Chapter 1.2, Problem 27E

To determine

To calculate: To find the value of the variable and the find the measure of the polygon

Answer to Problem 27E

Value of variable is $b^{\circ }={90}^{\circ }$

Measure of the polygon is ${540}^{\circ }$

Explanation of Solution

Given: Polygon is

  

Calculation:

Given the polygon as follows:

  

Here, sum of polygon is equal to ${540}^{\circ }$

Thus, equation is given as

  $b^{\circ }+{\left(b+45\right)}^{\circ }+{90}^{\circ }+{\left(2b-90\right)}^{\circ }+\frac{3}{2}{b}^{\circ }={540}^{\circ }$

On solving the equation:

  $\begin{array}{l}{b}^{\circ }+b^{\circ }+{45}^{\circ }+{90}^{\circ }+2b^{\circ }-{90}^{\circ }+\frac{3}{2}{b}^{\circ }={540}^{\circ }\\ \left(b^{\circ }+b^{\circ }+2b^{\circ }+\frac{3}{2}{b}^{\circ }\right)+\left(-{90}^{\circ }+{45}^{\circ }+{90}^{\circ }\right)={540}^{\circ }\\ \frac{11}{2}{b}^{\circ }+{45}^{\circ }={540}^{\circ }\\ \frac{11}{2}{b}^{\circ }={540}^{\circ }-{45}^{\circ }\\ 11b^{\circ }={495}^{\circ }\times 2\\ 11b^{\circ }={990}^{\circ }\\ b^{\circ }={90}^{\circ }\end{array}$

Substitute $b^{\circ }={90}^{\circ }$ in the left side of equation $b^{\circ }+{\left(b+45\right)}^{\circ }+{90}^{\circ }+{\left(2b-90\right)}^{\circ }+\frac{3}{2}{b}^{\circ }={540}^{\circ }$ to find the measure of polygon

  $\begin{array}{l}\Rightarrow {90}^{\circ }+{\left(90+45\right)}^{\circ }+{90}^{\circ }+{\left(2\left(90\right)-90\right)}^{\circ }+\frac{3}{2}{\left(90\right)}^{\circ }\\ \Rightarrow {90}^{\circ }+{135}^{\circ }+{90}^{\circ }+{\left(180-90\right)}^{\circ }+3{\left(45\right)}^{\circ }\\ \Rightarrow {90}^{\circ }+{135}^{\circ }+{90}^{\circ }+{90}^{\circ }+{135}^{\circ }\\ \Rightarrow {540}^{\circ }\end{array}$

Conclusion:

Hence, Value of variable is $b^{\circ }={90}^{\circ }$

Measure of the polygon is ${540}^{\circ }$