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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.

∵ TR is a line
∴ ∠PQT + ∠PQR = 180°
⇒    110° + ∠PQR = 180°
⇒ ∠PQR = 180° - 110° = 70°    ...(1)
∵ QS is a line
∴ ∠SPR + ∠QPR = 180°
⇒    135° + ∠QPR = 180°
⇒ ∠QPR = 180° - 135° = 45° ...(2)
In ∆PQR,
∠PQR + ∠QPR + ∠PRQ = 180°
| ∵ The sum of all the angles of a triangle is 180°
⇒ 70° + 45° + ∠PRQ = 180°
| Using (1) and (2)
⇒    115° + ∠PRQ = 180°
⇒ ∠PRQ = 180° - 115° = 65°.

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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

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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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