Question

let abcde be a convex pentagon with ( mangle a = 86^{circ} ), ( mangle b = 108^{circ} ), ( angle acongangle e ), and ( angle bcongangle d ). calculate ( mangle c ). ( mangle c=)____ degrees

Explanation:

Step1: Calculate the sum of interior angles

The sum of interior angles of a pentagon \(n = 5\) is \((n - 2)\times180^{\circ}=(5 - 2)\times180^{\circ}=540^{\circ}\).

Step2: Use the angle - equality relationship

Since \(\angle A\cong\angle E\) and \(\angle B\cong\angle D\), let \(m\angle C=x\). Then \(m\angle A + m\angle B+m\angle C+m\angle D+m\angle E=540^{\circ}\). Substituting \(m\angle A = 86^{\circ}\), \(m\angle B = 108^{\circ}\), and because \(\angle A\cong\angle E\), \(\angle B\cong\angle D\), we have \(86^{\circ}+108^{\circ}+x + 108^{\circ}+86^{\circ}=540^{\circ}\).

Step3: Solve the equation for \(x\)

Combine like terms: \((86 + 86)+(108 + 108)+x=540^{\circ}\), \(172^{\circ}+216^{\circ}+x=540^{\circ}\), \(388^{\circ}+x=540^{\circ}\). Then \(x=540^{\circ}-388^{\circ}=152^{\circ}\).

Answer:

\(152\)