Exercise 10 Page 368

Identify the pair of vertical angles. Use the Alternate Interior Angles Theorem to determine the second pair of congruent angles.

Statements	Reasons
1. MS ∥ RQ	1. Given
2. ∠ PMS ≅ ∠ PRQ	2. Alternate Interior Angles Theorem
3. ∠ SPM ≅ ∠ QPR	3. Vertical Angles Theorem
4. MS ≅ RQ	4. Given
5. △ MSP ≅ △ RQP	5. Angle-Angle-Side (AAS) Congruence Postulate

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We begin by noticing that ∠ SPM and ∠ QPR are vertical angles, and so by the Vertical Angles Theorem we conclude that ∠ SPM ≅ ∠ QPR.

Since MS ∥ RQ and MR is a transversal, by the Alternate Interior Angles Theorem we get that ∠ PMS ≅ ∠ PRQ.

Remember that we also have MS ≅ RQ, and so by applying the Angle-Angle-Side (AAS) Congruence Postulate we get △ MSP ≅ △ RQP. We summarize this proof in the following two-column table.

Statements	Reasons
1. MS ∥ RQ	1. Given
2. ∠ PMS ≅ ∠ PRQ	2. Alternate Interior Angles Theorem
3. ∠ SPM ≅ ∠ QPR	3. Vertical Angles Theorem
4. MS ≅ RQ	4. Given
5. △ MSP ≅ △ RQP	5. Angle-Angle-Side (AAS) Congruence Postulate

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