Question

given: lines p and q are parallel and r is a transversal.
prove: ∠2 ≅ ∠7
diagram of lines p (top) and q (bottom) with transversal r, 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: dropdown
b: options: angle 2 is congruent to angle 3. angle 2 is congruent to angle 6. angle 3 is congruent to angle 7. angle 3 is congruent to angle 6.

Explanation:

Step1: Analyze Statement A (Reason: vert. ∠’s ≅)

Vertical angles are congruent. For line \( p \) and transversal \( r \), \( \angle 2 \) and \( \angle 3 \) are not vertical angles. Wait, \( \angle 2 \) and \( \angle 3 \)? No, vertical angles with \( \angle 2 \) would be \( \angle 3 \)? Wait, no, \( \angle 1 \) and \( \angle 3 \) are vertical, \( \angle 2 \) and \( \angle 4 \) are vertical? Wait, no, looking at the diagram: when two lines intersect, vertical angles are opposite. So at the intersection of \( p \) and \( r \), \( \angle 2 \) and \( \angle 3 \)? Wait, no, \( \angle 1 \) and \( \angle 3 \) are vertical (opposite), \( \angle 2 \) and \( \angle 4 \) are vertical? Wait, maybe I made a mistake. Wait, the reason for A is "vert. ∠’s ≅" (vertical angles are congruent). So we need a pair of vertical angles. Let's see: \( \angle 2 \) and \( \angle 3 \)? No, \( \angle 1 \) and \( \angle 3 \) are vertical, \( \angle 2 \) and \( \angle 4 \)? Wait, maybe the correct vertical angle for \( \angle 2 \) is \( \angle 3 \)? No, maybe \( \angle 2 \) and \( \angle 3 \) are adjacent? Wait, no, the intersection of \( p \) and \( r \) creates four angles: \( \angle 1 \) (top left), \( \angle 2 \) (top right), \( \angle 3 \) (bottom left), \( \angle 4 \) (bottom right). So vertical angles: \( \angle 1 \cong \angle 3 \), \( \angle 2 \cong \angle 4 \)? Wait, no, vertical angles are opposite each other. So \( \angle 1 \) and \( \angle 3 \) are vertical (opposite), \( \angle 2 \) and \( \angle 4 \) are vertical (opposite). Wait, but the options for A: "Angle 2 is congruent to angle 3" – no, that's not vertical. Wait, maybe I misread. Wait, the options for A and B: let's list the options again. The dropdown for A and B has: "Angle 2 is congruent to angle 3", "Angle 2 is congruent to angle 6", "Angle 3 is congruent to angle 7", "Angle 3 is congruent to angle 6".

Wait, let's re-examine the proof structure:

1. Given \( p \parallel q \), \( r \) is transversal.

2. Statement A, Reason: vertical angles ≅. So A must be a pair of vertical angles. Let's see the angles: at the intersection of \( p \) and \( r \), vertical angles are \( \angle 2 \) and \( \angle 3 \)? No, \( \angle 1 \) and \( \angle 3 \), \( \angle 2 \) and \( \angle 4 \). Wait, maybe the diagram is different. Wait, the lines \( p \) and \( q \) are parallel, \( r \) is transversal. So \( \angle 2 \) and \( \angle 3 \) – no, maybe \( \angle 2 \) and \( \angle 3 \) are adjacent? Wait, no, vertical angles are opposite. Wait, maybe the correct vertical angle for \( \angle 2 \) is \( \angle 3 \)? No, that can't be. Wait, maybe the problem has a typo, but let's think about the next step.

Step 3: Statement B, Reason: corr. ∠’s thm (corresponding angles theorem). Corresponding angles are congruent when lines are parallel. So corresponding angles would be, for example, \( \angle 2 \) and \( \angle 6 \) (corresponding), or \( \angle 3 \) and \( \angle 7 \) (corresponding).

Step 4: \( \angle 2 \cong \angle 7 \) by transitive property. So we need \( \angle 2 \cong \) something, and that something \( \cong \angle 7 \).

Let's try A: "Angle 2 is congruent to angle 3" (vertical angles? Wait, no, \( \angle 2 \) and \( \angle 3 \) are adjacent, supplementary? Wait, maybe the diagram is such that \( \angle 2 \) and \( \angle 3 \) are vertical? No, that's not standard. Wait, maybe the intersection is labeled differently. Let's assume that at the intersection of \( p \) and \( r \), \( \angle 2 \) and \( \angle 3 \) are vertical (maybe the diagram has \( \angle 1 \) top, \( \angle 2 \) right, \( \angle 3 \) bottom, \( \angle 4 \) left? No, that's not standard. Wait, maybe the user's diagram has \( \angle 2 \) and \( \angle 3 \) as vertical. Let's proceed.

If A is \( \angle 2 \cong \angle 3 \) (vertical angles), then B needs to be \( \angle 3 \cong \angle 7 \) (corresponding angles, since \( p \parallel q \), transversal \( r \), so \( \angle 3 \) and \( \angle 7 \) are corresponding). Then by transitive property, \( \angle 2 \cong \angle 3 \) and \( \angle 3 \cong \angle 7 \), so \( \angle 2 \cong \angle 7 \). That works.

Let's check the options:

For A: "Angle 2 is congruent to angle 3" – that's a vertical angle pair (if the diagram is such that \( \angle 2 \) and \( \angle 3 \) are vertical). For B: "Angle 3 is congruent to angle 7" – that's corresponding angles (since \( p \parallel q \), transversal \( r \), so \( \angle 3 \) and \( \angle 7 \) are corresponding). Then transitive property: \( \angle 2 \cong \angle 3 \) and \( \angle 3 \cong \angle 7 \) implies \( \angle 2 \cong \angle 7 \).

So A: Angle 2 is congruent to angle 3; B: Angle 3 is congruent to angle 7.

Answer:

A: Angle 2 is congruent to angle 3
B: Angle 3 is congruent to angle 7