Chapter 12, Problem 10PTTS

To determine

To find: The measure of $\angle 1$ .

Answer to Problem 10PTTS

The value of $\angle 1$ is ${125}^{\circ }$ .

Explanation of Solution

Given information:

  

Calculation: The given angle ${55}^{\circ }\text{ and }\angle \text{2}$ are congruent angles which are equal in measure, also $\angle 1\text{ and }\angle \text{3}$ are equal and opposite.

Therefore, $\angle 2={55}^{\circ }$ and $\angle 1=\angle 3$

Sum of the four angles of the given figure is ${360}^{\circ }$ , because they completes a circle.

Hence, $\angle 1+\angle 2+\angle 3+{55}^{\circ }={360}^{\circ }$$\mathrm{......}(1)$

Let’s find the value of $\angle 1$ from equation $(1)$ ,

  $\begin{array}{c}\angle 1+\angle 2+\angle 3+{55}^{\circ }={360}^{\circ }\\ \angle 1+{55}^{\circ }+\angle 1+{55}^{\circ }={360}^{\circ }\\ 2\angle 1={360}^{\circ }-{110}^{\circ }\\ \angle 1=\frac{{250}^{\circ }}{2}\\ \angle 1={125}^{\circ }\end{array}$

Hence, The value of $\angle 1$ is ${125}^{\circ }$ .