Question 132

ABCD is a cyclic quadrilateral. The tangents to the circle at the points A and C on it, intersect at P. If $$\angle ABC = 98^\circ$$, then what is the measure of $$\angle APC$$?

Solution

ACD is a cyclic quadrilateral so,

$$\angle ABC + \angle ADC = 180\degree$$

$$\angle ADC = 180 - 98 = 82\degree$$

$$\angle AOC = 2 \times \angle ADC = 2 \times 82 = 164\degree$$

In quadrilateral AOCP-

$$\angle OAP + \angle APC + \angle PCO + \angle COA = 360\degree$$

$$\angle OAP = \angle PCO = 90\degree$$

($$\because$$ tangent angle)

$$\angle APC = 360 - 90 - 90 - 164 = 16\degree$$

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