Title 05-Congruence of triangles(1).pdf ✅

5 CHAPTER CONGRUENCE OF TRIANGLES 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 6 cm Q R S 6 cm Thus, line segment PQ = ~ line segment RS as PQ = RS = 6 cm. 2. Congruence of Angles. Two angles are congruent if they have the same measure. B A O' 40° P Q O 40° Thus, AO'B = ~ QOP, as m AO'B = m QOP = 40°. 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. A B 4. Congruence of squares. Two squares are congruent if they have same side length. P S Q R X T 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 B C P S Q R Rectangle ABCD = ~ Rectangle PQRS as AB = PQ and BC = QR. 6. Congruence of circles. Two circles are congruent if they have the same radius. A 2 cm B 2 cm Circle A = ~ Circle B, as radius of A = radius of B = 2 cm. ➢ CONGRUENCE OF TRIANGLES
Two triangles are congruent if they are copies of each other, and when superposed they cover each other exactly. A B C D E F ABC and DEF have the same size and shape. They are congruent. So we would express this as ABC = ~ DEF. This means that, when we place DEF on ABC, D falls on A, E falls on B and F falls on C, also DE falls along AB, EF falls along BC and DF falls along AC. Corresponding angles are : A and D, B and E, C and F. Corresponding vertices are : A and D, B and E, C and F. 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. ❖ EXAMPLES ❖ Ex.1 Write the correspondence between the vertices, sides and angles of the triangles XYZ and MLN, if XYZ ~ = MLN. Sol. By the order of letters, we find that X  M, Y  L and Z  N  XY = ML, YZ = LN, XZ = MN Also X = M, Y = L and Z = N. Ex.2 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° Sol. Let us draw the triangles and write the measures of their corresponding parts along with them. P Q R 4 cm 60° 6 cm 80° 40° 5 cm Z Y X 5 cm 4 cm 80° 40° 60° 6 cm From the above figures, we note that PQ = XZ, QR = YZ, PR = XY and P = X, Q = Z, R = Y  P  X, Q  Z and R  Y Hence, PQR = ~ XZY ➢ CRITERIA FOR 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 ❖ 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. 3 cm 5 cm 4 cm B C A 3 cm 5 cm 4 cm Q R P Examine the congruence of triangles by method of superposition. Also verify the congruence by equality of six corresponding elements of the triangles. Sol. 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. 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

P = M = 40° PO = OM = 6 cm and PQ = ML = 4 cm (given) Thus, by SAS congruence criteria POQ  MOL 3. ASA Congruence Criteria (Condition) 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 In the following pair of triangles figure, the measure of some parts are given. Verify if the two triangles are congruent. Sol. C A 5 cm B 50° 40° G E 5 cm F 50° 40° In triangles, ABC and EFG. Given, AB = EF = 5 cm A = E = 50° B = F = 40° Therefore, by ASA congruence condition ABC  EFG Ex.9 In figure, AO = BO and A = B. (i) Is AOC = BOD ? Why ? (ii) Is AOC  BOD by ASA congruence condition ? (iii) State the three facts you have used to answer (ii). (iv) Is ACO = BDO ? A C O D B Sol. (i) Yes, AOC = BOD [Vertically opposite angles] (ii) In AOC and BOD, we have AOC = BOD [Vertically opposite angles ] AO = BO [Given] OAC = DBO [Given] Therefore, by ASA congruence condition, we have AOC  BOD (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. ❖ EXAMPLES ❖ Ex.10 In figure, PQ = PS, PQ ⊥ QR and PS ⊥ RS. (i) Is PQR  PSR ? Why ? (ii) Is QR = RS ? Why ? Sol. Q P S R 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.