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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.

In ∆XYZ,
∠XYZ + ∠YZX + ∠ZXY = 180°
| ∵ The sum of all the angles of a triangle is 180°
⇒ 54° + ∠YZX + 62° = 180°
⇒ 116° + ∠YZX = 180°
⇒ ∠YZX = 180° - 116° = 64° ...(1)
∵ YO is the bisector of ∠XYZ

therefore space angle XYO equals angle OYZ equals 1 half angle XYZ space space space space space space space space space space space space space equals 1 half left parenthesis 54 degree right parenthesis equals 27 degree space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 2 right parenthesis

∵ ZO is the bisector of ∠YZX

therefore space space space space space angle XZO space equals angle OZY equals 1 half angle YZX space space space space space space space space space space space space space space space space space space equals space 1 half left parenthesis 64 degree right parenthesis equals 32 degree space space space space space space space space space space space space space space space space space.. left parenthesis 3 right parenthesis

| Using (1)

In ∆OYZ,
∠OYZ + ∠OZY + ∠YOZ = 180°
|∵ The sum of all the angles of a triangle is 180°
⇒ 27° + 32° + ∠YOZ = 180°
| Using (2) and (3)
⇒ 59° + ∠YOZ = 180°
⇒ ∠YOZ = 180° - 59° = 121°.

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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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