Question

Ina is given the theorem and diagram below. She constructs the outline of a proof below to show that lines w and x are parallel.

Consecutive Interior Angles Theorem: If two lines are parallel, the interior angles on the same side of a transversal that crosses the lines are supplementary.

Given: lines w and x intersected by line t;
∠
1
and
∠
5
are congruent.

Prove:
w
∥
x

Angles 3, 6 and 4, 5 are pairs of alternate interior angles. Angles 1, 8 and 2, 7 are pairs of alternate exterior angles.

The left box shows angle 1 is congruent to angle 5 and the reason A shows ‘given’. The first right box shows angle 1 and angle 3 are a linear pair and the reason B shows ‘Definition of linear pair’. A downward arrow points another box below it. The box shows angle 1 and angle 3 are supplementary and reason C shows ‘Linear pair postulate’. An arrow from left and an arrow from right points a box in the middle. The middle box shows angle 3 and angle 5 are supplementary and the reason D is unknown. An arrow below the middle box points another box that shows w parallel to x and reason E unknown.

Which pair of reasons should be listed in D and E to complete Ina's proof?

A.
D: Linear pair postulate
E: Converse of the Consecutive Interior Angles Theorem

B.
D: Linear pair postulate
E: Consecutive Interior Angles Theorem

C.
D: Congruent angles are supplementary to the same angle.
E: Converse of the Consecutive Interior Angles Theorem

D.
D: Congruent angles are supplementary to the same angle.
E: Consecutive Interior Angles Theorem

Answer 1

Correct answer: C.

Reasoning: From ∠1 ≅ ∠5 and ∠1 and ∠3 supplementary (linear pair), you get ∠3 and ∠5 supplementary by the fact that congruent angles are supplementary to the same angle. Since ∠3 and ∠5 are consecutive interior angles and they are supplementary, the converse of the Consecutive Interior Angles Theorem implies w ∥ x.