Chapter 1.2, Problem 19PSC

To determine

To Find : Whether $\angle ABD$ is a straight angle?

Explanation of Solution

Given information :

  

  $\begin{array}{l}m\angle ABC=m\angle CBD\\ m\left(\angle ABC\right)=\left(\frac{3x}{2}+2\right)°\\ m\left(\angle CBD\right)=\left(2x-29\frac{1}{4}\right)°\end{array}$

Concept used : Measure of straight angle is $180°$

Proof:

Here, we have

  $\begin{array}{l}m\angle ABC=m\angle CBD\\ m\left(\angle ABC\right)=\left(\frac{3x}{2}+2\right)°\\ m\left(\angle CBD\right)=\left(2x-29\frac{1}{4}\right)°\\ \Rightarrow \left(\frac{3x}{2}+2\right)°=\left(2x-29\frac{1}{4}\right)°\\ \Rightarrow -\frac{x}{2}=\left(-2-29\frac{1}{4}\right)°\\ \Rightarrow -\frac{x}{2}=\left(-\frac{125}{4}\right)°\\ \Rightarrow x=\left(\frac{125}{2}\right)°\\ \Rightarrow x=\left(62\frac{1}{2}\right)°\\ \Rightarrow \angle ABC+\angle CBD=\angle ABD\\ \Rightarrow \angle ABC+\angle CBD=\left(\frac{3}{2}\times \frac{125}{2}+2\right)°+\left(2\times \frac{125}{2}-29\frac{1}{4}\right)°\\ \Rightarrow \angle ABC+\angle CBD=\left(\frac{383}{4}\right)°+\left(125-\frac{117}{4}\right)°\\ \Rightarrow \angle ABC+\angle CBD=\left(\frac{383}{4}\right)°+\left(\frac{500-117}{4}\right)°\\ \Rightarrow \angle ABC+\angle CBD=\left(\frac{383}{4}\right)°+\left(\frac{383}{4}\right)°\\ \Rightarrow \angle ABC+\angle CBD=\left(\frac{383}{2}\right)°\ne 380°\end{array}$