Question

proving alternate exterior angles congruent given: lines p and q are parallel and r is a transversal. prove: ∠2 ≅ ∠7 diagram: lines p (top) and q (bottom) are parallel, transversal r intersects them, forming angles 1,2,3,4 (on p) and 5,6,7,8 (on q) statements reasons 1. p || q, r is a transv. 1. given 2. a 2. vert. ∠s ≅ 3. b 3. corr. ∠s thm 4. ∠2 ≅ ∠7 4. transitive prop which statements could complete the proof? a: __ b: __

Explanation:

Step1: Analyze Reason for A

The reason for statement A is "vert ∠’s ≅" (vertical angles are congruent). Vertical angles are opposite angles formed by intersecting lines. Looking at the diagram, ∠2 and ∠3? No, wait, ∠2 and ∠4? Wait, no, vertical angles with ∠7? Wait, ∠1 and ∠3 are vertical? No, let's see: when transversal r intersects line p, angles ∠1 and ∠3 are vertical? Wait, no, ∠2 and ∠4? Wait, no, maybe ∠2 and ∠3? Wait, no, let's think again. Wait, the vertical angles for ∠7: ∠7 and ∠5? No, ∠7 and ∠5 are adjacent? Wait, no, when transversal r intersects line q, angles ∠5, ∠6, ∠7, ∠8. ∠7 and ∠5 are adjacent, ∠7 and ∠8 are vertical? Wait, no, ∠7 and ∠5 are supplementary? Wait, maybe I made a mistake. Wait, the first step is given: p || q, r is transversal. Then statement A: reason is vertical angles congruent. So we need to find a pair of vertical angles. Let's look at the diagram: ∠2 and ∠4? No, ∠1 and ∠3? Wait, maybe ∠2 and ∠3? No, wait, ∠1 and ∠3 are vertical? Wait, no, when two lines intersect, vertical angles are opposite. So line r intersects line p, forming ∠1, ∠2, ∠3, ∠4. So ∠1 and ∠3 are vertical, ∠2 and ∠4 are vertical. Line r intersects line q, forming ∠5, ∠6, ∠7, ∠8. ∠5 and ∠7 are vertical, ∠6 and ∠8 are vertical. Ah! So ∠2 and ∠4? No, wait, the reason for A is vertical angles congruent, so maybe ∠2 ≅ ∠4? No, wait, maybe ∠1 ≅ ∠3? No, the goal is to prove ∠2 ≅ ∠7. Let's see the transitive property: if ∠2 ≅ x and x ≅ ∠7, then ∠2 ≅ ∠7. The third statement B has reason "corr ∠’s thm" (corresponding angles theorem), which says that if two parallel lines are cut by a transversal, corresponding angles are congruent. So corresponding angles: ∠2 and ∠6? No, ∠2 and ∠6? Wait, line p and q are parallel, transversal r. Corresponding angles: ∠2 and ∠6? No, ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8. Wait, ∠3 and ∠7 are corresponding angles. So if we use corresponding angles theorem, then ∠3 ≅ ∠7. Then, if we have ∠2 ≅ ∠3 (vertical angles), then by transitive property, ∠2 ≅ ∠7. Ah! So statement A should be ∠2 ≅ ∠3 (vertical angles congruent), and statement B should be ∠3 ≅ ∠7 (corresponding angles theorem). Let's check:

1. p || q, r is transversal (given)
2. ∠2 ≅ ∠3 (vertical angles congruent) - this is A
3. ∠3 ≅ ∠7 (corresponding angles theorem) - this is B
4. ∠2 ≅ ∠7 (transitive property)

Yes, that makes sense. So A is ∠2 ≅ ∠3, B is ∠3 ≅ ∠7.

Step2: Confirm A and B

For A: Vertical angles are congruent, so ∠2 and ∠3 are vertical angles (formed by transversal r and line p), so ∠2 ≅ ∠3.

For B: Corresponding angles theorem: since p || q and r is transversal, ∠3 (on line p, below transversal) and ∠7 (on line q, below transversal) are corresponding angles, so ∠3 ≅ ∠7.

Answer:

A: $\boldsymbol{\angle 2 \cong \angle 3}$
B: $\boldsymbol{\angle 3 \cong \angle 7}$