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1 SSS SIMILARITY AA SIMILARITY SAS SIMILARITY PARALLEL TO SIDE OF TRIANGLE PARALLEL TRANSVERSALS ARE POLYGONS SIMILAR? PROBLEM 1 PROBLEM 2 PROBLEM 3 PROBLEM 4 PROBLEM 5 STANDARDS 4 and 5 END SHOW JOINING MIDPOINTS IN A TRIANGLE PROBLEM 6 PROPORTIONS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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2 Standard 4: Students prove basic theorems involving congruence and similarity. Los estudiantes prueban teoremas básicos que involucran congruencia y semejanza. Standard 5: Students prove triangles are congruent or similar and are able to use the concept of corresponding parts of congruent triangles. Los estudiantes prueban que triángulos son congruentes o semejantes y son capaces de usar el concepto de partes correspondientes de triángulos congruentes. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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3 STANDARD 1.3 In a classroom there are 3 boys and 6 girls. What is the ratio of boys to girls? 3 to 6 or 3:63: = =0.5 (0.5) (100%) = 50% PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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4 D C = A B How do you express the ratio of A to B? A to B A : B A B How do you express the ratio of C to D? C to D C : D C D Now when we equal to ratios, we get a PROPORTION: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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5 A D A and D are the EXTREMES B C B and C are the MEANS Cross-multiplying: (A)(D)=(C)(B)The product of the MEANS is equal to the product of the EXTREMES = A D B C = A D B C = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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6 A B C K L M KL AB MK CA LM BC If A K and B L and C M then STANDARDS 4 and 5 Triangles are SIMILAR when the corresponding sides are proportional: SSS similarity CABMKL Both triangles are similar ( ) KL AB MK CA LM BC OR PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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7 4 6 X+6 2X+3 A S B R C T The triangles below are similar, find CA=? and TS=? STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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8 4 6 X+6 2X+3 4 BC 6 RT X+6 CA 2X+3 TS STANDARDS 4 and 5 4(2X+3) = 6(X+6) 8X +12 = 6X A S B R C T 8X = 6X X 2X = 24 2 X = 12 CA = X + 6 = + 6 = 18 TS = 2X + 3 = 2( ) = = 27 The triangles below are similar, find CA=? and TS=? PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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9 J R K S L T JK X+2 RS X+6 KL 10 ST 20 STANDARDS 4 and 5 What is the value of X and JL if JKL RST. JK= X+2, KL=10, RS=X+6, ST= 20, and JL = 5X + 2 = = 20(X+2) = 10(X+6) 20X +40 = 10X X = 10X X 10X = X = 2 JL = 5X + 2 = 5( ) = = 12 5X+2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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10 A B C X YZ ABCXYZ By AA similarity STANDARDS 4 and 5 and A X if C Z then PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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11 A B C X Y Z AB XY AC XZ and A X if then ABCXYZ By SAS similarity STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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12 E S F R G T U STANDARDS 4 and 5 2. A line perpendicular to one line is perpendicular to any line parallel to it. First Prove that all triangles in the figure are similar among them: 1. Two lines cut by a common perpendicular transversal are parallel. 3. Two perpendicular lines form 4 right angles. 4. Alternate interior angles are congruent. 5. Corresponding angles are congruent H If EG= 25, GF=15, EF= 20, FT = 10, UR= 3, and Given EG RT. Find RF, UF, and RS. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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13 E S F R G T U STANDARDS 4 and 5 H If EG= 25, GF=15, EF= 20, FT = 10, UR= 3, and Given EG RT. Find RF, UF, and RS. All highlighted triangles in figure are similar by AA SIMILARITY! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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14 E S F R G T U STANDARDS 4 and 5 If EG= 25, GF=15, EF= 20, FT = 10, UR= 3, and Given EG RT. Find RF, UF, and RS = UR GF RF EG = 3 15 RF 25 15RF = (3)(25) 15 15RF = 75 RF = 5 H All highlighted triangles in figure are similar by AA SIMILARITY! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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15 E S F R G T U STANDARDS 4 and 5 If EG= 25, GF=15, EF= 20, FT = 10, UR= 3, and Given EG RT. Find RF, UF, and RS. 3 = UR GF RF EG = 3 15 RF 25 15RF = (3)(25) 15 15RF = 75 RF = 5 H 5 RF = UF + UR = UF = UF UF = 16 2 UF = 4 Applying the Pythagorean Theorem PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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16 E S F R G T U STANDARDS 4 and 5 If EG= 25, GF=15, EF= 20, FT = 10, UR= 3, and Given EG RT. Find RF, UF, and RS. = UR GF RF EG = 3 15 RF 25 15RF = (3)(25) 15 15RF = 75 RF = 5 H 5 RF = UF + UR = UF = UF UF = 16 2 UF = 4 Applying the Pythagorean Theorem = RT EG RS GF = RS 15 25RS = (15)(15) 25 25RS = 225 RS = 9 15 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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17 E S F R G T U STANDARDS 4 and 5 If EG= 25, GF=15, EF= 20, FT = 10, UR= 3, and Given EG RT. Find RF, UF, and RS. = UR GF RF EG = 3 15 RF 25 15RF = (3)(25) 15 15RF = 75 RF = 5 H RF = UF + UR = UF = UF UF = 16 2 UF = 4 Applying the Pythagorean Theorem = RT EG RS GF = RS 15 25RS = (15)(15) 25 25RS = 225 RS = 9 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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18 A B C D E DB AD AE EC DE BC STANDARDS 4 and 5 then If PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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19 A B C D E DB AD AE EC DE BC STANDARDS 4 and 5 then If PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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20 STANDARDS 4 and 5 A S B C T X 18 Find the value for X = SB CS TA CT = 6 18 X 24 (24) = (6)(24) 18 X = X X = 8 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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21 A B C D EF DE EF AB BC STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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22 STANDARDS 4 and 5 Z Y 6 7 Y + 5 Z + 1 Find the values for Y and Z: = Y+5 7 Y 6 7Y = 6(Y+5) 7Y = 6Y Y Y = 30 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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23 STANDARDS 4 and 5 Z Y 6 7 Y + 5 Z + 1 Find the values for Y and Z: = Y+5 7 Y 6 7Y = 6(Y+5) 7Y = 6Y Y Y = Z Y Z+1 Y+5 35Z = 30(Z+1) 35Z = 30Z Z 5Z = 30 5 Z = 6 = Z 30 Z = Z 30 Z+1 35 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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24 K L M R S If KR RL KS SM and thenRSLM andRS=LM 1 2 STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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25 STANDARDS 4 and 5 K L M R S Find in the problem below the value for RS: then RS = (120) 1 2 RS = 60 RS = LM 1 2 If PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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X Y The two irregular polygons are similar find values for X and Y: STANDARDS 4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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27 X X Y Y 30 The two irregular polygons are similar find values for X and Y: STANDARDS 4 and 5 = = X 14.8 (30) = (15)(30) 10 Y = Y Y = X = X = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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