Geometry: Proving Segments and Angles Are Congruent

Proving Segments and Angles Are Congruent

After you have shown that two triangles are congruent, you can use the fact that CPOCTAC to establish that two line segments (corresponding sides) or two angles (corresponding angles) are congruent.

• Example 4: If ∠R and ∠V are right angles, and ∠RST ~= ∠VST (see Figure 12.11), write a two-column proof to show ¯RT ~= ¯TV.

• Solution: You need a game plan. If you could show that ΔRST ~= ΔVST, then you could use CPOCTAC to show that ¯RT ~= ¯TV. To show that ΔRST ~= ΔVST, you simply use the AAS Theorem.

	Statements	Reasons
1.	∠R and ∠V are right angles, and ∠RST ~= ∠VST	Given
2.	ΔRST ~= ΔVST	AAS Theorem
3.	¯RT; ~= ¯TV	CPOCTAC

• Example 5: Suppose that in Figure 12.12, →CB bisects ∠ACD and ¯BC ⊥ AD. Write a two-column proof to show that ∠A ~= ∠D.

• Solution: Because ¯BC ⊥ ¯AD, you know that ∠ABC ~= ∠DBC. Because →CB bisects ∠ACD , you know that ∠ACB ~= ∠DCB. Finally, ∠BC is congruent to itself, and you can use the ASA Postulate to show that ΔABC ~= ΔDBC. By CPOCTAC, you can conclude that ∠A ~= ∠D. Let's write it up.

	Statements	Reasons
1.	→CB bisects ∠ACD and ¯BC ⊥ ¯AD	Given
2.	∠ABC and ∠DBC are right angles	Definition of ⊥
3.	m∠ABC = 90º and m∠DBC = 90º	Definition of right angle
4.	m∠ABC = m∠DBC	Substitution
5.	∠ABC ~= ∠DBC	Definition of
6.	∠ACB ~= ∠DCB	Definition of angle bisector
7.	¯BC ~= ¯BC	Reflexive property of ~=
8.	ΔABC ~= ΔDBC	ASA Postulate
9.	∠A ~= ∠D	CPOCTAC