Section 8.5 Proving Triangles are Similar Chapter 8 Similarity Section 8.5 Proving Triangles are Similar USING SIMILARITY THEOREMS USING SIMILAR TRIANGLES IN REAL LIFE

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

USING SIMILARITY THEOREMS THEOREM 8.2 Side-Side-Side (SSS) Similarity Theorem If the corresponding sides of two triangles are proportional, then the triangles are similar. P Q R A B C If = = A B PQ BC QR CA RP then ABC ~ PQR.

Locate P on RS so that PS = LM. Proof of Theorem 8.2 M N L R T S GIVEN PROVE = = ST MN RS LM TR NL  RST ~  LMN P Q SOLUTION Paragraph Proof 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 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. Use the definition of congruent triangles and the AA Similarity Postulate to conclude that  RST ~  LMN.

Determine if the triangles are similar USING SIMILARITY THEOREMS Determine if the triangles are similar Compare Side Lengths of LKM and NOP Ratios Different, triangles not similar

Determine if the triangles are similar USING SIMILARITY THEOREMS Determine if the triangles are similar Compare Side Lengths of LKM and NOP Ratios Same, triangles are similar RQS ~ LKM

USING SIMILARITY THEOREMS THEOREM 8.3 Side-Angle-Side (SAS) Similarity Theorem X Z Y M P N 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. ZX PM XY MN If X M and = then XYZ ~ MNP.

USING SIMILARITY THEOREMS CED 44° 68° 20

USING SIMILARITY THEOREMS Statements Reasons

USING SIMILARITY THEOREMS Statements Reasons ~

Use similar triangles to estimate the height of the wall. Finding Distance Indirectly ROCK CLIMBING 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. Similar triangles can be used to find distances that are difficult to measure directly. Use similar triangles to estimate the height of the wall. 85 ft 6.5 ft 5 ft A B C E D Not drawn to scale

Use similar triangles to estimate the height of the wall. Finding Distance Indirectly Use similar triangles to estimate the height of the wall. SOLUTION Due to the reflective property of mirrors, you can reason that ACB  ECD. 85 ft 6.5 ft 5 ft A B C E D 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.

So, the height of the wall is about 65 feet. Finding Distance Indirectly Use similar triangles to estimate the height of the wall. SOLUTION = EC AC DE BA Ratios of lengths of corresponding sides are equal. 85 ft 6.5 ft 5 ft A B C E D So, the height of the wall is about 65 feet. DE 5 = 85 6.5 Substitute. Multiply each side by 5 and simplify.  DE 65.38

Finding Distance Indirectly The Tree is 72 feet tall

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

HW Pg :6;9;11;13-17;19-25;27-29;32-34;39-47