Congruence of Triangles

7 CONGRUENCE OF TRIANGLES 3. Congruence of plane figures. Two plane figures A and B are congruent as they superpose each other. We can write it as figure A ~ = figure B. CONTENTS • Congruent Figures • Congruence of Triangles • Criteria for Congruence of Triangles CONGRUENT FIGURES Two figures/objects are said to be congruent if they are exactly of the same shape and size. The relationship between two congruent figures is called congruence. We use the symbol ≅ for 'congruent to'. 1. Congruence among line segments. Two line segments are congruent if they have the same length. P B A Q 6 cm 4. Congruence of squares. Two squares are congruent if they have same side length. P S X T Q R Y Z Square PQRS ~ = Square XYZT as PQ = XY. 5. Congruence of rectangles. Two rectangles are said to be congruent if they have the same length and breadth. A D P S B C Q R S 6 cm R ~ line segment RS as Thus, line segment PQ = PQ = RS = 6 cm. 2. Congruence of Angles. Two angles are congruent if they have the same measure. Rectangle ABCD ~ = Rectangle PQRS as AB = PQ and BC = QR. 6. Congruence of circles. Two circles are congruent if they have the same radius. 40° Q Thus, ∠AO'B ~ = ∠QOP, as m ∠AO'B = m ∠QOP = 40°. O 2c m A O' 2c 40° m P B A B Circle A ~ = Circle B, as radius of A = radius of B = 2 cm. Let us draw the triangles and write the measures of their corresponding parts along with them. Sol. CONGRUENCE OF TRIANGLES Two triangles are congruent if they are copies of each other, and when superposed they cover each other exactly. A Z P D 4 cm 60° 80° Q B C E 5 cm 80° 6 cm 40° 5 cm 4 cm RY From the above figures, we note that PQ = XZ, QR = YZ, PR = XY and ∠P = ∠X, ∠Q = ∠Z, ∠R = ∠Y along BC and DF falls along AC . ∴ P ↔ X, Q ↔ Z and R ↔ Y Corresponding angles are : ∠A and ∠D, ∠B and ∠E, ∠C and ∠F. Hence, ∆PQR ~ = ∆XZY Corresponding vertices are : A and D, B and E, C and F. CRITERIA FOR CONGRUENCE OF TRIANGLES Corresponding sides are : AB and DE, BC and EF, AC and DF . Hence, three sides and three angles are the six matching parts for the congruence of triangles. 1. SSS Congruence Criteria (Condition) Two triangles are congruent, if three sides of one triangle are equal to the corresponding three sides of the other triangle. EXAMPLES Sol. X 6 cm F ∆ABC and ∆DEF have the same size and shape. They are congruent. So we would express this as ~ ∆DEF. This means that, when we place ∆ABC = ∆DEF on ∆ABC, D falls on A, E falls on B and F falls on C, also DE falls along AB, EF falls Ex.1 60° 40° Write the correspondence between the vertices, sides and angles of the triangles XYZ and MLN, if ∆XYZ ~ = ∆MLN. By the order of letters, we find that EXAMPLES Ex.3 Two triangles, ABC and PQR have been drawn such that AB = 3 cm, BC = 4 cm and AC = 5 cm. Also PR = 5 cm, QR = 4 cm and PQ = 3 cm. X ↔ M, Y ↔ L and Z ↔ N In following pairs of triangles, find the correspondence between the triangles so that they are congruent. In ∆PQR : PQ = 4 cm, QR = 5 cm, PR = 6 cm, ∠P = 60°, ∠Q = 80°, ∠R = 40°. In ∆XYZ : XY = 6 cm, ZY = 5 cm, XZ = 4 cm, ∠X = 60°, ∠Y = 40°, ∠Z = 80° B 3 cm Ex.2 3 cm Also ∠X = ∠M, ∠Y = ∠L and ∠Z = ∠N. P A ∴ XY = ML, YZ = LN, XZ = MN 5 cm 4 cm C Q 5 cm 4 cm R Examine the congruence of triangles by method of superposition. Also verify the congruence by equality of six corresponding elements of the triangles. Trace a copy of a ∆ABC and super-impose it on ∆PQR. We find that the triangles cover each other exactly, so that A ↔ P, B ↔ Q and C ↔ R i.e., ∆ABC ≅ ∆PQR. Sol. EXAMPLES Ex.5 Examine whether the two triangles are congruent or not by using the given information. Also measure the angles of the triangles and fill the information in the