Question

find the missing angle for each irregular polygon.

sum of the interior angles = \boxed{}^\circ

x = \boxed{}

m\angle p = \boxed{}^\circ

m\angle r = \boxed{}^\circ

m\angle t = \boxed{}^\circ

Explanation:

Step1: Find the sum of interior angles

The polygon is a hexagon (6 sides). The formula for the sum of interior angles of a polygon is \((n - 2)\times180^{\circ}\), where \(n\) is the number of sides. For \(n = 6\), we have \((6 - 2)\times180^{\circ}=4\times180^{\circ} = 720^{\circ}\).

Step2: Set up the equation for the sum of angles

The angles are \(115^{\circ}\), \(x + 75^{\circ}\), \(130^{\circ}\), \(x + 50^{\circ}\), \(135^{\circ}\), and \(x + 80^{\circ}\). Their sum should be \(720^{\circ}\). So:
\[115+(x + 75)+130+(x + 50)+135+(x + 80)=720\]
Simplify the left - hand side:
\[115+x + 75+130+x + 50+135+x + 80=720\]
\[ (115 + 75+130+50+135+80)+(x+x+x)=720\]
\[585 + 3x=720\]

Step3: Solve for \(x\)

Subtract 585 from both sides:
\[3x=720 - 585\]
\[3x = 135\]
Divide both sides by 3:
\[x=\frac{135}{3}=45\]

Step4: Find \(m\angle P\)

\(m\angle P=x + 80^{\circ}\). Substitute \(x = 45\):
\[m\angle P=45+80=125^{\circ}\]

Step5: Find \(m\angle R\)

\(m\angle R=x + 75^{\circ}\). Substitute \(x = 45\):
\[m\angle R=45+75 = 120^{\circ}\]

Step6: Find \(m\angle T\)

\(m\angle T=x + 50^{\circ}\). Substitute \(x = 45\):
\[m\angle T=45+50=95^{\circ}\]

Answer:

Sum of the Interior Angles = \(\boldsymbol{720}\)°
\(x=\boldsymbol{45}\)
\(m\angle P=\boldsymbol{125}\)°
\(m\angle R=\boldsymbol{120}\)°
\(m\angle T=\boldsymbol{95}\)°