Question

given: lines p and q are parallel and r is a transversal. prove: ∠2 ≅ ∠7 which statements could complete the proof? a: b: 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 vertical angles? Wait, no—wait, \( \angle 2 \) and \( \angle 3 \)? Wait, no, vertical angles are opposite each other when two lines intersect. So when \( r \) intersects \( p \), \( \angle 2 \) and \( \angle 3 \)? Wait, no, \( \angle 1 \) and \( \angle 3 \) are vertical? Wait, no, let's look at the diagram. Line \( r \) intersects \( p \) at a point, creating angles 1,2,3,4. So \( \angle 2 \) and \( \angle 3 \)? Wait, no, \( \angle 2 \) and \( \angle 3 \) are adjacent? Wait, no, vertical angles are \( \angle 1 \) and \( \angle 3 \), \( \angle 2 \) and \( \angle 4 \)? Wait, maybe I made a mistake. Wait, the reason for statement A is "vert. ∠’s ≅" (vertical angles are congruent). So we need a pair of vertical angles. Let's check the options for A. The options for A (from the dropdown) include "Angle 2 is congruent to angle 3"—no, wait, vertical angles: when two lines intersect, vertical angles are opposite. So \( \angle 2 \) and \( \angle 3 \) are adjacent? Wait, no, maybe the diagram: line \( r \) intersects \( p \), so angles 1 (top left), 2 (top right), 3 (bottom left), 4 (bottom right). So \( \angle 2 \) and \( \angle 3 \) are adjacent, but \( \angle 2 \) and \( \angle 4 \) are vertical? Wait, no, \( \angle 1 \) and \( \angle 3 \) are vertical, \( \angle 2 \) and \( \angle 4 \) are vertical. Wait, maybe the problem has a typo, or I missee. Wait, the reason for A is vertical angles, so statement A should be a vertical angle congruence. Let's check the options: "Angle 2 is congruent to angle 3"—no. Wait, maybe \( \angle 2 \) and \( \angle 3 \) are not vertical. Wait, maybe the correct vertical angle for \( \angle 2 \) is \( \angle 3 \)? No, that can't be. Wait, maybe the diagram is different. Wait, the lines: \( p \) and \( q \) are parallel, \( r \) is transversal. So for statement A, reason is vertical angles, so we need two vertical angles. Let's see the options for A: the first option is "Angle 2 is congruent to angle 3"—no, that's adjacent. Wait, maybe \( \angle 2 \) and \( \angle 3 \) are vertical? Wait, no, when two lines intersect, vertical angles are opposite. So if \( r \) intersects \( p \), the intersection creates four angles: 1 (top left), 2 (top right), 3 (bottom left), 4 (bottom right). So \( \angle 1 \) and \( \angle 3 \) are vertical (opposite), \( \angle 2 \) and \( \angle 4 \) are vertical. So maybe the statement A is \( \angle 2 \cong \angle 3 \)? No, that's not vertical. Wait, maybe the problem has \( \angle 2 \) and \( \angle 3 \) as vertical? No, that's adjacent. Wait, maybe I'm wrong. Let's move to statement B.

Step2: Analyze Statement B (Reason: corr. ∠’s thm.)

Corresponding angles theorem: if two parallel lines are cut by a transversal, corresponding angles are congruent. So we need a pair of corresponding angles. Let's see the options for B. The options include "Angle 3 is congruent to angle 7"—because \( p \parallel q \), transversal \( r \), so \( \angle 3 \) (on line \( p \), bottom left) and \( \angle 7 \) (on line \( q \), bottom left) are corresponding angles. So \( \angle 3 \cong \angle 7 \) by corr. ∠’s thm.

Now, let's back to statement A. The reason is vertical angles, so we need \( \angle 2 \cong \angle 3 \)? Wait, no, vertical angles: maybe \( \angle 2 \cong \angle 3 \) is not vertical. Wait, maybe the diagram has \( \angle 2 \) and \( \angle 3 \) as vertical? Wait, maybe the labels are different. Let's re-examine: the diagram shows line \( p \) with angles 1,2,3,4 (1 and 3 are left, 2 and 4 are right), line \( q \) with angles 5,6,7,8 (5 and 7 left, 6 and 8 right). So when \( r \) intersects \( p \), \( \angle 2 \) (top right) and \( \angle 3 \) (bottom left) are adjacent, but \( \angle 2 \) and \( \angle 4 \) are vertical. Wait, maybe the problem's statement A is \( \angle 2 \cong \angle 3 \) is incorrect, but the options for A: the first option is "Angle 2 is congruent to angle 3". Wait, maybe the reason is vertical angles, so we need \( \angle 2 \cong \angle 3 \) (even if they look adjacent, maybe the diagram is labeled differently). Alternatively, maybe \( \angle 2 \) and \( \angle 3 \) are vertical. Let's proceed.

For statement B, reason is "corr. ∠’s thm." (corresponding angles theorem). So we need two corresponding angles. Since \( p \parallel q \) and \( r \) is transversal, corresponding angles are congruent. So \( \angle 3 \) (on \( p \), bottom left) and \( \angle 7 \) (on \( q \), bottom left) are corresponding angles, so \( \angle 3 \cong \angle 7 \). So statement B should be \( \angle 3 \cong \angle 7 \).

Now, the transitive property: if \( \angle 2 \cong \angle 3 \) (statement A) and \( \angle 3 \cong \angle 7 \) (statement B), then \( \angle 2 \cong \angle 7 \) (statement 4), which matches the transitive property.

So:

- A: Angle 2 is congruent to angle 3 (vertical angles? Maybe the diagram labels \( \angle 2 \) and \( \angle 3 \) as vertical, perhaps I mislabeled)
- B: Angle 3 is congruent to angle 7 (corresponding angles, since \( p \parallel q \))

Answer:

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