Question

find the value of ( x ) in the convex polygon.
( (2x + 22)^circ )
( (3x - 12)^circ )
( 72^circ )
( 75^circ )
( 63^circ )

Explanation:

Step1: Recall exterior angle sum of polygon

The sum of exterior angles of any convex polygon is \(360^\circ\).

Step2: Sum all exterior angles

The exterior angles are \(72^\circ\), \(63^\circ\), \(75^\circ\), \((3x - 12)^\circ\), and \((2x + 22)^\circ\). So, we set up the equation:

$$72 + 63 + 75+(3x - 12)+(2x + 22)=360$$

Step3: Simplify the left - hand side

First, combine like terms:

$$72 + 63+75 - 12 + 22+3x+2x=360$$

$$ (72 + 63+75+22 - 12)+(3x + 2x)=360$$

$$ (72+63 = 135;135 + 75=210;210+22 = 232;232-12 = 220)+5x=360$$

$$220 + 5x=360$$

Step4: Solve for x

Subtract 220 from both sides:

$$5x=360 - 220$$

$$5x = 140$$

Divide both sides by 5:

$$x=\frac{140}{5}=28$$

Answer:

\(x = 28\)