Question

lines c and d are parallel lines cut by transversal p. which must be true by the corresponding angles theorem? ∠1 ≅ ∠7, ∠2 ≅ ∠6, ∠3 ≅ ∠5, ∠5 ≅ ∠7

Explanation:

Step1: Recall Corresponding Angles Theorem

The Corresponding Angles Theorem states that if two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. Corresponding angles are in the same relative position at each intersection where a straight line crosses two others.

Step2: Identify Corresponding Angles

• For \( \angle 1 \) and \( \angle 7 \): \( \angle 1 \) is at the top - left of the intersection with line \( c \), \( \angle 7 \) is at the bottom - right of the intersection with line \( d \). They are not corresponding angles.
• For \( \angle 2 \) and \( \angle 6 \): \( \angle 2 \) is at the top - right of the intersection with line \( c \), \( \angle 6 \) is at the top - right of the intersection with line \( d \). These are corresponding angles. By the Corresponding Angles Theorem, if \( c\parallel d \) and \( p \) is a transversal, \( \angle 2\cong\angle 6 \).
• For \( \angle 3 \) and \( \angle 5 \): \( \angle 3 \) is at the bottom - right of the intersection with line \( c \), \( \angle 5 \) is at the bottom - left of the intersection with line \( d \). They are not corresponding angles (they are alternate interior angles, but we are checking for corresponding angles here).
• For \( \angle 5 \) and \( \angle 7 \): \( \angle 5 \) and \( \angle 7 \) are vertical angles (they are formed by the intersection of line \( p \) and line \( d \) and are opposite each other), not corresponding angles.

Answer:

\( \angle 2 \cong \angle 6 \) (Option: \( \boldsymbol{\angle 2 \cong \angle 6} \))