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Lines and Angles

Short Answer Type

61. In figure, sides QP and RQ of ∆PQR are produced to points S and T respectively. If ∠SPR = 135° and ∠PQT = 110°. find ∠PRQ.

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62. In figure, ∠X = 62°, ∠XYZ = 54°. If YO and ZO are the bisectors of ∠XYZ and ∠XZY respectively of ∆XYZ, find ∠OZY and ∠YOZ.

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63. In figure, if AB || DE, ∠BAC = 35° and ∠CDE = 53°, find ∠DCE.

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64. In figure, if lines PQ and RS intersect at point T, such that ∠PRT = 40°, ∠RPT = 95° and ∠TSQ = 75°, find ∠SQT.

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65. In figure, if PQ ⊥ PS, PQ || SR, ∠SQR = 28° and ∠QRT = 65°, then find the values of x and y.

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

angle QTR equals 1 half angle QPR

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67. Prove that if one angle of a triangle is equal to the sum of the other two angles, the triangle is right angled.

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Long Answer Type

68. In figure, the bisectors of ∠ABC and ∠BCA intersect each other at the point O. Prove that $angle BOC equals 90 degree plus 1 half angle straight A$

∵ BO is the bisector of ∠ABC.

therefore space angle OBC equals 1 half angle ABC equals 1 half angle straight B space space space space space space space space space space space space space space space.... left parenthesis 1 right parenthesis

∵ CO is the bisector of ∠ACB

therefore space space angle OCB equals 1 half angle ACB equals 1 half angle straight C space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis

In ∆OBC, ∠BOC + ∠OBC + ∠OCB = 180°
| ∵ The sum of the three angles of a ∆ is 180°

$rightwards double arrow space space space space angle BOC plus 1 half angle straight B plus 1 half angle straight C equals 180 degree$

| From (1) and (2)

$rightwards double arrow space space angle BOC equals 180 degree minus 1 half left parenthesis angle straight B plus angle straight C right parenthesis space space space space space space space space space space.... left parenthesis 3 right parenthesis$

In ∠ABC, ∠A + ∠B + ∠C = 180°
| ∵ The sum of the three angles of a triangle is 180°

$rightwards double arrow space space angle straight B plus angle straight C equals 180 degree minus angle straight A rightwards double arrow 1 half left parenthesis angle straight B plus angle straight C right parenthesis equals fraction numerator 180 degree minus angle straight A over denominator 2 end fraction equals 90 degree minus 1 half angle straight A space space space space space space space space space space space space space space... left parenthesis 4 right parenthesis$

From (3) and (4), we have

$angle BOC equals 180 degree minus left parenthesis 90 degree minus 1 half angle straight A right parenthesis equals 90 degree plus 1 half angle straight A.$

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Short Answer Type

69.

The side EF, FD and DE of a triangle DEF are produced in order forming three exterior angles DFP, EDQ and FER respectively. Prove that
∠DFP + ∠EDQ + ∠FER = 360°.

Prove that the sum of the exterior angles of a triangl is 360°.

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70. In the figure, AD and CE are the angle bisectors of ∠A and ∠C respectively. If ∠ABC = 90° then find ∠AOC.

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