Chapter 8 Similarity Section 8.5 Proving Triangles are Similar U SING S IMILARITY T HEOREMS U SING S IMILAR T RIANGLES IN R EAL L IFE

Postulate A C B D F E  A  D and  C  F    ABC ~  DEF 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

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.

U SING S IMILARITY T HEOREMS Compare Side Lengths of  LKM and  NOP Ratios Different, triangles not similar Determine if the triangles are similar

U SING S IMILARITY T HEOREMS Compare Side Lengths of  LKM and  NOP Ratios Same, triangles are similar  RQS ~  LKM Determine if the triangles are similar

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

U SING S IMILARITY T HEOREMS  CED 44° 68° 20

U SING S IMILARITY T HEOREMS StatementsReasons

U SING S IMILARITY T HEOREMS StatementsReasons ~

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.

Finding Distance Indirectly The Tree is 72 feet tall

Finding Distance Indirectly The Tree is 72 feet tall 4 x 72 The mirror would need to be placed 36 feet from the tree

HW #16 Pg Odd, 29, 31-37, 39-47