Question

find the value of x in the convex polygon.
(3x - 23)°
111°
(2x + 19)°
119° (2x + 48)°

Explanation:

Step1: Determine the sum of interior angles

A convex pentagon (5 - sided polygon) has the sum of interior angles given by the formula \((n - 2)\times180^{\circ}\), where \(n = 5\). So, \((5 - 2)\times180^{\circ}=3\times180^{\circ} = 540^{\circ}\).

Step2: Set up the equation for the sum of angles

The angles of the pentagon are \(111^{\circ}\), \(119^{\circ}\), \((3x - 23)^{\circ}\), \((2x + 19)^{\circ}\), and \((2x + 48)^{\circ}\). The sum of these angles should be \(540^{\circ}\). So, we set up the equation:
\[111 + 119+(3x - 23)+(2x + 19)+(2x + 48)=540\]

Step3: Simplify the left - hand side of the equation

First, combine like terms:
\[

$$\begin{align*} 111+119 - 23+19 + 48+(3x+2x + 2x)&=540\\ (111 + 119)+(-23+19 + 48)+(7x)&=540\\ 230+(44)+7x&=540\\ 274+7x&=540 \end{align*}$$

\]

Step4: Solve for \(x\)

Subtract 274 from both sides of the equation:
\[7x=540 - 274\]
\[7x = 266\]
Divide both sides by 7:
\[x=\frac{266}{7}=38\]

Answer:

\(x = 38\)