following table : Triangle ABC Triangle PQR Difference ∠A = ∠P = ∠A – ∠P = ∠B = 90° ∠Q = 90° ∠B – ∠Q = 0 We find that in all cases the difference is either zero or very close to zero, which may be treated as zero. Given below are measures of some parts of two triangles. In ∆XYZ : XY = 6 cm, YZ = 8 cm, ∠Y = 47° In ∆PQR : QR = 6 cm, PR = 8 cm, ∠R = 47° Sol. Let us make a rough sketch of the triangles before examining their congruence. P Z So we have ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R Because all sides (given) and all the angles (observed) of ∆ABC are equal to the corresponding sides and angles of triangle PQR. ∴ ∆ABC ≅ ∆PQR Ex.4 X ABC and DBC are two triangles drawn on a common base BC such that AB = DC and DB = AC on the same side of BC. (See figure) Sol. 47° Y 6 cm 47° Q 6 cm R Clearly, here XY = QR = 6 cm, ZY = PR = 8 cm and ∠Y = ∠R = 47° (included angles). Thus, by SAS congruence criteria ∆XYZ ≅ ∆QRP. Are ∆ADB and ∆DAC congruent ? If yes, state the corresponding parts. Which condition did you use to establish the congruence ? 8 cm 8 cm Ex.6 Triangle PQR is isosceles with PQ = PR. Line segment PS bisects ∠P and meets the side QR at point S. In ∆ADB and ∆DAC, we have A (i) Is ∆PSQ ≅ ∆PSR ? D (ii) Can we say that QS = SR ? B Sol. C AB = DC (Given) BD = CA (Given) and AD = AD (Common side) ∴ ∆ADB ≅ ∆DAC In ∆PSR and ∆PSQ, the three pairs of equal parts (two sides and one angle) are as follows: P Also, A ↔ D, D ↔ A and B ↔ C Since, the three corresponding equal parts are the sides of the triangles, therefore, SSS congruence condition is used to prove the congruence. Q 2. SAS Congruence Criteria (Condition) When two sides and the included angle of one triangle is equal to the corresponding sides and the included angle of another triangle, the two triangles are congruent. This, condition of congruence is known as side-angle-side congruence. In short we write SAS condition. and S R PQ = PR (given) PS = PS (common) ∠QPS = ∠RPS (PS bisects ∠P) So (i) Yes, ∆PSQ ≅ ∆PSR (ii) Yes, QS = SR (corresponding sides of congruent triangles). Ex.7 In adjoining figure, prove that Ex.9 ∆POQ ≅ ∆MOL (i) Is ∠AOC = ∠BOD ? Why ? In ∆POQ and ∆MOL, we have Sol. (ii) Is ∆AOC ≅ ∆BOD by ASA congruence condition ? P 4 cm 40° In figure, AO = BO and ∠A = ∠B. L 6 cm (iii) State the three facts you have used to answer (ii). 4 cm O 40° 6 cm (iv) Is ∠ACO = ∠BDO ? Q M A ∠P = ∠M = 40° O PO = OM = 6 cm D C and PQ = ML = 4 cm (given) Thus, by SAS congruence criteria B ∆POQ ≅ ∆MOL Sol. (i) Yes, ∠AOC = ∠BOD 3. ASA Congruence Criteria (Condition) [Vertically opposite angles] Two triangles are congruent, if two angles and the included side of one is equal to the corresponding angles and side of the other. EXAMPLES Ex.8 (ii) In ∆AOC and ∆BOD, we have ∠AOC = ∠BOD [Vertically opposite angles ] In the following pair of triangles figure, the measure of some parts are given. Verify if the two triangles are congruent. G C AO = BO [Given] ∠OAC = ∠DBO [Given] Therefore, by ASA congruence condition, we have Sol. 50° A 50° 40° 5 cm B E 5 cm In triangles, ABC and EFG. Given, AB = EF = 5 cm ∠A = ∠E = 50° ∠B = ∠F = 40° Therefore, by ASA congruence condition ∆ABC ≅ ∆EFG ∆AOC ≅ ∆BOD 40° F (iii) AO = BO, ∠A = ∠B and ∠AOC = ∠BOD (iv) Yes, since ∆AOC ≅ ∆BOD 4. RHS Congruence Criteria (Condition) Two right triangles are congruent, if the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and a side of the other triangle. Ex.12 Complete the following statements : EXAMPLES Ex.10 In figure, PQ = PS, PQ ⊥ QR and PS ⊥ RS. (i) Two line segments are congruent if  . (i) Is ∆PQR ≅ ∆PSR ? Why ? (ii) Is QR = RS ? Why ? (ii) Among two congruent angles, one has a measure of 70°, the measure