Question

in the diagram below, quadrilateral $stuv$ is inscribed in circle $w$. solve for $x$ and $y$.

Explanation:

Step1: Use the property of cyclic quadrilaterals

In a cyclic quadrilateral, the sum of opposite angles is \(180^{\circ}\).
For angles \(\angle S\) and \(\angle V\):
\(121+(2x - 29)=180\)

Step2: Solve for \(x\)

Simplify the equation:
\(121+2x-29 = 180\)
\(92+2x=180\)
Subtract \(92\) from both sides:
\(2x=180 - 92\)
\(2x=88\)
Divide both sides by \(2\):
\(x = 44\)

Step3: Use the property of angles in a circle (another property of cyclic quadrilaterals)

For angles \(\angle T\) and \(\angle U\):
\(63+(2y - 29)=180\)

Step4: Solve for \(y\)

Simplify the equation:
\(63+2y-29 = 180\)
\(34+2y=180\)
Subtract \(34\) from both sides:
\(2y=180 - 34\)
\(2y=146\)
Divide both sides by \(2\):
\(y = 73\)

Answer:

\(x = 44\), \(y = 73\)