: If the triangles are similar (~), then all of the sides must be congruent proportional (create equal scale fractions). Example: A~ F Before you start your proof, it is important to plan! Setup the three fractions: Opp. A & D (skip A & D). Opp. B & C (skip B & C). 1st 2nd : A F A F EF A F AC DF Opp. C & F (skip C & F). A F Multiplication Property of Equality: The resulting step after a number, variable or object is multiplied to both sides of the equal sign. EF = AC DF A~ F EF = AC DF = Example: : R~ K, = 6, RT =, = & = 16. Prove: JK = 40 Plan first! 1st 2nd : R R~ K, R~ K, K = 6, RT =, = 6, RT =, = & = 16 = & = 16 R & H S & J T & K RT JK HK Substitute: =, RT=, =6, & =16 JK want HK don t want 6 16 have both Use what you want & what you have: JK = JK = JK = 6 16 240 = 6(JK) 40 = JK JK = 40 Div. Prop. = Sym. Prop. = or JK = 3 JK = JK = 6 16 0 = 3(JK) 40 = JK JK = 40 Div. Prop. = Sym. Prop. = Example: : WYZ~ M, WY = 3, WZ = 5, LM = 20, & KM = 25 Prove: = 1st 2nd : WYZ M WYZ~ M, WYZ~ M, W & K Y & L Z & M WY = 3, WZ = 5, WY = 3, WZ = 5, YZ WZ WY LM = 20, & KM = 25 LM = 20, & KM = 25 LM KM WZ Substitute: KM = WY WZ KM = WY LM=20, WZ=5, WY=3, 5 & KM=25 25 = 3 5 25 = 3 YZ 5 3 or 1 5() = 75 20 25 5 = 3 want both Use what you want & what you have: WZ KM = WY don t have want = Div. Prop. = =
1. : A~ FGH, =, =, AC = 9 & FG = Prove: = 6 2. : PQR~ G, PQ = 25, QR = 20, GH = 20, & GJ = Prove: = 16 3. : V~ J, = 55, = 33, SV = 44, & JL = 32 Prove: = 24 Page 2 of 6
4. : Q~ Z, QS =, ZA = 26, = 39, & ZB = 52 Prove: = 6 5. : F~ NPQ, = 45, DF = 36, NP = 50 & PQ = 20 Prove: = 40 6. : D~ V, CD = 6, =, = & SV = 16 Prove: = 9 Page 3 of 6
7. : A~ PQR, = 6, AC =, PQ =, & PR = 1 Prove: = 10. : VWY~ V, VW = 20, WY = 16, = 2, & SV = 42 Prove: = 24 1. Answers when NOT Simplifying the Fractions 2. A~ FGH, =, =, AC = 9 & FG = FG = AC = 9 () = 72 = 6 Div. Prop. = PQR~ G, PQ = 25, QR = 20, GH = 20 & GJ = PQ GH = QR 25 20 = 20 25() = 400 = 16 Div. Prop. = AC = FG 9 = 72 = () 6 = Div. Prop. = = 6 Symm. Prop. = QR = PQ GH 20 = 25 20 400 = 25() 16 = Div. Prop. = = 16 Symm. Prop. = Page 4 of 6
3. V~ J, = 55, = 33, SV = 44 & JL = 32 SV JL = 44 32 = 33 44() = 1056 = 24 Div. Prop. = 4. Q~ Z, QS =, ZA = 26, = 39 & ZB = 52 = QS ZB 39 = 52 52() = 3 = 6 Div. Prop. = = SV JL 33 = 44 32 1056 = 44() 24 = Div. Prop. = = 24 Symm. Prop. = QS ZB = 52 = 39 3 = 52() 6 = Div. Prop. = = 6 Symm. Prop. = 5. F~ NPQ, = 45, DF = 36, NP = 50 & PQ = 20 NP = DF 45 50 = 36 45() = 100 = 40 Div. Prop. = 6. D~ V, CD = 6, =, = & SV = 16 = CD = 6 () = 72 = 9 Div. Prop. = DF = NP 36 = 45 50 100 = 45() 40 = Div. Prop. = = 40 Symm. Prop. = CD = 6 = 72 = () 9 = Div. Prop. = = 9 Symm. Prop. = 7. A~ PQR, = 6, AC =, PQ = & PR = 1 PQ = AC PR = 1 1() = 10 = 10 Div. Prop. =. VWY~ V, VW = 20, WY = 16, = 2 & SV = 42 SV = WY 42 = 16 2 2() = 672 = 24 Div. Prop. = AC PR = PQ 1 = 10 = 1() 10 = Div. Prop. = = 10 Symm. Prop. = WY = SV 16 2 = 42 672 = 2() 24 = Div. Prop. = = 24 Symm. Prop. = Page 5 of 6
Answers when Simplifying the Fractions (without symmetric) 1. A~ FGH, =, =, AC = 9 & FG = FG = AC = 9 3 2 = 9 3() = 1 = 6 Div. Prop. = 2. PQR~ G, PQ = 25, QR = 20, GH = 20 & GJ = PQ GH = QR 25 20 = 20 5 4 = 20 5() = 0 = 16 Div. Prop. = 3. V~ J, = 55, = 33, SV = 44 & JL = 32 SV JL = 44 32 = 33 11 = 33 11() = 264 = 24 Div. Prop. = 4. Q~ Z, QS =, ZA = 26, = 39 & ZB = 52 = QS ZB 39 = 52 39 = 2 13 13() = 7 = 6 Div. Prop. = 5. F~ NPQ, = 45, DF = 36, NP = 50 & PQ = 20 NP = DF 45 50 = 36 9 10 = 36 9() = 360 = 40 Div. Prop. = 6. D~ V, CD = 6, =, = & SV = 16 = CD = 6 = 3 4 4() = 36 = 9 Div. Prop. = 7. A~ PQR, = 6, AC =, PQ = & PR = 1 PQ = AC PR = 1 = 2 3 3() = 30 = 10 Div. Prop. =. VWY~ V, VW = 20, WY = 16, = 2 & SV = 42 SV = WY 42 = 16 2 42 = 4 7 7() = 16 = 24 Div. Prop. = Page 6 of 6