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SIMILAR TRIANGLES Congruency Two shapes are congruent if one of the shapes fits exactly on top of the other shape. These three triangles are all congruent In congruent shapes: • corresponding angles are equal • corresponding lengths are equal To prove that two triangles are congruent you must show that they satisfy one of the following four sets of conditions: SSS: three sides are equal SAS: two sides and the included angle are the same ASA: two angles and the included side are the same RHS: right-angled triangle with hypotenuse and one other side the same Which triangles are congruent to triangle A? A Similar shapes R These two quadrilaterals are similar. S PQ QR RS SP AB BC CD DA C D In similar shapes: • corresponding angles are equal A • corresponding sides are in the same ratio To show that two triangles are similar it is sufficient to show that just one of the above conditions is satisfied. B P Q Which triangles are similar to triangle A? A Examples 1 The triangles are similar. Find the values of x and y. (All lengths are in cm.) x 5.5 y 9 8 12 Using ratio of corresponding sides: x 12 5.5 8 y 8 9 12 8x 66 12y 72 x 8.25 y6 Examples 2 The triangles are similar. Find the values of x and y. (All lengths are in cm.) 7 Turn one of the triangles so that you can see which are the corresponding sides. x 8 7 Using ratio of corresponding sides: y 15 12 x 8 15 12 12x 120 x 10 y 12 7 8 8y 84 y 10.5 Examples 3 Find the values of x and y. (All lengths are in cm.) x 6 Separate the two triangles. 9 2.5 2 Using ratio of corresponding sides: y y 8 9 6 x 2.5 8 x 6 6y 72 6(x 2.5) 8x y 12 6x 15 8x x +2.5 8 y 2x 15 x 7.5 x 6 9 6 Examples 4 Find the values of x and y. (All lengths are in cm.) 4 Turn the top triangle so that you can see which are the corresponding sides. Using ratio of corresponding sides: x 9 4 6 y 6 6 9 6x 36 9y 36 x6 y4 y x 6 9 y 4 6 x 6 9 x Examples 5 Find the values of x and y. (All lengths are in cm.) 6 Turn the top triangle so that you can see which are the corresponding sides. Using ratio of corresponding sides: x 12 20 16 y 16 6 12 16x 240 12y 96 x 15 y 8 12 y 16 20 6 12 x y 16 20