Question

ABCD is a cyclic quadrilateral. Tangents at A and C intersect at P. If ∠ABC = ${100^0}$ then what is the measure of angle APC?
A. ${20^0}$
B. ${30^0}$
C. ${35^0}$
D. ${40^0}$

Answer 1

Hint: In order to solve this problem we need to draw a rough diagram and use the concept that the sum of opposite angles of a quadrilateral is 180 degrees. Doing this will solve your problem.

Complete step-by-step answer:

The figure of this problem can be drawn as:

It is given that angle ABC = 100 degrees.
As we know that the sum of opposite angles in any quadrilateral is 180 degrees.
So, $\angle $ABC +$\angle $D = 180
That is
100 + $\angle $D = 180
$\angle $D = 80 degrees.
$\angle $AOC = 2$\angle $ADC = 2(80) = 160 degrees (Angle made by same base of the triangle AOC is half of the angle AOC)
Now in Quadrilateral PAOC $\angle $P and $\angle $AOC are opposite angles so,
$\angle $P + $\angle $AOC = 180 degrees
On putting the value of angle AOC we get,
$\angle $P + 160 = 180 degrees
So, $\angle $P = 20 degrees.
Hence, the correct option is A.

Note: In this question we have used the concept of sum of opposite angles of quadrilateral and also we have fully analysed the diagram. Take care of drawing the figure since figure plays an important role in drawing the diagram.