Question

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

Explanation:

Step1: Recall corresponding angles theorem

Corresponding angles are in the same relative position. When two parallel lines are cut by a transversal, corresponding angles are congruent.

Step2: Analyze each option

• For $\angle2\cong\angle6$: $\angle2$ and $\angle6$ are not in corresponding positions.
• For $\angle3\cong\angle5$: $\angle3$ and $\angle5$ are alternate - interior angles (not corresponding).
• For $\angle1\cong\angle7$: $\angle1$ and $\angle7$ are not in corresponding positions.
• For $\angle5\cong\angle7$: $\angle5$ and $\angle7$ are not in the correct relationship. Wait, no. Wait, actually, when lines \(c\) and \(d\) are parallel and cut by transversal \(p\), \(\angle3\) and \(\angle5\) are alternate - interior angles. \(\angle1\) and \(\angle5\) are corresponding angles (by the corresponding angles theorem). Wait, no, re - checking. The corresponding angles: \(\angle1\) and \(\angle5\) (top - left of each intersection), \(\angle2\) and \(\angle6\) (top - right), \(\angle3\) and \(\angle7\) (bottom - right), \(\angle4\) and \(\angle8\) (bottom - left). So \(\angle3\cong\angle7\) (corresponding angles). There is a mistake in the initial analysis. Let's start over.

Corresponding angles are pairs of angles that are in the same relative position. For parallel lines \(c\) and \(d\) cut by transversal \(p\):

• \(\angle1\) and \(\angle5\): top - left at each intersection (corresponding).
• \(\angle2\) and \(\angle6\): top - right (corresponding).
• \(\angle3\) and \(\angle7\): bottom - right (corresponding).
• \(\angle4\) and \(\angle8\): bottom - left (corresponding).

Answer:

\(\angle3\cong\angle7\) (Assuming there was a mis - label in the options and if we consider the correct corresponding angle pair from the given options' structure. If the options are as written \(\angle3\cong\angle5\) is wrong (alternate - interior), \(\angle2\cong\angle6\) is wrong (they are actually alternate - exterior if lines \(c\) and \(d\) are parallel. Wait, no, for parallel lines \(c\) and \(d\) cut by transversal \(p\), \(\angle2\) and \(\angle6\) are alternate - exterior angles (\(\angle2\) is outside the two parallel lines on one side of the transversal, \(\angle6\) is outside on the other side). \(\angle1\) and \(\angle5\) (corresponding), \(\angle3\) and \(\angle7\) (corresponding). If the options are \(\angle3\cong\angle5\) (alternate - interior), \(\angle2\cong\angle6\) (alternate - exterior), \(\angle1\cong\angle7\) (no), \(\angle5\cong\angle7\) (no). But if we assume a typo and the intended option was \(\angle3\cong\angle7\) (which is correct by corresponding angles theorem)