Given: g ∥ h and ∠2 ≅ ∠3

Prove: e ∥ f

Horizontal and parallel lines e and f are intersected by diagonal and parallel lines g and h. At the intersection of lines g and e, the bottom right angle is angle 2. At the intersection of lines h and e, the bottom right angle is angle 1. At the intersection of lines f and h, the top left angle is angle 3.

Statements Reasons

1. g || h 1. given

2. ∠1 ≅ ∠2 2. corresponding angles theorm

3. ∠2 ≅ ∠3 3. given

4. ∠1 ≅ ∠3 4. transitive property

5. e || f 5. ?

What is the missing reason in the proof?

vertical angles theorem

alternate exterior angles theorem

converse corresponding angles theorem

converse alternate interior angles theorem

In the image, you can observe a diagram representing this problem.

We know by given that and .

From the parallelism between line g and line h, we deduct several congruence between angles.

, by corresponding angles (same side of the transversal, one interior, the other exterior to parallels).

Now, to demonstrate , we must demonstrate a congruence between angle 2 and an angle on the intersection between line g and line f.

In the parallelogram formed, we know

Where is the angle at the intersection line g and line f.

But, we know and , so

Notice that we don't have a congruence, however there's theorem which states that the same-side interior angles of parallels are supplementary.

In this case, we use the corolary of that theorem, which states if two same-side interior angles are supplementary, then the lines are parallels.

However, according to the choices of the problem, the missin proof is "converse alternate interior angles theorem", because the problem was demonstrate using transitive property, to show that angles 1 and 3 are congruent, there by converse alternate interior angles theorem, lines e and f are parallels.

This is the same case we used, but using converse alternate interior angles theorem.