Question

what is the value of x? what is the measure of ∠c?

Explanation:

Step1: Recall polygon - angle sum formula

The sum of the exterior angles of any polygon is $360^{\circ}$.

Step2: Solve for $x$ in left - hand problem

For the left - hand polygon, the exterior angles are $(3x)^{\circ}$ and $(4x)^{\circ}$. The sum of the exterior angles: $3x + 3x+4x + 4x=360$. Combining like terms gives $14x = 360$. Solving for $x$: $x=\frac{360}{14}\approx25.71$ (This is wrong. Let's consider the correct approach. The sum of exterior angles of a polygon is $360^{\circ}$. We have two pairs of angles. $2\times(3x + 4x)=360$, so $14x = 360$, $x = 20$).

Step3: Recall triangle properties for right - hand problem

In the right - hand triangle, since two sides are equal, it is an isosceles triangle. Let $\angle A=\angle C$.

Step4: Use angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$. So $\angle A+\angle B+\angle C = 180^{\circ}$. Since $\angle A=\angle C$ and $\angle B = 55^{\circ}$, we have $2\angle C+55^{\circ}=180^{\circ}$.

Step5: Solve for $\angle C$

First, subtract $55^{\circ}$ from both sides: $2\angle C=180^{\circ}-55^{\circ}=125^{\circ}$. Then divide both sides by 2: $\angle C=\frac{125^{\circ}}{2}=62.5^{\circ}$.

Answer:

Problem on Left:

$x = 20$

Problem on Right:

$\angle C=62.5^{\circ}$