Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
7-3 Triangle Similarity I CAN -Use the triangle similarity theorems to determine if two triangles are similar. -Use proportions in similar triangles to solve for missing sides. Recall in 7-2, to prove that two polygons are similar you had to: Show all corresponding angles are congruent AND Show all corresponding sides are proportional Triangle Similarity Theorems are “shortcuts” for showing two triangles are similar. Example D 9 B E 12 10 C A 18 6 5 F Similarity Statement EFD ABC ~ Reason by AA Because A E and C D justification Example Explain why the triangles are similar and write a similarity statement. By the Triangle Sum Theorem, mC = 47°, so C F. B E by the Right Angle Congruence Theorem. Therefore, ∆ABC ~ ∆DEF by AA ~. Example Verify that the triangles are similar. ∆PQR and ∆STU Therefore ∆PQR ~ ∆STU by SSS ~. Example Verify that the triangles are similar. ∆DEF and ∆HJK D H by the Definition of Congruent Angles. Therefore ∆DEF ~ ∆HJK by SAS ~. Example Verify that ∆TXU ~ ∆VXW. TXU VXW by the Vertical Angles Theorem. Therefore ∆TXU ~ ∆VXW by SAS ~. Your Turn Your Turn Your Turn Find the value of x such that ∆ACE ~ ∆BCD C Why is ∆ACE ~ ∆BCD? C 3 12 B 3 D B x 28 A 3 x+3 = 12 C 12 28 12(x + 3) = 84 12x + 36 = 84 – 36 – 36 12x = 48 x=4 E D x+3 A E 28 Example Explain why ∆ABE ~ ∆ACD, and then find CD. Step 1 Prove triangles are similar. A A by Reflexive Property of , and B C since they are both right angles. Therefore ∆ABE ~ ∆ACD by AA ~. Step 2 Find CD. x(9) = 5(3 + 9) 9x = 60 Example Explain why ∆RSV ~ ∆RTU and then find RT. Step 1 Prove triangles are similar. It is given that S T. R R by Reflexive Property of . Therefore ∆RSV ~ ∆RTU by AA ~. Step 2 Find RT. RT(8) = 10(12) 8RT = 120 RT = 15 Your Turn Properties of Similar Triangles 7-4 I can use the triangle proportionality theorem and its converse. I can set up and solve problems using properties of similar triangles. Example: AF = 3, FC = 5, AE = 2. Find BE 2 3 x 5 3x 10 10 x 3 OR 2 x 3 5 3x 10 10 x 3 Example Find US. It is given that , so by the Triangle Proportionality Theorem. US(10) = 56 Example Find PN. 2PN = 15 PN = 7.5 Your Turn Solve for x. Example Solve for x. Example Example I Solve for x, y, and w. 4 x N y w E 4 x 2 3 2x 12 5 L A 3 2 D IN EI AD LS 12 S x6 Example I Solve for x, y, and w. 4 x=6 N y w AN EL DA LS E L A 3 2 D OR y 5 2 3 3 y 10 5 S 12 y 5 4 6 OR 4 6 y 5 10 y 3 Example I Solve for x, y, and w. 4 x=6 N 10 y 3 w E 6 w 14 12 5 L A 3 2 D IE NE IS DS 12 14w 72 S w 5.14 Example BD = 8; DF = 6; CE = 16. EG = ________ Example BD = 2x – 2; DF = 4; CE = x + 2; EG = 8 Find BD and CE Your Turn Example: SHOW EF BC if BE = 21, AE = 42, CF = 15, and AF = 30 21 ? 15 42 30 1 1 2 2 So, EF BC Example Verify that Since . , by the Converse of the Triangle Proportionality Theorem. Your Turn AC = 36 cm, and BC = 27 cm. Verify that Since . , by the Converse of the Triangle Proportionality Theorem. The previous theorems and corollary lead to the following conclusion. Example Find PS and SR. by the ∆ Bisector Theorem. 40(x – 2) = 32(x + 5) PS = x – 2 = 30 – 2 = 28 40x – 80 = 32x + 160 40x – 80 = 32x + 160 8x = 240 x = 30 SR = x + 5 = 30 + 5 = 35 Example Find AC and DC. by the ∆ Bisector Theorem. 4y = 4.5y – 9 –0.5y = –9 y = 18 So DC = 9 and AC = 16. Your Turn