# Lines and Angles

**Exercise 1**

**1. **In Fig. 6.13, lines AB and CD intersect at O. If ∠ AOC + ∠ BOE = 70 and ∠ BOD = 40^{0}, find ∠BOE and reflex ∠COE.

**2. **In Fig. 6.14, lines XY and MN intersect at O. If ∠POY = 90^{0} and a : b = 2 : 3, find c.

**3. **In the given figure, ∠PQR = ∠PRQ, then prove that ∠PQS = ∠PRT.

**4. **In Fig. 6.16, if x + y = w + z, then prove that AOB is a line.

**5.**In the given figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that

**Exercise 2**

**6. **It is given that ∠XYZ = 64^{0} and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects ∠ZYP , find ∠XYQ and reflex ∠QYP.

**7. **In the given figure, find the values of x and y and then show that AB || CD.

**8. **In the given figure, if AB || CD, CD || EF and y : z = 3 : 7, find x.

**9.**In the given figure, If AB || CD, EF ⊥ CD and ∠GED = 126

^{0}, find ∠AGE , ∠GEF and ∠FGE.

**10. ** In the given figure, if PQ || ST, ∠PQR = 110^{0} and ∠RST = 130^{0}, find ∠QRS.

[Hint: Draw a line parallel to ST through point R.]

**11. **In the given figure, if AB || CD, ∠APQ = 50^{0} and ∠PRD = 127^{0}, find x and y.

**12. **In the given figure, PQ and RS are two mirrors placed parallel to each other. An incident ray AB strikes the mirror PQ at B, the reflected ray moves along the path BC and strikes the mirror RS at C and again reflects back along CD. Prove that AB || CD.

**Exercise 3**

**13. ** In the given figure, sides QP and RQ of ΔPQR are produced to points S and T respectively. If ∠SPR = 135^{0} and ∠PQT = 110^{0}, find ∠PRQ.

**14. **In the given figure, ∠X = 62^{0}, ∠XYZ = 54^{0}. If YO and ZO are the bisectors of ∠XYZ and ∠XZY respectively of ΔXYZ, find ∠OZY and ∠YOZ.

**15.** In the given figure, if AB || DE, ∠ ∴ BAC = 35^{0} and ∠CDE = 53^{0}, find ∠DCE.

**16.** In the given figure, if lines PQ and RS intersect at point T, such that ∠PRT = 40^{0}, ∠RPT = 95^{0} and ∠TSQ = 75^{0}, find ∠SQT.

**17.**In the given figure, if PQ ⊥ PS , PQ || SR, ∠SQR = 28

^{0}and ∠QRT = 65

^{0}, then find the values of x and y.

**18.** In the given figure, the side QR of ΔPQR is produced to a point S. If the bisectors of ∠PQR and ∠PRS meet at point T, then prove that