Chapter 6, Problem 15CR

To determine

To calculate: The Measure of $\angle B$ .

Answer to Problem 15CR

Measure of $\angle B$ is $110°$ .

Explanation of Solution

Given information: Here, we have given that ABCD is a parallelogram, $\angle A=(3x+y)°$ , $\angle D=(5x+10)°$ and $\angle C=(5x+20)°$

Formula used : Opposite angles of parallelogram are equal and sum of adjacent angles of parallelogram is $180°$

Calculation: As, $\angle A$ and $\angle C$ are opposite to each other,

  $\begin{array}{l}\Rightarrow \angle A=\angle C\\ \Rightarrow (3x+y)°=(5y+20)°\\ \Rightarrow (3x+y-5y)°=20°\\ \Rightarrow (3x-4y)°=20°\\ \Rightarrow 3x=(20+4y)°\\ \Rightarrow x=\frac{(20+4y)°}{3}\end{array}$

Now, as $\angle D$ and $\angle C$ are adjacent to each other,

  $\begin{array}{l}\Rightarrow \angle D+\angle C=180°\\ \Rightarrow (5x+10)°+(5y+20)°=180°\\ \Rightarrow (5x+5y+30)°=180°\\ \Rightarrow 5(x+y+6)°=180°\\ \Rightarrow (x+y+6)°=36°\end{array}$

Now, putting value of $x$ in above equation, we get,

  $\begin{array}{l}\Rightarrow (x+y+6)°=36°\\ \Rightarrow \left(\frac{20+4y}{3}+y+6\right)=36°\\ \Rightarrow \frac{(20+4y+3y)°}{3}=30°\\ \Rightarrow (20+7y)°=3(30)°\\ \Rightarrow (20+7y)°=90°\\ \Rightarrow 7y=90°-20°\\ \Rightarrow 7y=70°\\ \Rightarrow y=10°\end{array}$

Now, putting value of y in $x=\frac{(20+4y)°}{3}$ , we get

  $\begin{array}{l}\Rightarrow x=\frac{[20+4(10)]°}{3}\\ \Rightarrow x=\frac{(20+40)°}{3}\\ \Rightarrow x=\frac{60°}{3}\\ \Rightarrow x=20°\end{array}$

  $\therefore \angle D=(5x+10)°=[5(20)+10]°=(100+10)°=110°$

As, $\angle D$ and $\angle B$ are opposite to each other,

  $\begin{array}{l}\Rightarrow \angle D=\angle B\\ \Rightarrow \angle B=110°\end{array}$