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# Chapter 1.3

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### Chapter 1.3

1. 1. What are you going tolearn? Properties of Two Congruent Figuresto identify two congruentor incongruent planefigures by showing therequirements11..33Examine two one-thousand-rupiah notes(paper money). You will find that they are thesame in both shape and size. In other words,they are said to be congruent.to determine congruenttrianglesto prove two congruenttriangles Now look at the quadrilaterals below.to determine theproportions of the sides oftwo congruent trianglesand to find their lengthsABCPQSDRFigure 1.7to show theconsequences of twocongruent trianglesto distinguish the conceptof similarity from that ofcongruencyKey term:• congruentDiscussiona. What are the side lengths of quadrilaterals ABCD and PQRS? Measure thecorresponding sides with a ruler.b. What are the sizes of the angles of quadrilaterals ABCD and PQRS?Measure the corresponding angles with a protractor.c. Are the two figures congruent? Explain.d. What are the requirements for two polygons to be congruent? Explain.e. Find objects around you that have congruent surfaces.Using the requirements for two polygons to be congruent, find the pairs ofcongruent figures below.Mathematics for Junior High School Grade 9 / 17
2. 2. Real Life SituationNow look at the patterns of ties below.Figure 1.9 shows examples of the geometrical patterns.(a) (b)Figure 1.8Figure 1.9DA BCEFGHThe ties above have triangular patterns. The shape and the size of thetriangles on each tie are the same. Such triangles are examples of congruenttriangles. For further clarification, examine the following explanation.ActivitiesDraw two rectangles as the figures onthe left. Move rectangle ABCD to theright so that point A coincides withpoint K, B with L, C with M, and D withN. Rectangle ABCD, then, will preciselycover rectangle KLMN. In such case,ABCD is said to be congruent to KLMN.It is symbolised by ABCD ≅ KLMN.A BCDK LMNFigure 1.1018 / Student’s Book – Similarity and Congruency
3. 3. PRQ FGFigure 1.11SCopy trapezium PQRS (Figure 1.11)on a piece of paper and cut it. Move itso that P coincides with E, Q with F,R with G, and S with H. Thentrapezium PQRS covers trapeziumEFGH. In other words, PQRS iscongruent to EFGH. This issymbolised by PQRS ≅ EFGH.EHCopy ΔABC (Figure 1.12) on a pieceof paper and cut it. Move it so that Acoincides with P, B with Q, and Cwith R. In other words, ΔABC coversΔPQR or ΔABC is congruent to ΔPQR.This is symbolised by ΔABC ≅ ΔPQR.The following are examples of congruent triangles.Look at Figure 1.13. Which triangles are congruent to ΔABC? Explain whatyou are going to do to ΔABC to make it cover the triangles congruent to it.Solution:The triangles congruent to ΔABC are ΔJIHand ΔMKL.ΔABC will precisely cover ΔJIH if youmake the right movement.ΔABC will also precisely cover ΔMKL ifyou make the right movement.ACQ PRBFigure 1.12Mathematics for Junior High School Grade 9 / 19
4. 4. Real life exampleLook at the figure of a tent below.The front part of the tent has a triangularshape.APC MIs ΔACP ≅ ΔAMP? Explain.Solution:ΔACP ≅ ΔAMP, because ΔACP canprecisely cover ΔAMP by reflectingΔACP on AP .20 / Student’s Book – Similarity and Congruency
5. 5. Characteristics of Two Congruent TrianglesLook at the figure of a bridge on theleft. To make it stronger, the bridge issupported by iron bars formingtriangles. Look at ΔMPO and ΔNQK. Ifthe picture is redrawn and enlarged,ΔMPO and ΔNQK will look like thetriangles in Figure 1.15.Figure 1.14PNMO Q KFigure 1.15Move ΔMPO so that it precisely covers ΔNQK. Then the two trianglesare congruent. PO coincides with QK, PM with QN, and OM with KN. Thesides that coincide are called the corresponding sides. So, sides PO andQK are corresponding sides, PM and QN are corresponding sides, and OMand KN are corresponding sides.Two congruent triangles have thecorresponding sides of the same lengthTwo congruenttrianglesMathematics for Junior High School Grade 9 / 21
