Question

use the figure to the right to answer questions #1 - 6

1. what is the relationship between < 2 and < 7?

a. 2 + 7 = 90
b. 2 + 7 = 180
c. 2 = 7
d. no relationship

1. what is the relationship between < 8 and < 6?

a. 5 + 8 = 90
b. 5 + 8 = 180
c. 5 = 8
d. no relationship

1. what is the relationship between < 3 and < 5?

a. 3 + 5 = 90
b. 3 + 5 = 180
c. 3 = 5
d. no relationship

1. what is the relationship between < 3 and < 6?

a. 3 + 6 = 90
b. 3 + 6 = 180
c. 3 = 6
d. no relationship

1. what is the relationship between < 6 and < 8?

a. 6 + 8 = 90
b. 6 + 8 = 180
c. 6 = 8
d. no relationship

1. what is the relationship between < 1 and < 5?

a. 1 + 5 = 90
b. 1 + 5 = 180
c. 1 = 5
d. no relationship

Explanation:

Step1: Recall the properties of parallel lines and transversals

When two parallel lines \(m\) and \(n\) are cut by a transversal, corresponding angles are equal, alternate - interior angles are equal, alternate - exterior angles are equal, and consecutive interior angles are supplementary.

Step2: Analyze each pair of angles

1. For \(\angle2\) and \(\angle7\):

• \(\angle2\) and \(\angle7\) are not complementary (\(2 + 7

eq90\)), not supplementary (\(2 + 7
eq180\)), and not equal. So, they have no relationship.

1. For \(\angle5\) and \(\angle8\):

• \(\angle5\) and \(\angle8\) are vertical angles. By the vertical - angle theorem, \(\angle5=\angle8\).

1. For \(\angle3\) and \(\angle5\):

• \(\angle3\) and \(\angle5\) are consecutive interior angles. When two parallel lines \(m\) and \(n\) are cut by a transversal, consecutive interior angles are supplementary. So, \(\angle3+\angle5 = 180^{\circ}\).

1. For \(\angle3\) and \(\angle6\):

• \(\angle3\) and \(\angle6\) are alternate - interior angles. When two parallel lines \(m\) and \(n\) are cut by a transversal, alternate - interior angles are equal. So, \(\angle3=\angle6\).

1. For \(\angle6\) and \(\angle8\):

• \(\angle6\) and \(\angle8\) are not complementary (\(6 + 8

eq90\)), not supplementary (\(6 + 8
eq180\)), and not equal. But wait, \(\angle6\) and \(\angle8\) are vertical angles. By the vertical - angle theorem, \(\angle6=\angle8\).

1. For \(\angle1\) and \(\angle5\):

• \(\angle1\) and \(\angle5\) are corresponding angles. When two parallel lines \(m\) and \(n\) are cut by a transversal, corresponding angles are equal. So, \(\angle1=\angle5\).

Answer:

1. d. No Relationship
2. c. \(5 = 8\)
3. b. \(3+5 = 180\)
4. c. \(3 = 6\)
5. c. \(6 = 8\)
6. c. \(1 = 5\)