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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

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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.

Given: AD and CE are the angle bisectors of ∠A and ∠C respectively. ∠ABC = 90°
To Determine: ∠AOC
Determination: In ∆ABC,
∠A + ∠B + ∠C = 180°
| Angles sum property of a triangle
⇒ ∠A + 90° + ∠C = 180°
⇒ ∠A + ∠C = 90° ...(1)
In ∆AOC,
∠OAC + ∠OCA + ∠AOC = 180°
| Angle sum property of a triangle

$rightwards double arrow space space space space 1 half angle straight A plus 1 half angle straight C plus angle AOC equals 180 degree space space space space space space space space space space space space space left enclose table row cell because space AD space and space CE space are space the space angle space bisectors end cell row cell space Of space angle straight A space and space angle straight C space respectively end cell end table end enclose rightwards double arrow space space space space space space 1 half left parenthesis angle straight A plus angle straight C right parenthesis plus angle AOC equals 180 degree rightwards double arrow space space space space space space 1 half left parenthesis 90 degree right parenthesis plus angle AOC equals 180 degree rightwards double arrow space space space space space space space 45 degree plus angle AOC equals 180 degree rightwards double arrow space space space space space space space angle AOC equals 180 degree minus 45 degree equals 135 degree$

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