Question

quadrilateral cdef is inscribed in circle a. if m∠def = (3x + 7)° and m∠dcf = (2x + 8)°, what is the value of x?

Explanation:

Step1: Use the property of cyclic quadrilaterals

In a cyclic quadrilateral, the sum of opposite angles is \(180^{\circ}\). Here, \(\angle DEF\) and \(\angle DCF\) are supplementary. So, \(m\angle DEF + m\angle DCF=180^{\circ}\).

Step2: Substitute the given angle expressions

Substitute \(m\angle DEF=(3x + 7)^{\circ}\) and \(m\angle DCF=(2x + 8)^{\circ}\) into the equation: \((3x + 7)+(2x + 8)=180\).

Step3: Simplify the left - hand side

Combine like terms: \(3x+2x+7 + 8=180\), which simplifies to \(5x+15 = 180\).

Step4: Solve for \(x\)

Subtract 15 from both sides: \(5x=180 - 15\), so \(5x=165\). Then divide both sides by 5: \(x=\frac{165}{5}=33\).

Answer:

\(x = 33\)