Chapter 8 Similarity

Section 5 Proving Triangles are Similar

How many attending retreat tomorrow? Look to the class blackboard for solutions to today’s exit ticket. It will help you with the quiz.

Example 2: Proving that Two Triangles are Similar Color variations in the tourmaline crystal shown lie along the sides of isosceles triangles. In the triangles each vertex angle measures 52°. Explain why the triangles are similar. The vertex angle measures 52 ° and each triangle is isosceles. Recall the base angle of an isosceles triangle are congruent. If the angles at the vertex of each isosceles triangle are congruent, then the triangles are similar. In this case 180 – 52 = 128. Remaining angles are 128/2 = 64 °.

Example 5: Using Scale Factors Find the length of the altitude QS. ∆𝑵𝑷𝑸 ~ ∆ 𝑻𝑸𝑷 ∠ RQT ≅∠PQN - vertical angles and ∠NPQ ≅ ∠TQR - opposite angles (MS is perpendicular to Both NP and TR so the segments are parallel) By AA ∆𝑵𝑷𝑸 ~ ∆ 𝑻𝑸𝑷 The triangles are similar so the ratio of MQ:QS is proportional to the ration NP:TR. 𝟏𝟐+𝟏𝟐 𝟖+𝟖 = 𝟐𝟒 𝟏𝟔 = 𝟑 𝟐 therefore 𝑴𝑸 𝑻𝑸 = 𝟑 𝟐 𝟔 𝑻𝑸 = 𝟑 𝟐 means TQ = 4

GOAL 1: Using Similarity Theorems

Example 1: Proof of Theorem 8.2 . PQ Divides RST into two similar triangles. PSQ is congruent to LMN. So LMN ~ RST

Example 2: Using the SSS Similarity Theorem Which of the following three triangles are similar? ABC and FED 12/8 = 9/6 = 6/4 = 3/2

Example 3: Using the SAS Similarity Theorem Use the given lengths to prove that ∆𝑅𝑆𝑇 ~ ∆𝑃𝑆𝑄. SAS: angle S and RS proportional to PS, TS to QS 16/4 = 20/5 = 4

GOAL 2: Using Similar Triangles in Real Life Example 4: Using a Pantograph

Example 4: Using a Pantograph (continued) a) How can you show that ∆𝑃𝑅𝑄 ~ ∆𝑃𝑇𝑆? b) In the diagram, PR is 10 inches and RT is 10 inches. The length of the cat, RQ, in the original print is 2.4 inches. Find the length of TS in the enlargement.

https://youtu.be/zbjwY1Zz4jo How? https://youtu.be/zbjwY1Zz4jo

Example 5: Finding Distance Indirectly

Example 6: Finding Distance Indirectly RST ~ RPQ by AA so ST/PQ = TR/QR QR is distance across river 9/63 = 12/ QR 9QR = 12 x 63 QR = 84 You WILL see this type of problem again

. ∆ABC has sides of lengths 4, 2, and 7. The ratios of the lengths of ∆DEF are 6:3:21. Are the triangles similar? In other words, does 4/6 = 2/3 = 7/21? You WILL see this type of problem again

EXIT SLIP Now you try with exit slip, Quiz tomorrow