6. 6. Congruency of figures can be determined from the size of correspondingangles. Look at the following congruent triangles.CBARPQSince ΔABC ≅ ΔPQR, both triangles can cover one to each other.Consequently, point A coincides with P, point B with Q, and point C with R,and ∠CAB = ∠RPQ, ∠ABC = ∠PQR, and ∠ACB = ∠PRQ. Therefore, ∠CAB and∠RPQ are corresponding angles, ∠ABC and ∠PQR are corresponding angles,and ∠ACB and ∠PRQ are corresponding angles.Two congruent triangles have thecorresponding angles of the same sizeTwo congruenttrianglesEXAMPLE 2ΔUVW and ΔDEF below are congruent. Find the sides of thesame length and the angles of the same size.Solution:Since ΔUVW and ΔDEF are congruent, thecorresponding sides are the same length, orUV = DE, UW = DF dan VW = EF.Furthermore, the corresponding angles are thesame size, or∠U = ∠D, ∠V= ∠E, and ∠W = ∠F.DEFUVWFigure 1.1622 / Student’s Book – Similarity and Congruency
7. 7. Requirements and Consequences of CongruencyLook at the following figures.AB = PQ, AC = PR dan BC = QR.Move ΔABC so that point A willcoincide with P, point B with Q,and point C with R, so that ΔABCwill precisely cover ΔPQR.Therefore, ΔABC ≅ ΔPQR.Two triangles will be congruent ifthe three corresponding sides arethe same length.What are the consequences if two triangles are congruent according to(s, s, s)? Look at the two figures below.Two triangles are congruent if the three sides ofthe first triangle are the same length as thecorresponding sides of the second triangle (s, s, s)Requirements fortwo congruenttrianglesABPQC RFigure 1.17RC**ABPQFigure 1.18Mathematics for Junior High School Grade 9 / 23
8. 8. The figures show that ΔABC and ΔPQR have two sides of the samelength, and the angles between the sides are the same size. ThenAB = PQ, AC = PR, and ∠A = ∠PSuppose we move ΔABC such that point A coincide with P.Because ∠A = ∠P, then ∠A coincides with ∠P. Because AC = PR, pointC coincides with R, and because AB = PQ, point B coincides with Q.ΔABC will then precisely cover ΔPQR. Therefore, ΔABC and ΔPQR arecongruent. Two triangles will be congruent if the two sides are the samelength and the angles between the sides are the same size.Two triangles will be congruent if the two sidesof the first triangle are the same size as thecorresponding sides of the second triangle, and theangles between the sides are the same size (s, a, s).Requirements fortwo congruenttrianglesWhat are the consequences of the two triangles being congruentaccording to (s, a, s)?We have identified two requirements for two congruent triangles. Forthe third requirement, we are going to look at two triangles that have onecorresponding side of the same length and the two corresponding angleson the corresponding sides are the same size.Look at Figure 1.19 below.ABCxPQRxyyFigure 1.1924 / Student’s Book – Similarity and Congruency
9. 9. The figure shows that∠A = ∠P, AB = PQ, and ∠B = ∠Q.AB is the side where ∠A and ∠B are located.PQ is the side where ∠P and ∠Q are located.Since the sum of the three angles in a triangle is 180°,∠A + ∠B + ∠C = 180° and∠P + ∠Q + ∠R = 180°Therefore,∠C = 180° - ∠A – ∠B, and∠R = 180° - ∠P – ∠Q.Because ∠A = ∠P and ∠B = ∠Q, ∠R = 180° - ∠A - ∠BSo, ∠C = ∠RFinally, we have the following relationship∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R.Therefore, the three corresponding angles in the two triangles are thesame size.By definition, the two triangles are similar. Since the two triangles aresimilar, the proportions of the corresponding sides are the same, namelyPQAB =QRBC =PRACSince AB = PQ (given), thenPQAB =QRBC =PRAC = 1Accordingly, AB = PQ, BC = QR and AC = PR. This means that the threecorresponding sides in the two triangles are the same length. Byrequirement (s, s, s) that we have learned, then ΔABC ≅ ΔPQR. What arethe consequences?Mathematics for Junior High School Grade 9 / 25
