Question

1. (127) select the pair of lines in the diagram below that must be parallel if ∠6 ≅ ∠8. if there are no lines that must be parallel, select “none.”

a. k || j
b. d || e
c. k || i
d. d || f
e. none

1. (118) find x and y.

a. x = 15; y = 45
b. x = 20; y = 60
c. x = 18; y = 72
d. x = 21; y = 85
e. none of the above

1. (131) find y.

a. y = 95
b. y = 121
c. y = 180
d. y = 139
e. none of the above

1. (130) find x and y.

a. x = 48; y = 100
b. x = 56; y = 110
c. x = 42; y = 85
d. x = 45; y = 90
e. none of the above

1. (131) find x and y.

a. x = 34; y = 90
b. x = 42; y = 84
c. x = 25; y = 45
d. x = 26; y = 92
e. none of the above

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. We know that \(\angle6\) and \(\angle8\) are not related by any of the parallel - line angle relationships (corresponding angles, alternate interior angles, same - side interior angles) for the given line pairs.
For two lines to be parallel, if we have a transversal, the following must hold:

• Corresponding angles: \(\angle a=\angle b\) (if two lines are parallel and cut by a transversal, corresponding angles are equal and vice - versa)
• Alternate interior angles: \(\angle c=\angle d\) (if two lines are parallel and cut by a transversal, alternate interior angles are equal and vice - versa)
• Same - side interior angles: \(\angle e+\angle f = 180^{\circ}\) (if two lines are parallel and cut by a transversal, same - side interior angles are supplementary and vice - versa)

For pair \(k\parallel j\): The transversal would need to relate \(\angle6\) and \(\angle8\) in a way that is not the case. For pair \(d\parallel e\): \(\angle6\) and \(\angle8\) are not in a position (corresponding, alternate interior, same - side interior) to prove \(d\parallel e\). For pair \(k\parallel i\): \(\angle6\) and \(\angle8\) are not relevant. For pair \(d\parallel f\): \(\angle6\) and \(\angle8\) are not in a position (corresponding, alternate interior, same - side interior) to prove \(d\parallel f\)

Answer:

E. None