Question

in the diagram below, quadrilateral \\(qrst\\) is inscribed in circle \\(u\\). solve for \\(x\\) and \\(y\\).

Explanation:

Identify opposite angles in the cyclic quadrilateral

Inscribed quadrilateral \(QRST\) has opposite angles that sum to \(180^\circ\).
The opposite angle pairs are:

$$\angle Q \text{ and } \angle S$$

$$\angle R \text{ and } \angle T$$

Set up the equation for x

Using the opposite angles \(\angle R\) and \(\angle T\):

$$(5x + 29) + 71 = 180$$

Solve for x

Using the Linear Equations knowledge point

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Set up the equation for y

Using the opposite angles \(\angle Q\) and \(\angle S\):

$$124 + (y - 90) = 180$$

Solve for y

Using the Linear Equations knowledge point

$$
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Answer:

\(x = 16\), \(y = 146\)