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.

Figure 12.11∠R and ∠V are right angles, and ∠RST ~= ∠VST.

• 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.

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

Figure 12.12→CB bisects ∠ACD and ¯BC ⊥ ¯AD.

• 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.