Exercise 9 Page 637

How should we prove congruence?

From the diagram, we see two right triangles, △ KFG and △ HGF. Also, in addition to the right angles, we know one more pair of congruent corresponding angles. If we can prove congruence between these right triangles, we should be able to prove that ∠ 1≅ ∠ 2 by the Congruent Complements theorem.

Plan for proving congruence

Our plan includes 3 general steps.

1. Show that FG and FG are congruent by the Reflexive Property of Congruence.
2. Show that △ FHG and △ GKF are congruent by the AAS Congruence Theorem.
3. Show that ∠ 1 and ∠ 2 are congruent by the Congruent Complements theorem.

Proof

Two-Column Proof

We can then prove ∠ 1 ≅ ∠ 2 using a two-column proof.

Statement	Reason
1. ∠ GKF ≅ ∠ FHG, ∠ KFG ≅ ∠ HGF	1. Given
2. FG≅ FG	2. Reflexive Property of Congruence
3. △ KFG ≅ △ HGF.	3. AAS Congruence Theorem
4. ∠ HFG ≅ ∠ KGF	4. Corresponding parts of congruent triangles are congruent
5. ∠ 1 ≅ ∠ 2	5. Congruent Complements Theorem