Chapter 5.3, Problem 22PSB

To determine

To Find: The value of $m\angle 1$ .

Answer to Problem 22PSB

  $m\angle 1=70°$

Explanation of Solution

Given:

  

  $\begin{array}{l}a\parallel b\\ \angle 1=\left(x+3y\right)°\\ \angle 2=\left(2x+30\right)°\\ \angle 3=\left(5y+20\right)°\end{array}$

Concept Used:

When two lines are parallel

• Corresponding angles are equal
• Interior angles on the same side of transversal are supplementary.

Calculation:

Here, we have

  $\begin{array}{l}a\parallel b\\ \angle 1\text{ and }\angle 3\text{ are corresponding angles}\\ \angle 1=\angle 3\\ \Rightarrow \left(x+3y\right)°=\left(5y+20\right)°\\ \Rightarrow x-2y=20\left(1\right)\\ \angle 2\text{ and }\angle 3\text{ are interior angles on the same side of transversal}\\ \Rightarrow \angle 2+\angle 3=180°\\ \Rightarrow \left(2x+30\right)°+\left(5y+20\right)°=180°\\ \Rightarrow 2x+5y=180-50\\ \Rightarrow 2x+5y=130\left(2\right)\\ \text{Operate 2}\times \left(1\right)-\left(2\right)\\ \Rightarrow -9y=-90\\ \Rightarrow y=10\left(3\right)\\ \text{Using }\left(3\right)\text{in}\left(2\right)\\ x-2\times 10=20\\ \Rightarrow x=40\\ \Rightarrow m\angle 1=\left(40+3\times 10\right)°=70°\\ \therefore m\angle 1=70°\end{array}$