U SING S IMILARITY T HEOREMS THEOREM S THEOREM 8.2 Side-Side-Side (SSS) Similarity Theorem If the corresponding sides of two triangles are proportional, then the triangles are similar. If = = A B PQ BC QR CA RP then  ABC ~  PQR. A BC P Q R

U SING S IMILARITY T HEOREMS THEOREM S THEOREM 8.3 Side-Angle-Side (SAS) Similarity Theorem If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. then  XYZ ~  MNP. ZX PM XY MN If XM and= X ZY M PN

Proof of Theorem 8.2 GIVEN PROVE = ST MN RS LM TR NL  RST ~  LMN S OLUTION Paragraph Proof M NL RT S PQ Locate P on RS so that PS = LM. Draw PQ so that PQ RT. Then  RST ~  PSQ, by the AA Similarity Postulate, and. = ST SQ RS PS TR QP Use the definition of congruent triangles and the AA Similarity Postulate to conclude that  RST ~  LMN. Because PS = LM, you can substitute in the given proportion and find that SQ = MN and QP = NL. By the SSS Congruence Theorem, it follows that  PSQ   LMN.

E FD A C B GJ H Using the SSS Similarity Theorem Which of the following three triangles are similar? S OLUTION To decide which of the triangles are similar, consider the ratios of the lengths of corresponding sides. Ratios of Side Lengths of  ABC and  DEF = =, 6 4 AB DE Shortest sides = =, 12 8 CA FD Longest sides = BC EF Remaining sides Because all of the ratios are equal,  ABC ~  DEF

= =, CA JG Longest sides E FD Using the SSS Similarity Theorem A C B GJ H Which of the following three triangles are similar? S OLUTION To decide which of the triangles are similar, consider the ratios of the lengths of corresponding sides. Ratios of Side Lengths of  ABC and  GHJ = = 1, AB GH Shortest sides = 9 10 BC HJ Remaining sides Because all of the ratios are not equal,  ABC and  DEF are not similar. E FD A C B GJ H Since  ABC is similar to  DEF and  ABC is not similar to  GHJ,  DEF is not similar to  GHJ.

Using the SAS Similarity Theorem Use the given lengths to prove that  RST ~  PSQ. S OLUTION PROVE  RST ~  PSQ GIVEN SP = 4, PR = 12, SQ = 5, QT = 15 Paragraph Proof Use the SAS Similarity Theorem. Find the ratios of the lengths of the corresponding sides. = = = = 4 SR SP 16 4 SP + PR SP = = = = 4 ST SQ 20 5 SQ + QT SQ Because S is the included angle in both triangles, use the SAS Similarity Theorem to conclude that  RST ~  PSQ. The side lengths SR and ST are proportional to the corresponding side lengths of  PSQ PQ S RT

U SING S IMILAR T RIANGLES IN R EAL L IFE S CALE D RAWING As you move the tracing pin of a pantograph along a figure, the pencil attached to the far end draws an enlargement. Using a Pantograph P R T S Q

U SING S IMILAR T RIANGLES IN R EAL L IFE Using a Pantograph As the pantograph expands and contracts, the three brads and the tracing pin always form the vertices of a parallelogram. P R T S Q

U SING S IMILAR T RIANGLES IN R EAL L IFE Using a Pantograph The ratio of PR to PT is always equal to the ratio of PQ to PS. Also, the suction cup, the tracing pin, and the pencil remain collinear. P R T S Q

You know that. Because P  P, you can apply the SAS Similarity Theorem to conclude that  PRQ ~  PTS. = PQ PS PR PT Using a Pantograph S R Q T P How can you show that  PRQ ~  PTS ? S OLUTION

Because the triangles are similar, you can set up a proportion to find the length of the cat in the enlarged drawing. Using a Pantograph 10 " 2.4 " 10 " S R Q T P = RQ TS PR PT Write proportion. = TS = 4.8 Substitute. Solve for TS. So, the length of the cat in the enlarged drawing is 4.8 inches. 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 TS in the enlargement.

Finding Distance Indirectly Similar triangles can be used to find distances that are difficult to measure directly. R OCK C LIMBING You are at an indoor climbing wall. To estimate the height of the wall, you place a mirror on the floor 85 feet from the base of the wall. Then you walk backward until you can see the top of the wall centered in the mirror. You are 6.5 feet from the mirror and your eyes are 5 feet above the ground. 85 ft6.5 ft 5 ft A B C E D Use similar triangles to estimate the height of the wall. Not drawn to scale

Finding Distance Indirectly 85 ft6.5 ft 5 ft A B C E D Use similar triangles to estimate the height of the wall. S OLUTION Using the fact that  ABC and  EDC are right triangles, you can apply the AA Similarity Postulate to conclude that these two triangles are similar. Due to the reflective property of mirrors, you can reason that ACB  ECD.

85 ft6.5 ft 5 ft A B C E D  DE Finding Distance Indirectly Use similar triangles to estimate the height of the wall. S OLUTION = EC AC DE BA Ratios of lengths of corresponding sides are equal. Substitute. Multiply each side by 5 and simplify. DE 5 = So, the height of the wall is about 65 feet.