Question

quadrilateral cdef is inscribed in circle a. if m∠c = 9x° and m∠e = 7x + 4°, what is the measure of ∠e?

Explanation:

Step1: Recall property of cyclic quadrilateral

In a cyclic quadrilateral, opposite - angles are supplementary. That is, if \(CDEF\) is a cyclic quadrilateral, then \(m\angle C+m\angle E = 180^{\circ}\).

Step2: Set up the equation

We know that \(m\angle C = 9x^{\circ}\) and \(m\angle E=(7x + 4)^{\circ}\). So, \(9x+(7x + 4)=180\).

Step3: Simplify the left - hand side of the equation

Combine like terms: \(9x+7x+4 = 16x+4\). So the equation becomes \(16x+4 = 180\).

Step4: Solve for \(x\)

Subtract 4 from both sides: \(16x=180 - 4=176\). Then divide both sides by 16: \(x=\frac{176}{16}=11\).

Step5: Find the measure of \(\angle E\)

Substitute \(x = 11\) into the expression for \(m\angle E\). \(m\angle E=7x + 4\). So \(m\angle E=7\times11+4=77 + 4=81^{\circ}\).

Answer:

\(81^{\circ}\)