7 CONGRUENCE OF TRIANGLES

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Cameron Davis
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1 7 CONGRUENCE OF TRIANGLES Exercise 7.1 Q.1. Complete the following statements : (a) Two line segments are congruent if. (b) Among two congruent angles, one has a measure 70 ; the measure of the other angle is. (c) When we write A = B, we actually mean. Ans. (a) they have same length (b) 70 (c) measure of A = measure of B, i.e., m A = m B Q.2. Give any two real-life examples for congruent shapes. Ans. (i) Pages of this book are congruent. (ii) Coins of same denomination are congruent. Q.3. If ΔABC ΔFED under the correspondence ABC FED, write all the corresponding congruent parts of the triangles. Ans. We have, ΔABC ΔFED, also there is correspondence ABC FED It means that vertices of ΔABC are in correspondence to the ΔFED, i.e., A F, B E, C D and sides AB FE, BC ED, AC FD Congruent Parts (Angles) : A = F, B = E, C = D Congruent Parts (Sides) : AB = FE, BC = ED, AC = FD 1

2 Q.4. If ΔDEF ΔBCA, write the part (s) of ΔBCA that correspond to (i) E (ii) EF (iii) F (iv) DF Ans. (i) (ii) (iii) (iv) E = C EF = CA F = A DF = BA Exercise 7.2 Q.1. Which congruence criterion do you use in the following? (a) Given : AC = DF AB = DE So, (b) Given : So, BC = EF ΔABC ΔDEF ZX = RP RQ = ZY PRQ = XZY ΔPQR ΔXYZ (c) Given : MLN = FGH NML = GFH ML = GF So, (d) Given : ΔLMN ΔGFH EB = DB 2

3 AE = BC So, A = C = 90 ΔABE ΔCDB Ans. (a) In ΔABC and ΔDEF, we have AC = DF AB = DE BC = EF Hence, ΔABC ΔDEF (By SSS Congruence criterion) (b) In ΔPQR and ΔXYZ, we have ZX = RP RQ = ZY PRQ = XZY Hence, ΔPQR ΔXYZ (By SAS Congruence criterion) (c) In ΔLMN and ΔGFH, we have MLN = FGH NML = GFH ML = GF Hence, ΔLMN = ΔGFH (By ASA Congruence criterion) (d) In ΔABE and ΔCDB, we have EB = DB AE = BC A = C = 90 Hence, ΔABE ΔCDB (By RHS Congruence criterion) Q.2. You want to show that ΔART ΔPEN, (a) If you have to use SSS criterion, then 3

4 you need to show (i) AR = (ii) RT = (iii) AT = (b) If it is given that T = N and you are to use SAS criterion, you need to have (i) RT = and (ii) PN = (c) If it is given that AT = PN and you are to use ASA criterion, you need to have (i)? (ii)? Ans. (a) In ΔART and ΔPEN, we have (i) (ii) (iii) AR = PE RT = EN AT = PN Hence, ΔART ΔPEN (By SSS Congruence criterion) (b) In ΔART and ΔPEN, we have (i) (ii) So, T = N RT = EN AT = PN PN = AT Hence, ΔART ΔPEN (By SAS Congruence criterion) (c) In ΔART and ΔPEN, we have (i) (ii) AT = PN RAT = EPN A = P RTA = ENP T = N Hence, ΔART ΔPEN (By ASA Congruence criterion) Q.3. You have to show that ΔAMP ΔAMQ. In the following proof, supply the missing reasons. 4

5 Ans. In ΔAMP ΔAMQ Steps Reasons (i) PM = QM (ii) PMA = QMA (iii) AM = AM (Common) (iv) ΔAMP = ΔAMQ (SAS Congruence criterion) Q.4. In ΔABC, A = 30, B = 40 and C = 110 In ΔPQR, P = 30, Q = 40 and R = 110 A student says that ΔABC ΔPQR by AAA congruence criterion. Is he justified? Why or why not? Ans. No, he is not justified because AAA is not a criterion for congruence of triangles. In such a correspondence, one of them can be an enlarge copy of the other. (They would be congruent only if they are exact copies of one another. Q.5. In the figure, the two triangles are congruent. The corresponding parts are marked. We can write ΔRAT? Ans. In ΔRAT and ΔWON, we have RA = WO 5

