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 RST ~= VST, then you could use CPOCTAC to show that ¯RT ~= ¯TV. To show that RST ~= VST, you simply use the AAS Theorem.

	Statements	Reasons
1.	R and V are right angles, and RST ~= VST	Given
2.	RST ~= VST	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 ABC ~= DBC. 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.	mABC = 90º and mDBC = 90º	Definition of right angle
4.	mABC = mDBC	Substitution
5.	ABC ~= DBC	Definition of
6.	ACB ~= DCB	Definition of angle bisector
7.	¯BC ~= ¯BC	Reflexive property of ~=
8.	ABC ~= DBC	ASA Postulate
9.	A ~= D	CPOCTAC