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Parallel Lines and Similar Triangles

publish date: 2026/05/14 21:40:16.296794 UTC

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In the figure below, \(\overline{PR} \parallel \overline{MN}\). Are \(\triangle PQR\) and \(\triangle NQM\) similar triangles?

Yes — by the AAA similarity theorem, since vertical angles and alternate interior angles are congruent.

No — parallel lines do not create similar triangles.

Yes — but only because the triangles share vertex Q.

Cannot be determined without knowing the side lengths.

Correct Answer

Yes — by the AAA similarity theorem, since vertical angles and alternate interior angles are congruent.

Explanation

Three pairs of congruent angles are established:
① \(\angle PQR \cong \angle NQM\) (vertical angles are congruent).
② \(\angle RPQ \cong \angle MNQ\) (alternate interior angles, \(PR \parallel MN\)).
③ \(\angle QRP \cong \angle QMN\) (alternate interior angles).
By the AAA similarity theorem: \(\triangle PQR \sim \triangle NQM\).

Reference

Mathematics for college students

Quizzes you can take where this question appears

Determine whether two triangles are similar