Question

Quadrilateral OPQR is inscribed inside a circle as shown below. What is the measure of angle O? You must show all work and calculations to receive credit. (10 points)

Answer 1

A quadrilateral inscribed in a circle is called a "cyclic quadrilateral."

A quadrilateral is cyclic, only if the sum of the measures of the opposite angles is 180°. 

Also, the reverse is true: If we know a quadrilateral to be cyclic, than its opposite angles must be supplementary (add up to 180°) .


Thus, we must have:

$m(\angle O)+m(\angle Q)=180^{\circ}$

2x+(2x+4)=180

4x+4=180 (next dividing by 4:)

x+1=45

Thus, x=44, which means that the measure of angle O is 2x degrees, that is 88 degrees.



Answer: 88°