Page 345 - Start Up Mathematics_7

13. OP = OQ (midpoint), ∠A = ∠B (given), 3. (a) AD = CF ⇒ AD + DC = DC + CF

∠AOP = ∠BOQ (Vertically opposite) ⇒ AC = DF
∆ PAO ≅ ∆ QBO (by AAS) CB = DE (given), ∠C = ∠D (each 90°)
14. OD = OS (given), ∠K = ∠A (given),  ∆ ACB ≅ ∆ FDE (by SAS) ⇒ ∠A = ∠F (CPCT)
which are alternate angles hence AB || EF
∠KOD = ∠AOS (Vertically opposite)
(b) AB = EF (CPCT)
∆ KOD ≅ ∆ AOS (by AAS)
4. (a) AB = AC and PB = PC (isosceles triangle)
15. ∠P = ∠T, PR = TI (CPCT),
AP = AP (common)
PQ = TS (CPCT) ⇒ 2PA = 2TB ⇒ PA = TB
∆ APB ≅ ∆ APC (by SSS)
∆ PAR ≅ ∆ TBI (by SAS) (b) ∠APB = ∠APC (CPCT)
Hence, RA = IB (CPCT) 5. (a) FA = CT and TF = AC

Review Exercises (opposite sides of a parallelogram)
Multiple Choice Questions AT = AT (Common)
1. c 2. b 3. c 4. d 5. a  ∆ FAT ≅ ∆ CTA (by SSS)
6. d 7. d 8. c 9. a (b) ∠ATF = ∠TAC (CPCT)
Solve Mentally Thinking Skills
True or False 1. (a) ∠RPQ = ∠SQP (given), RP = SQ (given)
1. True 2. False 3. True 4. False PQ = PQ (common),  ∆ RPQ ≅ ∆ SQP (by SAS)
5. True 6. True 7. True 8. False (b) RQ = SP (CPCT)

Fill in the Blanks 2. (a) CE = AF (given), DC = BA (sides of a square)
1. RT 2. hypotenuse 3. equal ∠C = ∠A (each 90°),  ∆ DCE ≅ ∆ BAF (by SAS)
4. ASA 5. congruent, SSS 6. ≅ 7. (b) DE = BF (CPCT), AD = BC (opposite sides of square)
8. 9. AF = CE (given) ⇒ AD – AF = BC – CE
Answer in One Word or a Line ⇒ FD = EB hence DEBF is a parallelogram.
1. Two triangles are said to be congruent if they have same 3. (a) ∠E = ∠D (each 90°), CE = BD (given),
shape and size.
BC = BC (common)
2. SSS, SAS, ASA and RHS
 ∆ BCE ≅ ∆ CBD (by RHS)
3. If two figures cover each other exactly, they are said to (b) BE = CD (CPCT)
be congruent.
4. FA = MU and ∠A = ∠U (CPCT)
4. In congruent triangles, corresponding equal sides lie
opposite to equal angles and corresponding equal angles ∠E = ∠S (each 90°), ∆ FEA ≅ ∆ MSU (by AAS)
lie opposite to equal sides. FE = MS (by CPCT)
5. Two squares having equal sides are congruent. 5. OY = OZ (given), ∠XOY = ∠XOZ (each 90°)
6. Two circles having equal radii are congruent. XO = XO (common),  ∆ XOY ≅ ∆ XOZ (by SAS)
Let’s Evaluate ∠Y = ∠Z (CPCT), ∠OAY = ∠OBZ (each 90°),
1. (a) Yes, ASA congruency criterion OY = OZ (given)
(b) ∠MPO (c) No  ∆ AOY ≅ ∆ BOZ (by AAS), OA = OB (CPCT)
2. (a) No, insufficient data
Exemplar Problems
(b) Additional data Congruency criterion 1. c 2. ∆ DRQ 3. False

AX = AY AAS 4. ∠A = ∠L, ∠B = ∠M, ∠C = ∠N;
SA = PA ASA AB = LM, BC = MN, AC = LN

XS = YP AAS 5. (a) LM = ON, LN = OM, MN = NM
(b) Yes, by SSS criterion.

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