10. 10. Two triangles will be congruent if two angles inthe first triangle are the same size as thecorresponding angles in the second triangle,and the common leg of the two angles are thesame length (a, s, a).Requirementsfor twocongruenttrianglesInvestigate whether ΔRQT and ΔSQT in Figure1.20 are congruent. Look closely at ΔRQT andΔSQT. What are the consequences?AnswerThe figure shows that RT = ST, RQ = SQ, andTQ = TQ, or the three corresponding sides inthe two triangles are the same length. By therequirement (s, s, s), ΔRQT ≅ ΔSQT.Consequently, ∠R = ∠ S, ∠RTQ = ∠STQ, and∠TQR = ∠TQS.∴Investigate whether ΔDAC and ΔBAC arecongruent. What are the consequences?AnswerLook closely at ΔDAC and ΔBAC. It showsDA = BA, ∠DAC = ∠BAC, and AC = AC. Bythe requirement (s,a,s), ΔDAC ≅ ΔBAC.Consequently, CD = BC, ∠ADC = ∠ABC, and∠DCA = ∠BCAT6 m 6 mQR S2 m 2 mFigure 1.20AFigure 1.21BDC3 cm3 cmO O26 / Student’s Book – Similarity and Congruency
11. 11. Figure 1.22 shows that ∠A = ∠M and ∠B =∠L, such that ΔABC ≅ ΔMLK.KReason:Because ∠A = ∠M, AB = ML, and ∠B = ∠L,by the requirement (a, s, a), ΔABC ≅ ΔMLK.As a result,∠B = ∠K, BC = KL, and AC = KM.BCALM4 cm4 cmFigure 1.22Given parallelogram ERIT on the right.Show that TP = RO.Solution:ERTIPOTo show that TP = RO, complete the blankspaces on the left column.Statements ReasonsLook at ΔTIE and ΔREI1. IT = ER, ET = IR, EI = IE 1. Given2 a. ΔTIE ≅ Δ . . .b. ∠TIE = ∠ … and∠TEI = ∠ …2 a. (s,s,s)b. corresponding anglesNow look at ΔTPE and ΔROI.3. ∠ TPE =∠ … 3. both having a size of 9004. ∠TEP = ∠ … 4. according to 2b5. ∠PTE = 900 - ∠TEP 5. the sum of three angles in a trianglebeing 18006. ∠ORI = 900 - ∠ … 6. the sum of three angles in a trianglebeing 18007. ∠PTE = ∠ORI 7. according to 5 dan 6∠TEP = ∠RIO, ET = RI, and ∠PTE = ∠ORI. By the requirement (…, … , …),ΔTDE ≅ ΔROI. Because TP and RO are corresponding sides, then TP = RO.Mathematics for Junior High School Grade 9 / 27
12. 12. Similarity and CongruencyLook at the following two equilateral triangles.CA BRP QFigure 1.23a. Are ABC andΔΔ PQR similar?Explain.b. Are ABC andΔΔ PQR congruent?Explain.c. Are two similartriangles alwayscongruent? Explain.Look at the following two trianglesACQRBPa. Are ΔABC and Δ PQRsimilar? Explain.b. Are ΔABC and Δ PQRcongruent? Explain.c. Are two congruenttriangles always similar?Explain.Figure 1.24If two triangles are congruent, then they are similar.If two triangles are similar, they are not necessarily congruent28 / Student’s Book – Similarity and Congruency
13. 13. Find the pairs of congruent triangles and the pairs of similar triangles in Figure1.25 below.1 2 345678910Figure 1.251. Measure the following figures then determine whether the triangles in eachpair are congruent. If they are, give your reasons and find thecorresponding sides and the corresponding angles.a. b.CABOMRKWVUTL2. Are the triangles in each pair below congruent? If they are, give yourreasons. What are the consequences?a. A G b. 3T 77N 5 3 5RMMathematics for Junior High School Grade 9 / 29
14. 14. c. A d.GECG EAC BD3. Explain why ΔBDF ≅ ΔMKH, then find the values of m, s, and n.KMst872OnO32OHDB9t872OnOmOF4. Are ΔFKL and ΔKFG congruent? Give your reasons. If they are, find thecorresponding sides and the corresponding angles.F GLK5. PQRS is a kite. Show ΔPQR and ΔPSR are congruent triangles. Find thesides of the same length and the angles of the same size.PSRQ30 / Student’s Book – Similarity and Congruency
15. 15. For questions 6-13, use requirements (s,s,s), (s,a,s) or (a,s,a) to verify eachstatement.6. AB = CB 7. ∠OME = ∠EROBM OA CE RD8. ∠TSP = ∠TOP 9. KP =LMS O K LTP P MY10. ∠ORE = ∠OPE 11. CT = RPOR P C RENT PMathematics for Junior High School Grade 9 / 31
16. 16. 12. If line l is perpendicular to AB and CA = CB, show that PA=PB.lA BPC13. Suppose ABCD is a parallelogram. Show that ΔABC ≅ ΔCDA.D CAB32 / Student’s Book – Similarity and Congruency