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SIMILARITYSIMILARITY Similarity means same shape but different size, Δ ABC Δ DEF Δ GHI Similarity means same shape but different size, Δ ABC Δ DEF Δ GHI A C BDE F GH I

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Computing Ratios

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Using Proportions An equation that equates two ratios is a proportion a = c b d This proportion is read as a is to b as c is to d The numbers represented by a and d are known as the extremes The numbers represented by b and c are known as the means Properties of Proportions 1.The product of the extremes equals the product of the means, ad = bc 2.If 2 ratios are equal, then their reciprocals are also equal An equation that equates two ratios is a proportion a = c b d This proportion is read as a is to b as c is to d The numbers represented by a and d are known as the extremes The numbers represented by b and c are known as the means Properties of Proportions 1.The product of the extremes equals the product of the means, ad = bc 2.If 2 ratios are equal, then their reciprocals are also equal a = c b d b = d a c Reciprocal Property, where each fraction is reversed Cross Product Property, multiply opposite numerators x denominators

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Solving Proportions, using cross product property

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Interchange Means Property of Proportions Interchange Means Property, where the Means switch places Ifa=cThena=b bdcd a=cIf and only ifa + b=c + d bdbd

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Solving a Proportion Problem Trip from point A to point B is 2000 miles. With a full tank of gas you go 380 miles where you fill the tank again with 14 gallons at $ 1.25 per gallon. How much should you plan to spend for fuel for the rest of the trip? Solution: Set up a proportion by taking the ratio of total fuel cost to amount already spent equal to the ratio of total miles to the number of miles already traveled. Verbal model Fuel cost for 2000 miles=2000 miles Fuel cost for 380 miles380 miles Labels Fuel cost for 2000 miles=x Fuel cost for 380 miles=(14) (1.25) Equation x=2000 (14) (1.25)380 Simplify x=100 17.519 Multiply both sides by 17.5 x (17.5)=100 (17.5) 17.519 Simplifyx=92.11 Total – spent = balance$ 92 – $ 17.50=$ 74.50

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Solving a Proportion Problem The gear ratio of two gears is the ratio of the number of teeth of the larger gear to the number of teeth of the smaller gear. For the gears shown here, the ratio of gear A to gear B is equal to the ratio of gear B to gear C. Gear A has 18 teeth and gear C has 8 teeth. How many teeth does gear B have? a b

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Identifying Similar Polygons Definition of Similar ( ) Polygons Two polygons are similar if their corresponding angles are congruent and the lengths of corresponding sides are proportional. The symbol is read is similar to A D CB F E 3 5 4 610 8 AB = 5 = 1 DE102 BC = 4 = 1 EF82 CA = 3 = 1 FD62 Triangles are similar because angles are congruent and corresponding sides are proportional Scale Factor The common ratio of ½ or 1:2 is called the Scale Factor or Δ ABC to Δ DEF

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Identifying Similar Polygons If Δ ABC Δ DEF, then the following statements are true: < A < D, < B < E and < C < F and F E D A B C AB = BC = CA DEAFFD Angle-Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Angle-Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

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SSS Similarity Theorem: If corresponding sides of two triangles are proportional, then the two triangles are similar C A B F E D I H G 16 12 8 12 9 6 14 10 6 AB = 8 = 4 DE63 CA = 16 = 4 FD123 BC = 12 = 4 EF93 Ratios of side lengths of Δ ABC and Δ DEF Ratios of side lengths of Δ ABC and Δ GHI AB = 8 = 4 GH63 CA = 16 = 8 IG147 BC = 12 = 6 HI105 Because the ratios are equal, the triangles are similar Because the ratios are not equal, the triangles are not similar

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SAS Similarity Theorem: If an angle of 1 triangle is congruent to an angle of a 2 nd triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. S R P Q T Given: PQ is a midsegment of Δ RST Prove: Δ RST Δ PSQ 1. P is midpoint of RS Q is midpoint of ST 1. Definition of Midpoint 2.PS = ½ RS or PS/RS = ½ SQ = ½ ST or SQ/ST = ½ 2. Implied 3. < S < S 3. Reflexive 4. Δ RST Δ PSQ 4. SAS Similarity

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Proportions in Similar Triangles Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Triangle Proportionality Converse Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. A C D B E 1 3 2 4 5 > > P R Q T S 6 4 2 > > Finding the Length of a Segment In the figure to the left, QP TS. You are given that QT = 4, TR = 2 and PS = 6. What is the length of SR ? TR = SR Triangle Proportionality Theorem QTPS 2 = SR Substitution 46 3=SRmultiply 6 to both sides

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Using Proportionality Theorems 3 Parallel lines Proportionality Theorem If three parallel lines intersect two transversals, then they divide the transversals proportionally Bisect Angle Proportionality Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. 1 3 2 U P R Q T S Using 3 Parallel Lines Proportionality Theorems In the figure to the left < 1 < 2 < 3. You are given that PQ = 9, QR = 15, and ST =11. What is the length of TU ? PQ = ST 3 Proportionality Theorem QRTU 9 = 11 Substitution 15TU 9 (TU)=15(11)Cross Product Property TU=18 1/3divide both sides by 9

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