
âˆ 3 and âˆ 6 are alternate interior angles.

Parallel Lines and Transversals Quiz Answer Key Pdf Question 10.
Name the relationship between âˆ 4 and âˆ 6.
Angles âˆ 4 and âˆ 6 are same-side interior angles.

Find each angle measure.

Question 11.
mâˆ AGE when mâˆ FHD = 30Â° _________
âˆ AGE and âˆ FHD are alternate exterior angles. Therefore,
mâˆ AGE = mâˆ FHD = 30Â°
mâˆ AGE = 30Â°

Question 12.
mâˆ AGH when mâˆ CHF = 150Â° __________
mâˆ AGH and âˆ CHF are corresponding angles. Therefore,
mâˆ AGH = mâˆ CHF = 150Â°
mâˆ AGH = 150Â°

Question 13.
mâˆ CHF when mâˆ BGE = 110Â° ___________
âˆ CHF and âˆ BGE are alternate exterior angles. Therefore,
mâˆ CHF = mâˆ BGE = 110Â°
mâˆ CHF = 110Â°

Question 14.
mâˆ CHG when mâˆ HGA = 120Â° _____________
âˆ CHG is supplementary to âˆ HGA because they are same-side interior angles. Therefore,
mâˆ CHG + mâˆ HGA = 180Â°
mâˆ CHG + 120Â° = 180Â°
mâˆ CHG + 120Â° – 120Â° = 180Â° – 120Â°
mâˆ CHG = 60Â°

Question 15.
mâˆ BGH = _____________

âˆ BGH and âˆ GHD are supplementary because they are same-side interior angles. So,
âˆ BGH + âˆ GHD = 180Â°
3x + (2x + 50)Â° = 180Â°
5x = 180Â° – 50Â° = 130Â°
x = $$\frac{130}{5}$$ = 26Â°
âˆ BGH = 3xÂ° = 3 Ã— 26Â° = 78Â°
âˆ GHD = (2x + 50)Â° = (2 Ã— 26Â° + 50) = 102Â°

Lesson 7.1 How to Solve Parallel Lines Cut by a Transversal Question 16.
mâˆ GHD = __________
âˆ BGH is supplementary to âˆ GHD because they are same-side interior angles. Therefore,
mâˆ BGH + mâˆ GHD = 180Â°
3xÂ° + (2x + 50)Â° = 180Â°
3x + 2x + 50 – 50 = 180 – 50
5x = 130
$$\frac{5 x}{5}=\frac{130}{5}$$
x = 26
mâˆ GHD = (2x + 50)Â° = (2 âˆ™ 26 + 50)Â° = (52 + 50)Â° = 102Â°
mâˆ GHD = 102Â°

Question 17.
The Cross Country Bike Trail follows a straight line where it crosses 350th and 360th Streets. The two streets are parallel to each other. What is the measure of the larger angle formed at the intersection of the bike trail and 360th Street? Explain.

For easier calculations, we are going to use the following schema. It is given that mâˆ 3 = 48Â°

The larger angle formed at the intersection of the bike trail and 360th Street is the angle 5 in our schema. âˆ 5 is supplementary to âˆ 3 because they are same-side interior angles. Therefore,
mâˆ 5 + mâˆ 3 = 180Â°
mâˆ 5 + 48Â°= 180Â°
mâˆ 5 + 48Â° – 48Â° = 180Â° – 48Â°
mâˆ 5 = 132Â°
The larger angle formed at the intersection of the bike trail and 360th Street is 132Â°

Question 18.
Critical Thinking How many different angles would be formed by a transversal intersecting three parallel lines? How many different angle measures would there be?

As we can see from the above schema, there are 12 different angles formed by a transversal intersecting three parallel lines.
There are 2 different angle measures:

• mâˆ 1 = mâˆ 4 = mâˆ 5 = mâˆ 8 = mâˆ 9 = mâˆ 12
• mâˆ 2 = mâˆ 3 = mâˆ 6 = mâˆ 7 = mâˆ 10 = mâˆ 11

Question 19.
Communicate Mathematical Ideas In the diagram at the right, suppose mâˆ 6 = 125Â°. Explain how to find the measures of each of the other seven numbered angles.

• mâˆ 2 = mâˆ 6 = 125Â° because âˆ 2 and âˆ 6 are corresponding angles.
• mâˆ 3 = mâˆ 2 = 125Â° because âˆ 3 and âˆ 2 are vertical angles.
• mâˆ 7 = mâˆ 3 = 125Â° because âˆ 7 and âˆ 3 are corresponding angles.
• âˆ 4 is supplementary to âˆ 6 because they are same-side interior angles. Therefore,

mâˆ 4 + mâˆ 6 = 180Â°
mâˆ 4 + 125Â° = 180Â°
mâˆ 4 + 125Â° – 125Â° = 180Â° – 125Â°
mâˆ 4 = 55Â°

• mâˆ 8 = mâˆ 4 = 55Â° because âˆ 8 and âˆ 4 are corresponding angles.
• mâˆ 1 = mâˆ 4 = 55Â° because âˆ 1 and âˆ 4 are vertical angles
• mâˆ 5 = mâˆ 1 = 55Â° because âˆ 5 and âˆ 1 are corresponding angles

H.O.T. Focus on Higher Order Thinking

Question 20.
Draw Conclusions In a diagram showing two parallel lines cut by a transversal, the measures of two same-side interior angles are both given as 3xÂ°. Without writing and solving an equation, can you determine the measures of both angles? Explain. Then write and solve an equation to find the measures.
We are given that the measures of both angles are equal. We also know that the sum of two same-side interior angles is 180Â°. Therefore, without writing and solving any equation, we divide 180 by 2 and we find that each angle measures 90Â°.
If we use an equation to find the measures of the angle, we have:
mâˆ 1 + mâˆ 2 = 180Â°
3x + 3x = 180Â°
6x = 180Â°
$$\frac{6 x}{6}=\frac{180}{6}$$
x = 30
mâˆ 1 = mâˆ 2 = 3x = 3 âˆ™ 30 = 90Â°

Question 21.
Make a Conjecture Draw two parallel lines and a transversal. Choose one of the eight angles that are formed. How many of the other seven angles are congruent to the angle you selected? How many of the other seven angles are supplementary to your angle? Will your answer change if you select a different angle?
Select angle âˆ 1, for example. There are 3 congruent angles to âˆ 1 : âˆ 3, âˆ 5, and âˆ 7
âˆ 3 = âˆ 1 because they are vertical angles,
âˆ 7 = âˆ 1 because they are alternate exterior angles.
And, there are 4 supplementary angles: âˆ 2, âˆ 4, âˆ 6 and âˆ 8
No matter what angle we select, we will always have the same answer – 3 congruent angles, and 4 supplementary.

Three of the other seven are congruent to the selected angle.

• Four are supplementary.
• The answer will not change.

Lesson 7.1 How to Find Angles of Parallel Lines Cut by a Transversal Question 22.
Critique Reasoning In the diagram at the right, âˆ 2, âˆ 3, âˆ 5, and âˆ 8 are all congruent, and âˆ 1, âˆ 4, âˆ 6, and âˆ 7 are all congruent. Aiden says that this is enough information to conclude that the diagram shows two parallel lines cut by a transversal. Is he correct? Justify your answer.