Exercise 12 Page 637

How should we prove congruence?

Examining the diagram, we see that △ ABE and △ CBE have two congruent corresponding angles, ∠ CEB and ∠ AEB. Additionally we see that BE⊥ AC which means ∠ EBA is a right angle as well. Since the triangles share BE as a side, we have enough information to prove by the ASA Congruence Theorem that

△ ABE≅ △ CBE.
As ∠ EAB and ∠ ECB are congruent corresponding angles in △ ABE and △ CBE, we can show by the Congruence Complements Theorem that ∠ EAF≅ ∠ ECD. This gives us enough information to show

△ AEF ≅ △ CED

where ∠ 1 and ∠ 2 happens to be corresponding angles.

Plan for proving congruence

Our plan includes 4 general steps.

1. Show that △ ABE and △ CBE are congruent by the ASA Congruence Theorem.
2. Show that ∠ EAF and ∠ ECD are congruent by the Congruent Complements Theorem.
3. Show that △ AEF and △ CED are congruent by the SAS Congruence Theorem.
4. Show that ∠ 1 and ∠ 2 are corresponding angles in △ AEF and △ CED.