Question

in the diagram of circle p, ( mangle xyz ) is ( 72^{circ} ). what is the value of ( x )? ( 108^{circ} ) ( 144^{circ} ) ( 216^{circ} ) ( 252^{circ} )

Explanation:

Step1: Use the formula for the measure of an angle formed by two tangents

The measure of an angle formed by two tangents outside a circle is half the difference of the measures of the intercepted arcs. The formula is \(m\angle XYZ=\frac{1}{2}[(360 - x)-x]\).

Step2: Substitute the given angle measure into the formula

Given \(m\angle XYZ = 72^{\circ}\), we substitute into the formula: \(72=\frac{1}{2}(360 - 2x)\).

Step3: Solve the equation

Multiply both sides by \(2\): \(144=360 - 2x\).
Subtract \(360\) from both sides: \(144-360=- 2x\).
So, \(-216=-2x\).
Divide both sides by \(-2\): \(x = 108\).

Answer:

\(108^{\circ}\)