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1. Congruence of Plane Figures Description Congruence of line segments Congruence of angles Reflect and Review Teasers See below the table. Two line segments are said to be congruent if they have the same length. A B C D If ̅̅̅̅ ̅̅̅̅, then ̅̅̅̅ and ̅̅̅̅ have the same length, and we write it as: AB = CD. Two angles are said to be congruent if 1) Draw an angle they have same measure. P A congruent to PQR. Answers 1) ̅̅̅̅ ̅̅̅̅̅ P B Congruence of triangles C Q If ABC PQR, then ABC and PQR have same measure, and we write it as: mABC = mPQR. Q 1) Two triangles ∆EFG and ∆LMN are congruent by the correspondence ∆GEF ∆MNL. Write the parts of ∆LMN that corresponds to a) E b) ̅̅̅̅ If two triangles are congruent then their six elements (three sides and three angles) of one triangle are congruent to the six elements of the other triangle. A B R X C Y R 1) a) N b) Z 1 Consider two triangles, Δ ABC and Δ XYZ, in which A = X, B = Y and C = Z, and AB = XY, BC = YZ and AC = XZ. Here, we say that the two triangles are congruent under the correspondence A X, B Y and C Z. This can be written as ABC XYZ. If any two triangles, ΔABC and ΔPQR are congruent under the correspondence ABC QRP, then A = Q, B = R, C = P, and ̅̅̅̅= ̅̅̅̅, ̅̅̅̅ = ̅̅̅̅ and ̅̅̅̅ = ̅̅̅̅ . Teasers 1) Find the line segments that are congruent to ̅̅̅̅. Y B X A M D Q C N P 2. Criteria for Congruence of Triangles Description Reflect and Review SSS Congruence criterion If the three sides of one triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent. 1) If ABCDE is a regular pentagon, prove that ∆ACD is isosceles. A B D A C In ∆ABC and ∆ADC, AB = AD (Property of rhombus) BC = DC (Property of rhombus) AC = AC (Common side) Thus, ∆ABC ∆ADC (SSS congruence 2 Teasers E B D C rule). 1) In the fig, AB = DE, AC = DF and BAC = EDF. Prove that if F and C are midpoints of BC and EF respectively. SAS Congruence criterion If two sides and the angle included between them of one triangle are respectively equal to two sides and the angle included between them of another triangle, then the two triangles are congruent. D E B A A C B F In ∆ABC and ∆ADE, AB = AD (given) AC = AE (given) BAC = EAD (vertically opposite angles) C Thus, ∆ABC ∆ADE (SAS congruence rule). ASA Congruence criterion D E 1) In the trapezium ABCD, if AD = BC, OAB = OBA and OCD = ODC then prove that OA = OB and OC = OD. W L V If two angles and the included side of a triangle N M are equal to two corresponding angles and In ∆LMN and ∆WVU, the included side of another triangle, then the LMN = WVU (given) LNM = WUV and (given) triangles are congruent. MN = VU (given) U O A Thus, ∆LMN ∆WVU (ASA congruence rule). If the hypotenuse and one side of a right-angled triangle are respectively equal to the hypotenuse and one side of another right-angled triangle, then the right triangles are congruent. A C B 1) In the quadrilateral PQRS, PS = QR and PQS = RSQ = 900. Prove that PQRS is a parallelogram. X RHS Congruence criterion C D Y Z B 3 In the right angled triangles ACB and XYZ, ACB = XYZ AB = XZ and AC = XY Thus, ∆ACB ∆XYZ (RHS congruence rule). 4