Exercise 11 Page 637

How should we prove congruence?

From the diagram, we are given a pair of congruent corresponding sides and a pair of congruent corresponding angles in △ STR and △ QTP. With another congruent angle, we can show congruence of △ PTS and △ RTQ where ∠ 1 and ∠ 2 happen to be corresponding angles.

Plan for proving congruence

Our plan includes 3 steps.

1. Show that △ STR and △ QTP are congruent by the ASA Congruence Theorem.
2. Show that △ STP and △ QTR are congruent by the SAS Congruence Theorem.
3. Show that ∠ 1 and ∠ 2 are corresponding angles in △ STP and △ QTR.

Proof

Two-Column Proof

Finally, we can prove ∠ 1 ≅ ∠ 2 using a two-column proof.

Statement	Reason
1. ∠ RST ≅ ∠ PQT, ST ≅ TQ	1. Given
2. & ∠ PTQ and ∠ RTS are vertical angles & ∠ PTS and ∠ RTQ are vertical angles	2. Definition of vertical angles
3. & ∠ PTQ ≅ ∠ RTS & ∠ PTS ≅ ∠ RTQ	3. Vertical Angles Congruence Theorem
4. △ STR ≅ △ QTP	4. ASA Congruence Theorem
5. PT ≅ RT	5. Corresponding parts of congruent triangles are congruent.
6. △ STP ≅ △ QTR	6. SAS Congruence Theorem
7. ∠ 1 ≅ ∠ 2	7. Corresponding parts of congruent triangles are congruent.