T x Identify E the pairs of congruent corresponding angles and the corresponding sides.

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1 7.1 Similar Figures If 2 figures are similar then: (1) ORRESPONING NGLES RE (2) ORRESPONING SIES RE THE REUE RTIO OF 2 ORR. SIES IS LLE THE. IF 2 FIGURES RE SIMILR, THEN THE RTIO OF THEIR IS = TO THE. SIMILRITY STTEMENT: Ex. IG ~ MN 1.) I G 2) I IG MN Ex. 3 G 12 8 GWT ~ GIH I 14 H Scale Factor: y 10 W T x Identify E the pairs of congruent corresponding angles and the corresponding sides etermine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. If not, explain why not. 3. parallelograms EFGH and TUVW 4. E and LMN Tell whether the polygons must be similar based on the information given in the figures

2 Identifying Similar Triangles ngle-ngle Similarity Postulate (~) If two angles are congruent to two angles of another triangle, then the triangles are similar. Side-Side-Side Similarity Theorem (SSS~)If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar. Side-ngle-Side Similarity Theorem (SS~) If the measures of two pair of corresponding sides of two triangles are proportional and the included angle is congruent, then the triangles are similar. etermine whether each pair of triangles is similar. If so, state whether it is similar by ~, SSS~ or SS~. 1) 2) 3) 4) 2

3 5) 6) 7) 8) PRTIE PROLEMS SSS~) Write a similarity statement. Explain why the triangles are similar (~, SS~, T ~ SN ~ OR ~ Reason: Reason: Reason: Solve for the missing sides x = a = x = y = b = y = z = c = z = etermine whether triangles are similar, if so state the reason and give a similarity statement Hint: Find WO and FL using Pythagorean Theorem Reason: Reason: Reason: OW ~ GE ~ GOT ~ Helpful Hints for Similar Triangle Proofs: 1. Look for parallel lines 2. Once triangles are similar a. Use orresponding angles are congruent OR b. Use orresponding sides are similar OR c. Then you can use ross Product Property 3

4 Similarity Proofs: o on a separate sheet 1. Given: Prove: NM M = LM M For # 1, M N 2. Given: N // L Prove: N LM = L NM L E F 3. Given: GH GI Prove: E H EF HI E EF 4. Given: ; E H GH HI F EF Prove: GI HI For # 3, 4 E F V H G I 5. Given: Prove: 6. Given: VW VZ VX VY WZ // XY VW VZ VY VX For # 5, 6 X W 2 1 Z Y Which one(s) of the following must be true? (1) VWZ ~ VXY (2) WZ // XY (3) 1 Y X 7. What does the greek word ge stand for in the word geometry? #8 8. Given: ~ XYZ, and XW are altitudes. Prove: XW XY #9 9. Given: H EH ; G H Prove: E G = G HE E 10. Given: 1 2 G #10 Y W Z Prove: () 2 = 11. Given: EF // RS Prove: FX RS= XS EF #11 F S 1 2 #13 P 12. What does the greek word metry stand for in the word geometry? X E R Q U T 13. Given: QT // RS QU UT Prove: RV VS R V S 4

5 More Review Proofs: 14. Given: 1 2 JG GI Prove: JY YZ I 2 Y J 1 G Z JL KL 15. Given: NL ML Prove: J N J L K M N 16. Given: SR R S Prove: // R S R 7.4 Triangle Proportions Theorem: If a line is parallel to one side of a and intersects the other two sides, then it. Ex. Given: E // onclusion: nd ; E E SIE-SPLITTER THEOREM: If a line is // to one side of a and intersects the other 2 sides, then it. Ex. Given: E // E 5

6 Use diagram above to the right: Ex 1. If = 1, = 3, and E = 2.5, find E. Ex. 2 If = 6, = 10, and = 14, find E. Theorem: On any 2 transversals, parallel lines. Given: Lines l // m // k l a c onclusion: m b d k Ex. 1: If a = 5, b = 7, and d = 9, find c. Ex. 2: If a = 2, b = 3, and c + d = 20, find d. ngle isector Theorem: If a ray bisects an of a, then it divides the opposite side into segments proportional to. Given: bisects in. onclusion: Use diagram above: Ex. 1 If = 12, = 24, and = 10, find. Ex. 2: If = x, = 8, = 2 and = 7, find x. Ex. 3 Find the value of x in the diagram to the right: x 22 6

7 Find each length. 1. H 2. MV Verify that the given segments are parallel. 3. PQ and NM 4. WX and E Find each length. 5. SR and RQ 6. E and E 7. In, bisects and. Tell what kind of must be. 2. The perimeter of is 128 miles. Find X and Y. 3. Find KN and LM. 7.5 Using Proportional Relationships Refer to the figure for Exercises 1 3. city is planning an outdoor concert for an Independence ay celebration. To hold speakers and lights, a crew of technicians sets up a scaffold with two platforms by the stage. The first platform is 8 feet 2 inches off the ground. The second platform is 7 feet 6 inches above the first platform. The shadow of the first platform stretches 6 feet 3 inches across the ground. 1. Explain why is similar to E. (Hint: The sun s rays are parallel.) 2. Find the length of the shadow of the second platform in feet and inches to the nearest inch foot-8-inch-tall technician is standing on top of the second platform. Find the length of the shadow the scaffold and the technician cast in feet and inches to the nearest inch. 7

8 Refer to the figure for Exercises 4 6. Ramona wants to renovate the kitchen in her house. The figure shows a blueprint of the new kitchen drawn to a scale of 1 cm : 2 ft. Use a centimeter ruler and the figure to find each actual measure in feet. 4. width of the kitchen 5. length of the kitchen 6. width of the sink 7. area of the pantry Given that EFG WXYZ, find each of the following. 8. perimeter of WXYZ 9. area of WXYZ 8

9 Geometry Geometric Mean and Similarity in Right Triangles If a, b, and x are positive numbers and a x = x b, then x is called the geometric mean between a and b. Therefore x = ab. Find the geometric mean between the following numbers: 1. 2 and and and and 24 The figure at right represents a right triangle with an ltitude drawn from the right angle. This creates a figure where: 1. m is the geometric mean between and. 2. y is the geometric mean between and. 3. w is the geometric mean between and. 9

10 Refer to the diagram below to find the indicated lengths. raw a separate diagram for each problem. 1. c = 12; m = 6; a = 6. c = 12; m = 4; h = 2. c = 8; n = 6; b = 7. c = 16; m = 7; b = 3. m = 4; n = 6.25; h = 8. c = 18; n = 10; a = 4. n = 4; m = 15.75; h = 9. h = 12; m = 4; c = 5. c = 24; n = 6; h = 10. b = 8; m = 12; c = 10