Given , a second triangle Given

In some instances, we wish to prove a relationship that takes us beyond the simi- larity of triangles. The following consequences of the definition of similarity are often cited as reasons in a proof. The first fact, abbreviated CSSTP, is used in Example 2. Al- though the CSSTP statement involves triangles, the corresponding sides of any two similar polygons are proportional. That is, the ratio of any pair of corresponding sides equals the ratio of another pair of corresponding sides. The second fact, abbreviated CASTC, is used in Example 4. D A E C B 1 2 Figure 5.12 D A E C B Figure 5.13 EXAMPLE 1 Provide a two-column proof of the following problem. GIVEN: in Figure 5.12 PROVE: 䉭ABC 䉭EDC AB 7 DE PROOF Statements Reasons

1. 2.

3. 4. 䉭ABC 䉭EDC ∠1 ⬵ ∠2 ∠A ⬵ ∠E AB 7 DE

1. Given 2. If two

lines are cut by a transversal, the alternate interior angles are 3. Vertical angles are 4. AA ⬵ ⬵ 7 쮿 General Rule: Although there will be three methods of proof AA, SAS~, and SSS~ for similar triangles, we use AA whenever possible. This leads to a more efficient proof. Illustration: See lines 1 and 2 in the proof of Example 2. Notice that line 3 follows by the reason AA. STRATEGY FOR PROOF 왘 Proving That Two Triangles Are Similar General Rule: First prove that triangles are similar. Then apply CSSTP. Illustration: See statements 3 and 4 of Example 2. STRATEGY FOR PROOF 왘 Proving a Proportion Corresponding sides of similar triangles are proportional. CSSTP Corresponding angles of similar triangles are congruent. CASTC Exs. 1–4 EXAMPLE 2 Complete the following two-column proof. GIVEN: in Figure 5.13 PROVE: DE BC = AE AC ∠ADE ⬵ ∠B The proof of this theorem is left to the student; see Exercise 33. Note that this proof also requires the use of CSSTP. General Rule: First prove that two triangles are similar. Then form a proportion involv- ing the lengths of corresponding sides. Finally, apply the Means-Extremes Property. Illustration: See the following proof and Example 3 an alternative form of the proof. STRATEGY FOR PROOF 왘 Proving Products of Lengths Equal The lengths of the corresponding altitudes of similar triangles have the same ratio as the lengths of any pair of corresponding sides. THEOREM 5.3.2 PROOF Statements Reasons

1. 2.

3. 4. DE BC = AE AC 䉭ADE ~ 䉭ABC ∠A ⬵ ∠A ∠ADE ⬵ ∠B 1. Given 2. Identity 3. AA 4. CSSTP 쮿 NOTE: In this proof, DE appears above BC because the sides with these names lie opposite in the two similar triangles. AE and AC are the lengths of the sides opposite the congruent and corresponding angles and . That is, corresponding sides of similar triangles always lie opposite corresponding angles. ∠B ∠ADE ∠A The paragraph style of proof is generally used in upper-level mathematics classes. These paragraph proofs are no more than modified two-column proofs. Compare the following two-column proof to the paragraph proof found in Example 3. GIVEN: in Figure 5.14 PROVE: NP QR = RP MN ∠M ⬵ ∠Q PROOF Statements Reasons

1. 2.

3. 4.

5. NP QR = RP MN NP RP = MN QR 䉭MPN 䉭QPR ∠1 ⬵ ∠2 ∠M ⬵ ∠Q 1. Given hypothesis 2. Vertical angles are 3. AA 4. CSSTP 5. Means-Extremes Property ⬵ EXAMPLE 3 Use a paragraph proof to complete this problem. GIVEN: in Figure 5.14 PROVE: NP QR = RP MN ∠M ⬵ ∠Q P M R N Q 1 2 Figure 5.14