Question

12.
find the value of x.
(there is a pentagon with angles 105, 95, 95, 125 and an exterior angle x)

Explanation:

Step1: Recall the formula for the sum of interior angles of a polygon

The sum of interior angles of an \(n -\)sided polygon is \((n - 2)\times180^{\circ}\). For a pentagon (\(n=5\)), the sum of interior angles \(S=(5 - 2)\times180^{\circ}=540^{\circ}\).

Step2: Let the unknown interior angle adjacent to \(x\) be \(y\). Then \(y + x=180^{\circ}\) (linear - pair of angles).

The sum of the known interior angles is \(105^{\circ}+95^{\circ}+95^{\circ}+125^{\circ}=420^{\circ}\).

Step3: Find the value of \(y\)

Since the sum of interior angles of the pentagon is \(540^{\circ}\), we have \(y+420^{\circ}=540^{\circ}\). Solving for \(y\), we get \(y = 540^{\circ}-420^{\circ}=120^{\circ}\).

Step4: Find the value of \(x\)

Since \(y + x=180^{\circ}\) and \(y = 120^{\circ}\), then \(x=180^{\circ}-y\). Substituting \(y = 120^{\circ}\), we get \(x = 60^{\circ}\).

Answer:

\(60^{\circ}\)