Question

refer to the figure at the right. what is ( mangle p )?
( mangle p = square^circ )

Explanation:

Step1: Recall the sum of interior angles of a pentagon.

The sum of the interior angles of a pentagon is given by the formula \((n - 2)\times180^{\circ}\), where \(n = 5\). So, \((5 - 2)\times180^{\circ}= 3\times180^{\circ}= 540^{\circ}\).

Step2: Sum the given angles and set equal to 540.

The angles of the pentagon are \((4x + 2)^{\circ}\), \((6x - 6)^{\circ}\), \((4x - 10)^{\circ}\), \((5x - 11)^{\circ}\), and \((5x + 13)^{\circ}\). Summing them up:
\[

$$\begin{align*} (4x + 2)+(6x - 6)+(4x - 10)+(5x - 11)+(5x + 13)&= 540\\ 4x + 2 + 6x - 6 + 4x - 10 + 5x - 11 + 5x + 13&= 540\\ (4x + 6x + 4x + 5x + 5x)+(2 - 6 - 10 - 11 + 13)&= 540\\ 24x - 12&= 540 \end{align*}$$

\]

Step3: Solve for \(x\).

Add 12 to both sides:
\[
24x = 540 + 12\\
24x = 552
\]
Divide both sides by 24:
\[
x=\frac{552}{24}= 23
\]

Step4: Find \(m\angle P\).

Substitute \(x = 23\) into the expression for \(\angle P\), which is \((4x + 2)^{\circ}\):
\[
4\times23 + 2 = 92 + 2 = 94
\]

Answer:

\(94\)