Question

in quadrilateral qrst, (angle rst) measures ((5x + 15)^{circ}). angle tqr measures ((4x+3)^{circ}). what is the measure of angle rst? (15^{circ}) (75^{circ}) (105^{circ}) (165^{circ})

Explanation:

Step1: Use the property of cyclic quadrilaterals

In a cyclic quadrilateral, the sum of opposite angles is \(180^{\circ}\). So, \(\angle RST+\angle TQR = 180^{\circ}\).
Given \(\angle RST=(5x + 15)^{\circ}\) and \(\angle TQR=(4x + 3)^{\circ}\), we have the equation \((5x+15)+(4x + 3)=180\).

Step2: Solve the equation for \(x\)

Combine like - terms: \(5x+4x+15 + 3=180\), which simplifies to \(9x+18 = 180\).
Subtract 18 from both sides: \(9x=180 - 18=162\).
Divide both sides by 9: \(x=\frac{162}{9}=18\).

Step3: Find the measure of \(\angle RST\)

Substitute \(x = 18\) into the expression for \(\angle RST\): \(\angle RST=(5x+15)^{\circ}\).
\(\angle RST=5\times18+15\).
First, calculate \(5\times18 = 90\), then \(90+15=105\).

Answer:

\(105^{\circ}\) (the third option)