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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.

Let in ∆ABC,
∠A = ∠B + ∠C    ...(1)
We know that,
∠A + ∠B + ∠C = 180°
| ∵ The sum of the three angles of a triangle is 180°
⇒    ∠A + ∠A = 180° | From (1)
⇒    2 ∠A = 180°

$rightwards double arrow space space space space space space space space space space space space space space space space space space space space angle straight A equals fraction numerator 180 degree over denominator 2 end fraction equals 90 degree$
Hence triangle ABC is right angled triangle.

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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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