of the other angle is  . Q Sol. (iii) When we write ∠A = ∠B, we actually mean  . R P (iv) Two circles C1 and C2 are congruent, then their radii will be  . S In ∆PQR and ∆PSR, we have PQ = PS (given) ∠PQR = ∠PSR (both are right angles) PR = PR (common side) (i) ∴ By RHS congruence condition, we have ∆PQR ≅ ∆PSR (ii) Yes, QR = RS, because they are corresponding parts of congruent triangles. Sol. (i) Two line segments are congruent if they have the same length. (ii) Among two congruent angles, one has a measure of 70°, the measure of the other angle is 70°. (iii) When we write ∠A = ∠B, we actually mean m ∠A = m ∠B. Ex.11 AX is the bisector of ∠BAC, P is any point on AX. Prove that the perpendicular drawn from P to AB and AC are equal. (iv) Two circles C1 and C2 are congruent, then their radii will be equal. Given : An angle BAC bisected by AX. From any point P on AX, PM and PN are perpendiculars drawn to AB and AC respectively. Ex.13 If ∆ABC ~ = ∆FED under the correspondence ABC ↔ FED, write all the corresponding congruent parts of the triangles. Sol. Sol. C As ∆ABC ~ = ∆FED M A So, ∠A ↔ ∠F, ∠B → ∠E, ∠C ↔ ∠D. 1 2 P X N AB ↔ FE , BC ↔ ED , AC ↔ FD . B To Prove : PM = PN Proof : Ex.14 If ∆DEF ~ = ∆BCA, write the part(s) of ∆BCA that correspond to In ∆AMP and ∆ANP ∠M = ∠N [Each 90°] ∠1 = ∠2 (i) ∠E (ii) EF (iii) ∠F (iv) DF [AX is bisector of ∠BAC] AP = AP [Common] ∆AMP ≅ ∆ANP [By AAS congruence condition] Sol. ~ ∆BCA, then D ↔ B, E ↔ C, F ↔ A If ∆DEF = (i) ∠E = ∠C (ii) EF = CA PM = PN [Corresponding parts of congruent triangles] (iii) ∠F = ∠A (iv) DF = BA Ex.15 In the figures given below, lengths of the sides of the triangles are indicated. By applying SSS congruence rule, state which pairs of triangles are congruent. In case of congruent triangles, write the result in symbolic form. A Ex.16 In figure, AB = AC and D is the mid point of BC . (i) State the three pairs of equal parts in ∆ADB and ∆ADC. (ii) Is ∆ADB = ∆ADC ? Give reason P 1.5 cm (iii) Is ∠B = ∠C ? Why ? 1.5 cm 2.2 cm 2.2 cm (i) In ∆ADB and ∆ADC Sol. A B C 2.5 cm Q 2.5 cm R (i) D L B 3.2 cm 3.5 cm E 3 cm 3 cm F M C D 3.5 cm 3.2 cm N AB = AC (given) AD = AD (common) BD = DC (ii) (Q D is mid point of BC) ~ ∆ADC (ii) ∆ADB = (by SSS property) A (iii) Yes, ∠B = ∠C of congruent triangles) 3.5 cm 3.5 cm B (by corresponding parts Ex.17 In figures, measures of some parts of the triangles are indicated. By applying SAS congruence rule, state the pairs of congruent triangles, if any, in each case. In case of congruent triangles, write them in symbolic form. 2.5 cm D 2.5 cm C (iii) Sol. (i) In ∆ABC and ∆PQR D A AB = PQ = 1.5 cm BC = QR = 2.5 cm 80° 2.5 cm 2.8 cm 2.5 cm C E 70° 2.8 cm CA = RP = 2.2 cm ∴ ∆ABC ~ = ∆PQR (by SSS) B (ii) DE ≠ LM, EF ≠ MN (i) So, ∆DEF ~ =/ ∆LMN. 2.5 cm 2.5 cm (common) 35° 35° AB = AC = 3.5 cm B BD = CD = 2.5 cm ~ ∆ADC ∆ADB = R A (iii) In ∆ADB and ∆ADC AD = AD F 3 cm P C (ii) (by SSS) Sol. (i) In ∆ABC and ∆DEF 3 cm Q AB = DE = 2.5 cm (Q 80° ≠ 70°) ∠A ≠ ∠D AC = DF = 2.8 cm So, ∆ABC ~ =/ ∆DEF ∠A = ∠F (40° each) AB = EF (3.5 cm each) and ∠B = ∠E (60° each) ∴ ∆ABC ≅ ∆FED (by ASA) As (ii) In ∆ACB and ∆RPQ AC = RP = 2.5 cm ∠C = ∠P = 35° CB = PQ = 3 cm ∴ ∆ACB ≅ ∆RPQ (by SAS) Ex.18 In figure, AB and CD bisect each other at O. (i) State the three pairs of equal parts in two triangles AOC and BOD. (ii) Which of the following statements are true (a) ∆AOC ≅ ∆DOB (b) ∆AOC ≅ ∆BOD ? (ii) ∆PQR ≅/ ∆DEF as ∠Q = ∠D = 90° ∠E = ∠R = 50° PR ≠ EF Ex.20 In figure, ray AZ bisect ∠DAB as well as ∠DCB. D B Sol. (Q 3.3 cm ≠ 3.5 cm) C O C A Z D A (i) AO = OB CO = OD ∠AOC = ∠BOD (vertically opposite angles) (ii) ∆AOC ≅ ∆BOD (by SAS) Hence, (b) is true B (i) State the three pairs of equal