Question

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

Answer 1

You can inscribe a quadrilateral inside a circle if and only if its opposite angles are supplementar. In your case, this means that

$\hat{O} + \hat{Q} = \hat{R} + \hat{P} = 180^\circ$

Since we're interested in the angle R, let's use the second part of the equalities above to substitute the given values for R and P:

$\hat{R} + \hat{P} = 3y+8+y = 180^\circ$

The left hand side can be simplified to $4y+8$, so if we subtract 8 from both sides the equation becomes

$4y = 172 \implies y = \cfrac{172}{4} = 43$

Since $y = 3$, we can compute the angle R by plugging this value into its formula:

$R = 3y+8 = 3\cdot 43 + 8 = 129+8 = 137$