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.

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

Proof

Two-Column Proof

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

Statement	Reason
1. &∠ CEB ≅ ∠ AEB &AF ≅ CD & AC ⊥ BE & ∠ FAB is a right angle & ∠ EBC is a right angle & ∠ DCB is a right angle	1. Given
2. BE≅ BE	2. Reflexive Property of Congruence
3. ∠ EBA is a right angle	3. Definition of perpendicular lines
4. ∠ EBA ≅ ∠ EBC	4. Right Angles Congruence Theorem
5. △ ABE ≅ △ CBE	5. ASA Congruence Theorem
6. ∠ EAB ≅ ∠ ECB	6. Corresponding parts of congruent triangles are congruent.
7. ∠ EAF ≅ ∠ ECD	7. Congruence Complements Theorem
8. △ AEF ≅ △ CED	8. SAS Congruence Theorem
9. ∠ 1 ≅ ∠ 2	9. Corresponding parts of congruent triangles are congruent.

Hints

Check the answer

Practice exercises

Asses your skill

How should we prove congruence?

Plan for proving congruence

Proof