Question

given: in the diagram, r || s and both lines are cut by transversal t.
prove: both pairs of alternate interior angles are congruent.
match each statement to its corresponding reason. drag the items on the left to the correct location on the right.
∠3 ≅ ∠7
∠4 ≅ ∠8
∠3 ≅ ∠6
∠4 ≅ ∠5
lines r and s are parallel
and cut by transversal t.
∠7 ≅ ∠6
∠8 ≅ ∠5
given
corresponding angles theorem
vertical angles theorem
transitive property of congruence

Explanation:

Step1: Match "Lines \(r\) and \(s\) are parallel and cut by transversal \(t\)."

This is the given information in the problem statement.

Step2: Match "\(\angle3\cong\angle7\), \(\angle4\cong\angle8\)"

By the Corresponding Angles Theorem (when two parallel lines are cut by a transversal, corresponding angles are congruent).

Step3: Match "\(\angle7\cong\angle6\), \(\angle8\cong\angle5\)"

By the Vertical Angles Theorem (vertical angles are congruent).

Step4: Match "\(\angle3\cong\angle6\), \(\angle4\cong\angle5\)"

By the Transitive Property of Congruence (\(\angle3\cong\angle7\) and \(\angle7\cong\angle6\) implies \(\angle3\cong\angle6\); \(\angle4\cong\angle8\) and \(\angle8\cong\angle5\) implies \(\angle4\cong\angle5\)).

Answer:

• Lines \(r\) and \(s\) are parallel and cut by transversal \(t\): given
• \(\angle3\cong\angle7\), \(\angle4\cong\angle8\): Corresponding Angles Theorem
• \(\angle7\cong\angle6\), \(\angle8\cong\angle5\): Vertical Angles Theorem
• \(\angle3\cong\angle6\), \(\angle4\cong\angle5\): Transitive Property of Congruence