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Similar is a mathematical word meaning the same shape. We say that two triangles , triangle FDE and triangle LMK, are similar if the ratio of each side is similar. FD : DE : EF = LM : MK : KL This equation of two three term ratios can also be written in fraction form: FD DE EF = = LM MK KL NOTE: All sides of one triangle must be either all in the numerator or denominator. Corresponding sides are equal In similar triangles, corresponding angles are equal. F = L D=M E=K IMPORTANT: If you know two triangles are similar, then their corresponding angles are equal. Conversely, if two triangles have equal corresponding angles, then the triangles are similar. Prove the two triangles are similar? SOLUTION: Angles: A = D B=E C=F Sides: AB BC AC = = DE EF DF 8 6 = 12 9 = 5 7.5 540 540 540 = = 810 810 810 Since corresponding angles are equal, then corresponding sides are equal. Therefore the two triangles are similar. Prove the two triangles are similar? SOLUTION: Angles: Sides: Prove the two triangles are similar? SOLUTION: Angles: M = O L=P N=Q Sides: ML MN LN = = OP OQ PQ 4 6 3 = = 12 18 9 12 12 = 36 36 = Since corresponding angles are equal, then corresponding sides are equal. Therefore the two triangles are similar. 12 36 Find the measures of A, B, C. SOLUTION: Sides: Since the ratios of the sides are all equal, the triangles are similar, that also means the corresponding angles are equal. FH FG GH = = AC AB BC 3.5 3 = 7 6 = 5.3 10.6 1 2 = 1 2 = 1 2 Therefore: F = A A = 100o H=C C = 38o G=B B = 42o Find the measures of M, L, K. SOLUTION: Sides: Find the measures of M, L, K. SOLUTION: Sides: Since the ratios of the sides are all equal, the triangles are similar, that also means the corresponding angles are equal. ML MK LK = = AB AC BC 16 20 18 = = 4 5 4.5 4 Therefore: M = A = 4 = 4 M = 105o L=B L = 17o K=C K = 58o Triangle ABC is similar to Triangle RPQ. Find the lengths of RP and AC SOLUTION: Sides: AB BC AC = = RP PQ RQ 8 x 8 x = 12 16.8 = 12 y = 16.8 21 12 y = 16.8 21 12x = 8(16.8) 16.8y = 12(21) x= y = 252 134.4 12 RP = 11.2 16.8 AC = 15 Triangle SIP is similar to Triangle MAT. Find the lengths of MA and SP SOLUTION: Sides: Triangle SIP is similar to Triangle MAT. Find the lengths of MA and SP SOLUTION: Sides: SI SP IP = = MA MT AT 7 x 7 x = 13 25 = y 13 = 17.3 25 13 25 = y 17.3 13x = 7(25) 25y = 13(17.3) x= y = 224.9 175 13 MA = 13.5 25 SP = 9 For the figure below, show that Triangle DEF is similar to Triangle DGH. SOLUTION: If two triangles have equal corresponding angles, then the triangles are similar. Therefore: D = D ( common angle) E = G (Corresponding angles) F = H ( corresponding angles ) Therefore: Triangle DEF is similar to Triangle DGH. Class work • Check solutions to Lesson 18. • Copy down notes and examples • Do Lesson 18(2) worksheet.