6 Hence, AT = ON A = O ΔRAT ΔWON (By SAS Congruence criterion) Q.6. Complete the congruence statement : Ans. In ΔBCA and ΔBTA Hence, In ΔQRS and ΔQPT Hence, BA = BA (Common) BT = BC AT = AC ΔBCA ΔBTA(By SSS Congruence criterion) QT = QS QR = PT RQS = PTQ ΔQRS QPT (By SAS Congruence criterion) Q.7. In a squared sheet, draw two triangles of equal areas such that (i) the triangles are congruent. (ii) the triangles are not congruent. What can you say about their perimeters? Ans. (i) 6

7 There are two congruent triangles with equal areas, whose perimeters are equal. Actually, the congruent triangles are exact copies of each other. (ii) There are two triangles with equal areas, but not congruent, whose perimeters are either equal or not. Q.8. Draw a rough sketch of two triangles such that they have five pairs of congruent parts but still the triangles are not congruent. Ans. (i) In ΔABC and ΔPQR, AB = PQ = 5 cm AC = PR = 5 cm A = R = 45 B = P = 65 C = Q = 70 and BC QR Clearly, in two triangles they have five pairs of Congruent parts are equal but they are not Congruent. Q.9. If ΔABC and ΔPQR are to be congruent, name one additional pair of corresponding parts. What criterion did you use? Ans. (i) In ΔABC and ΔPQR B = Q = 90 C = R BC = QR 7

8 Hence, ΔABC ΔPQR (By ASA Congruence criterion) Q.10. Explain, why ΔABC ΔFED. Ans. In ΔABC and ΔFED, we have ABC = FED = 90 BAC = DFE BC = ED Hence, ΔABC ΔFED (By ASA Congruence criterion) 8

9 10 PRACTICAL GEOMETRY Exercise 10.1 Q.1. Draw a line, say AB, take a point C outside it. Through C, draw a line parallel to AB using ruler and compasses only. (1) Take a line AB and a point C outside AB. (2) Take any point D on line AB and join D to C. (3) With D as centre and a convenient radius, draw an arc cutting AB at E and DC at F. (4) Now with C as centre and the same radius as in step 3, draw an arc GH cutting CD at K. (5) Place the pointed tip of the compares at E and adjust the opening so that the pencil tip is at F. (6) With the same opening as in step 5, and with K as centre, draw an arc, cutting the are GH at L. (7) Now, join CL to draw a line PQ. Note, that 1 and 2 are alternate interior angles. So, AB PQ. 1

10 Q.2. Draw a line l. Draw a perpendicular to l at any point on l. On this perpendicular choose a point X, 4 cm away from l. Through X, draw a line m parallel to l. (1) Draw a line l. and take a point A on it. (2) At A, draw, AB perpendicular to l. (3) From AB, cut off AX = 4 cm. (4) At X, draw a line m which is perpendicular to AB, m is the required line. Q.3. Let l be a line and P be a point not on l. Through P draw a line m parallel to l. Now join P to any point Q on l. Choose any other point R on m. Through R, draw a line parallel to PQ. Let this meet l at S. What shape do the two sets of parallel lines enclose? (1) Draw a line l and take a point P outside l. (2) Through P, draw a line m, which is parallel to l. (3) Take any point or on l and and join PQ. (4) Take any point R on m. (5) Through R, draw a line RS PQ, which meets l at 5. The two sets of parallel lines enclose a parallelogram P or SR. 2

11 Exercise 10.2 Q.1. Construct ΔXYZ in which XY = 4.5 cm, YZ = 5 cm and ZX = 6 cm. (1) Draw a line segment ZX of length 6 cm. (2) With Z as a centre, draw an arc of radius 5.0 cm. (3) With X as a centre, draw another arc of radius 4.5 cm intersecting the arc draw in step 2 at Y. (4) Join ZY and YX to obtain the desired triangle. Q.2. Construct an equilateral triangle of side 5.5 cm. Ans. Steps of construction : (1) Draw a line segment YZ of length 5.5 cm. (2) With Y as centre, draw an arc of radius 5.5 cm. (3) With Z as a centre, draw another arc of radius 5.5 cm which cuts the previous arc at the point X. Join ZX and YX to (4) Obtain the required ΔXYZ. 3

