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SIMILARITIES AND CONGRUENCY Similarity ( all lengths in examples and in exercises are in cm) Two figures having the same shape are known as similar figures. The figures A and B show is a example of similar figure. So in short similarity is a relation between angles and sides. A B Two congruent figures must also be similar. When a figure is enlarged or reduced, the new figure is similar to the original one. e.g. find out the figures similar to figure A: A B C D SOLUTION: D only. Similar Triangles If two triangles are similar, then i. their corresponding angles are equal; ii. their corresponding sides are proportional. B A Y C X Z In the figure, if ABC = XYZ, then ‹A = ‹X, ‹B = ‹Y, ‹C = ‹Z and AB = BC = AC XY YZ XZ Example In the figure, ABC is similar to RPQ. Find the value of the unknowns. B 39 x A 52 z C Solution: Since ABC = RPQ, ‹B = ‹P So x = 90o. Also, AB = BC RP PQ Also, 39 = 58 y 48 so AC = BC RQ PQ z = 52 60 48 48 * 39 52 =y so Y = 36 60 * 52 48 =z z = 65 CONDITIONS FOR TRIANGLES TO BE SIMILAR A) THREE ANGLES EQUAL. If two triangles have three pairs of equal corresponding angles, then they must be similar. (Reference: AAA) Example: Show that ABC and PQR in the figures are similar. R A P 75o 35o 35o 75o B C Q Solution: In ABC and PQR as shown, ‹B = ‹Q, ‹C = ‹R, ‹A = 180o – 35o – 75o = 70o. ‹P = 180o – 35o – 75o = 70o so ‹A = ‹P therefore ABC is similar to PQR (AAA). B) THREE SIDES PROPORTIONAL if the three pairs of side of two triangles are proportional, then the two triangles must be similar. (Reference: 3 sides proportional) a b d e c f Example: Show that PQR and LMN in the figure are similar. L 3 R Q 16 2 4 8 P N 12 M In PQR and LMN as shown, LM = 8 = 4, MN = 12 = 4, LN = 16 = 4 PQ 2 QR 3 PR 4 So LM = MN = LN PQ QR PR Therefore PQR is similar to LMN (3 sides proportional) Example 2: Are the two triangles in the figure similar? Give reasons. A X 6 2 12 4 Y 4.5 Z C B 9 Solution: AB = 4 =2, AC = 12 = 2, BC = 9 = 2 XZ 2 XY 6 YZ 4.5 Yes, ABC is proportional to XZY as their 3 sides are proportional. C) TWO SIDES PROPORTIONAL AND THEIR INCLUDED ANGLE EQUAL. If two pairs of sides of two triangles are proportional and their includes angles are equal, then the two triangles are similar. (Reference: ratio of two sides, and the angle between two sides equal) p r P = q, x = y R s s q Example: Show that ABC and FED in the figure are similar. D A 4.5 E 45o 4 450 2 B 9 C F Solution: ‹B = ‹E‹‹‹ AB = 4 = 2, BC = 9 = 2 EF 2 DE 4.5 Therefore, AB = BC EF DE Therefore, ABC is similar to sides equal) FED (ratio of two sides, the angle between two are Example 2: Are the two triangles ZYX and CBA similar? Give reasons. Solution ‹ZYX = 180o – 78o – 40o = 62o, ‹ZYX = ‹CBA = 62o. BC = 6 = 2, AB = 4 = 2 YZ 3 XY 2 Yes, ZYX is similar to equal) CBA (ratio of two sides, the angle between two are sides Similar polygons Two polygons are said to be similar if there is correspondence between their vertices such that: a. Corresponding angles are congruent b. Lengths of corresponding sides are in proportion. A D V Y W B X C Angle A is similar to angle V Angle B is similar to angel W Angle C is similar to angle X Angel D is similar to angel Y AB = BC = CD = DA VW WX XY VY If the above interpretation is accepted the polygons are similar. Perimeter of similar polygons: If two polygons are similar, the ratio of their perimeter equals to the ratio of the lengths of corresponding sides. So for the above diagram the perimeter can be found like this. P = AB P2 VW AREA A B W Z a C Ka b D X Kb Y The two figures are similar the ratio of corresponding sides as k. Area of ABCD = ab Area of WXYZ = Ka * Kb Therefore K2ab = K2 ab This a general rule for finding area for similar figures. Exercise for similarities. Find the sides marked with letters in the questions. A 1) 5 D E a 4 C b 6 F B 2) 3) 16 y x 6 3 6 L 4) A m a M 15 N 6 m B 9 5) X 4cm 9cm2 8cm 8cm 1.5cm 6) Given triangle XYZ with line VW parallel to it find rations of : Y a) YW to WZ 7 b) ZY to WY V W c) ZY to ZW 4 X Z CONGRUENCE Two figures having the same shape and size are called congruent figures. Example of congruent shapes: C If two figures are congruent they will fit exactly on each other e.g. find out the congruent figures among the following: . D B C A E F G G In the above diagram only figure D, E and A, G are similar. Congruent Triangles when two triangles are congruent, all their corresponding sides and corresponding angles are equal. C E.g. A X B Z In the figure, if ‹A = ‹X ‹B = ‹Y ‹C = ‹Z ABC is congruent to XYZ then, AB = XY and BC = YZ CA = ZX e.g. name the pair of congruent triangles in the figure. B 12 13 P 5 R C 12 13 5 A Q So automatically ‹C =‹P From the figure we see that e.g. 2 : given that ABC is congruent to RPQ. Y ABC is congruent to XYZ find the unknowns p, q and r. q r 50o Z 6 X Solution For two congruent triangles, for their corresponding sides and angles are equal. Therefore p = 6, q = 5 and r = 50o. CONDITINS FOR TRIANGLES TO BE CONGRUENT A) THREE SIDES EQUAL. If AB = XY, BC = YZ and CA = ZX then the triangles are congruent. Y A X B C Z Reference: SSS. It means when three sides of one triangle is equal to the three sides of the other triangle, the two triangles are congruent. E.g. 5 Determine which of the pairs of triangles in the following triangles. 8 3 4 5 7 (i) (ii) 7 5 4 5 3 8 (iv) (iii) Solution: (i) (ii) and (iv) are a pair of congruent triangles, and (iii) are a pair of congruent triangles. Example 2 Each of these pairs of triangles are congruent. Which of them are congruent because of SSS? Solution B B) TWO SIDES EQUAL AND THEIR INCLUDED ANGLE EQUAL If AB = XY, ‹B = ‹Y and BC = YC then B ABC is congruent to XYZ Y A C X Z Reference: SAS. This means that if two sides of one triangle are equal to two sides of the other triangle and the angle between this pair of sides is the same in both triangles, then the two triangles are congruent. Example 1) Determine which of the following pairs are congruent (i) (ii) (iii) 7 6 50o 5 50o 6 3 7 1200 (iv) 3 5 120o Solution In the figure because of SAS, (i) and (iii) are congruent triangles (ii) and (iv) are congruent triangles example 2 in each of the following, equal sides and equal angles are indicated with the same markings. Write down a pair of congruent triangles, and give reasons. A) A B D A C B C Solution D E ABC is congruent to ACB iss congruent to CDA (SSS) ECD (SAS) C) TWO ANGLES AND ONE SIDE EQUAL. If ‹A = ‹X, AB = XY and ‹B= ‹Y, then ABC is congruent to XYZ Reference: ASA. It means that if two angles of one triangle are equal to two corresponding angles of another triangles and the side between each pair of angles is the same length in both triangles, then the two triangles are congruent. And even if ‹A= ‹X, ‹B = ‹Y and BC = YZ then, ABC is congruent to XYZ. Reference: AAS Determine which pairs of triangles in the following are congruent: (i) (ii) (iii) 30o 30o 7 (iv) 1350 70o 7 8 45o 8 1350 70 0 45o Solution In this figure because of ASA (i) and (iv) are congruent triangles. (ii) and (iii) are congruent triangles. Example 2 In the figure, equal angles are indicated with the same markings. Write down a pair of congruent triangles, and give reasons. Solution ABD is congruent to ACD. Example 2 Determine which pairs of triangles in the following are congruent (i) (ii) (iii) (iv) 45o 4 3 4 o 45 3 60o 100o 100o 55o Solution In the figure, because AAS (i) and (ii) are a pair of congruent triangles (iii) and (iv) are another pair of congruent triangles. 60o 55o Example 2 In the figure, equal angles are indicated with the same marking. Write down a pair of congruent triangles, and give reasons. Solution ABD is congruent to CBD (AAS) D) TWO RIGHT ANGLED TRIANGLES WITH EQUAL HYPOTHESIS AND ANOTHER PAIR OF EQUAL SIZES. A C X B Z ‹C = ‹Z = 90o, AB = XY and BC = YZ, then Y ACB is congruent to XZY. Reference: RHS Example 1: In the figure, ‹DAB and ‹BCD are both right angles and AD = BC. Judge whether ABD and CBD are congruent, and give reasons for your answer. A B Solution Yes, ABD is congruent to CBD (RHS) Congruency questions 1)Say whether the triangles are congruent or not. If they are give a reason. a) 3cm 85o b) 450 3cm 45o 85o 5cm 500 3cm 50o 3cm 5cm c) 3cm d) 3cm 80o SIMILAR 3D SHAPES o 60similar Volume of 3d objects. 7cm 3 cm o 40 When objects are similar, and is an enlargement of another. If the two are similar and the 3. 3cm ratio of corresponding side is k, the ratio 4cm 4cmof volume will be k e.g. 6 V=70cm3 8 v= X 7cm 70 = 6 V2 8 3 70 = 216 V2 512 V2= 512 * 70 216 V2 = 166 cm3 Exercise 1) Find out the missing volume V 200 5 10 Answers Similarities 1) a = 2.5, b = 3 2) c = 16 3) x = 12, y = 8 4) m = 10, a = 6.6. 5) x = 64cm2 6) a) 7 4 b) 11 7 C) 11 4 Congruency 1) a as AA corresponding S c as SAS d as RHS 3D shapes 1) 62.5 ASSESSMENT - SIMILARITY AND CONGRUENCY Writing Drawin Formatti Polygo skill g skill ng skill n Triangle 3 2 3 2 2 creativit picture page always- theore y flow s paragrap similar ms neatnes labelin hs math polygo applns s g equation ns s appln 1 0 1 1 2 3D shapes 3 perimet er vol, SA 2 Question s 3 3D shapes congruen cy variety 2 Exampl es 2 ex+ans 1 20 10