parts in ∆BAC and ∆DAC. (ii) Is ∆BAC ≅ ∆DAC ? Give reasons. Ex.19 In figures, measures of some parts are indicated. By applying ASA congruence rule, state which pairs of triangles are congruent. In case of congruence, write the result in symbolic form. 3.5 cm A 40° 60° (iv) Is CD = CB ? Give reasons. Sol. D B (iii) Is AB = AD ? Justify your answer. (i) In ∆BAC and ∆DAC, ∠BAC = ∠DAC [Q AZ bisects ∠DAB] AC = AC E C 60° 40° F ∠BCA = ∠DCA [Q AZ bisects ∠DCB] 3.5 cm (i) (ii) Yes, ∆BAC ≅ ∆DAC P (iii) Yes, D 90° 3.3 cm 90° Q 50° (iv) Yes, 3.5 cm R (by ASA) AB = AD (corresponding parts of congruent triangles) 50° E (common) CD = CB F (corresponding parts of congruent triangles) (ii) Sol. (i) In ∆ABC and ∆EFD. Ex.21 In figure, measures of some parts of triangles are given. By applying R.H.S. congruence rule, state which pairs of triangles are congruent. In case of congruent triangles, write the result in symbolic form. ∠PRQ = ∠XZY RQ = ZY So, P D 6 cm R Z Q 2.5 cm 3 cm 6 cm ∆PQR ≅ ∆XYZ Y P 90° X 90° Q R (c) Given EB = DB E F AE = BC (i) ∠A = ∠C = 90° D C 90° So, ∆ABE ≅ ∆CDB 90° 2 cm 2 cm A D E B 3.5 cm (ii) Sol. (i) In ∆PQR and ∆DEF A as PR = DF = 6 cm Sol. ∠Q = ∠E = 90° C B (a) ∆ABC ≅ ∆DEF (by SSS) (b) ∆PQR ≅ ∆XYZ (by SAS) But PQ ≠ DE (as 3 cm ≠ 2.5 cm) So, ∆PQR ≅/ ∆DEF (c) ∆EAB ≅ ∆DCB (by RHS) Ex.23 If ∆ABC and ∆PQR are to be congruent, name one additional pair of corresponding parts. What criterion did you use ? (ii) In ∆CAB and ∆DBA ∠C = ∠D = 90° each Sol. AB = AB = 3.5 cm To prove ∆ABC ≅ ∆PQR, R A CA = DB = 2 cm ∆CAB ≅ ∆DBA (by RHS) Ex.22 Which congruence criterion do you use in the following ? (a) Given AC = DF AB = DE B BC = EF P Q We need one additional pair of corresponding parts which is So, ∆ABC ≅ ∆DEF BC = QR D A C As, if ∠ABC = ∠PQR (90° each) BC = QR B (b) Given C E RP = ZX F ∠ACB = ∠PRQ (given) ∆ABC ≅ ∆PQR (by ASA) Ex.24 Explain why ∆ABC ≅ ∆FED. Sol. In ∆ABC and ∆FED D A E ∠B = ∠E (90° each) (given) ⇒ ∠C = ∠D ….(i) (third ∠C = third ∠D) So, now ∠B = ∠E = 90° C B ∠A = ∠F F BC = DE (given) ∠C = ∠D [From (i)] ∆ABC = ∆FED (given) (by ASA) IMPORTANT POINTS TO BE REMEMBERED (1) Two figures are congruent, if they have the same shape and size. (2) Two line segments say AB and CD are congruent if they have equal lengths, we write this as AB ≅ CD. (3) Two squares are congruent if measure of their side is same. (4) Two rectangles are congruent if they have the same length and breadth. (5) Two circles are congruent if they have same radius. (6) (7) Two triangles are congruent if the three sides and three angels of one triangle are equal to the corresponding sides and angles of the other triangle. Two triangles are congruent if three sides of one triangle are equal to corresponding three sides of another triangle (SSS congruence condition). (8) Two triangles are congruent if two sides and the included angle of one triangle are equal to corresponding sides and included angle of the other triangle (SAS congruence condition). ‘Triangle’ can be denoted as ‘∆’. (9) Two triangles are congruent if two angles and included side of one triangle are equal to the corresponding angles and included side of the other (ASA congruence condition). (10) Two right triangles are congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and corresponding side of other triangle. (11) Two congruent figures are equal in area, but the figures having equal area may not be congruent. (12) There is no such thing as AAA congruence of two triangles. (13) Two triangles with equal corresponding angles need not be congruent. In such a correspondence, one of them can be enlarged copy of the other. (They would be congruent only if they are exact copies of one another).