Prove || lines › Alt Ext angles congruence

Question

This is urgent!! Please help this is due tommorow!!

Answer 1

The specific figure did not come through with your question, so I will prove the Alternate Exterior Angles Theorem in its standard form. Whatever labels your worksheet uses, the structure below is the proof — just swap in your angle numbers.

The setup and the standard labeling

Two parallel lines cut by a transversal create eight angles. Using the usual numbering, angles 1–4 sit at the first intersection (1 upper-left, 2 upper-right, 3 lower-left, 4 lower-right) and 5–8 sit the same way at the second.

The exterior angles are the four outside the two parallel lines: ∠1, ∠2 at the top and ∠7, ∠8 at the bottom. Alternate means on opposite sides of the transversal. So the two alternate exterior pairs are ∠1 and ∠8, and ∠2 and ∠7.

I will prove ∠1 ≅ ∠8.

The idea first

There is no postulate that hands you alternate exterior angles directly, so you bridge through one you do have. ∠1 and ∠4 are vertical angles, so they are congruent for free — no parallel lines needed. And ∠4 and ∠8 are corresponding angles, which the Corresponding Angles Postulate makes congruent because the lines are parallel. Chain those two and ∠1 reaches ∠8.

The two-column proof

Given: line l ∥ line m, cut by transversal t

Prove: ∠1 ≅ ∠8

1. l ∥ m, t is a transversal — Given

2. ∠1 ≅ ∠4 — Vertical Angles Theorem

3. ∠4 ≅ ∠8 — Corresponding Angles Postulate

4. ∠1 ≅ ∠8 — Transitive Property of Congruence ∎

The proof for ∠2 ≅ ∠7 is identical with the mirrored angles: ∠2 ≅ ∠3 (vertical), ∠3 ≅ ∠7 (corresponding), so ∠2 ≅ ∠7 (transitive).

The converse, in case that is what you were assigned

Your title reads “Prove || lines,” which might mean you are going the other direction: given that alternate exterior angles are congruent, prove the lines are parallel. That proof runs backwards through the same chain:

1. ∠1 ≅ ∠8 — Given

2. ∠1 ≅ ∠4 — Vertical Angles Theorem

3. ∠4 ≅ ∠8 — Transitive Property

4. l ∥ m — Converse of the Corresponding Angles Postulate ∎

Note carefully what changed: step 4 now cites the converse postulate. Using the forward postulate to conclude that lines are parallel is circular reasoning, and it is the most common error on this type of problem — graders look for exactly this.

A check worth running on any proof like this

Only step 3 in the forward proof actually uses the parallel hypothesis. The vertical angle step is true for any two intersecting lines. If you ever write a proof about parallel lines and never invoke the parallelism anywhere, something has gone wrong — your argument would be claiming to hold for non-parallel lines too.