12 Q.3. Draw ΔPQR with PQ = 4 cm, QR = 3.5 cm and PR = 4 cm. What type of triangle is this? (1) Draw a line segment PQ of length 4 cm. (2) With P as centre and radius 4 cm, draw an arc. (3) With Q as centre and radius 3.5 cm draw another arc, intersecting the previous arc at R. (4) Join PR and QR. (5) ΔPQR is the required triangle. ΔPQR is an isosceles triangle in which PQ = PR = 4 cm. Q.4. Construct ΔABC such that AB = 2.5 cm, BC = 6 cm and AC = 6.5 cm. Measure B. (1) Draw a line segment BC of length 6 cm. (2) With C as centre, draw an arc of radius 6.5 cm. (3) With B as centre, draw another arc of radius 2.5 cm which intersects the previous are ata. (4) Join AB and AC. On measuring B, we find that it is equal to 90. Exercise

13 Q.1. Construct ΔDEF such that DE = 5 cm., DF = 3 cm and m EDF = 90 (1) Draw a line segment DE = 5 cm. (2) At D, draw EDX = 90 (3) From DX, cut off DF = 3 cm. (4) Join E and F. ΔDEF is the required triangle. Q.2. Construct an isosceles triangle in which the length of each of its equal sides is 6.5 cm and the angle between them is 110. (1) Draw a line segment BC of length 6.5 cm. (2) At B, draw, a ray BX so that CBX = 110. (3) From BX, cut of BA = 6.5 cm. (4) Join AC to obtain the required triangle ABC. Q.3. Construct ΔABC with BC = 7.5 cm, AC = 5 cm and m C = 60. (1) Draw a line segment BC = 7.5 cm. (2) At C, draw a ray CX which makes an angle of 60 to BC. (3) From CX, cut off CA = 5 cm. (4) Join AB to obtain the required ΔABC. Exercise

14 Q.1. Construct ΔABC, given m A = 60, m B = 30 and AB = 5.8 cm. (1) Draw a line segment AB = 5 cm. (2) At B, draw ray BX which makes an angle of 30 to AB. (3) At A, draw ray AY which makes an angle of 60 to AB. (4) Both rays intersect at point C. ΔABC is the required triangle. Q.2. Construct ΔPQR if PQ = 5 cm, m PQR = 105 and m QRP = 40. Ans. In ΔPQR, QPR = 180 ( PQR + QRP) = 180 ( ) = = 35 Steps of construction : (1) Draw a line segment PQ = 5 cm. (2) Draw a ray PM which makes an angle of 35 to PQ. (3) Draw ray QN which makes an angle of 105 to PQ. (4) Both rays intersect at R. ΔPQR is the required triangle. Q.3. Examine whether you can construct ΔDEF such that EF = 7.2 cm, m E = 110 and m F = 80. Justify your answer. 6

15 Ans. In ΔDEF, EF = 7.2 cm and E + F + D = 180 But, D 180 or, D 180 Which is not possible as the sum of three angles of a triangle is 180. So this triangle cannot be constructed. Exercise 10.5 Q.1. Construct the right-angled ΔPQR, where m Q = 90, QR = 8 cm and PR = 10 cm. (1) Draw a line segment QR = 8 cm (2) Draw a ray QY which makes an angle of 90 to QR. (3) With R as a centre, draw an arc of radius 10 cm, which cuts QY at P. (4) Join PR to obtain required ΔPQR. Q.2. Construct a right-angled triangle whose hypotenuse is 6 cm long and one of the legs is 4 cm long. (1) Draw a line segment AB. (2) With B as centre, draw a ray BY which makes an angle of 90 to AB. (3) With A as a centre, draw an arc of radius 6 cm which cuts BY at C. (5) Join AC to obtain required ΔABC. 7

16 Q.3. Construct an isosceles right-angled triangle ΔABC, where m ACB = 90 and AC = 6 cm. (1) Draw a line segment AC = 6 cm. (2) At C, draw ACY = 90. (3) From AY, cut off CB = 6 cm. (4) Join AB to get the required triangle. 8

Math-2. Lesson 5-2. Triangle Congruence

Math-2. Lesson 5-2. Triangle Congruence Math-2 Lesson 5-2 Triangle Congruence Naming Triangles Triangles are named using a small triangle symbol and the three vertices of the triangles. The order of the vertices does not matter for NAMING a