Question

Find the measures of $\angle \mathbf{ABC}$ and $\angle \mathbf{CBD}$.

Short answer

The measures of $\angle ABC$and $\angle CBD$ are $\mathbf{111}\mathbf{°}$and $\mathbf{69}\mathbf{°}$ respectively.

Step-by-step solution

­- Observe the given diagram.

The given diagram is:

­- Description of step.

From the given diagram it can be noticed that:

$m\angle ABC=x°$and $m\angle CBD=\left(\frac{2}{3}x-5\right)°$

From the given diagram it can be noticed that the angles $\angle ABC$and $\angle CBD$forms the linear pair.

Therefore, the sum of the angles $\angle ABC$and $\angle CBD$ is $180°$.

That implies, $\angle ABC+\angle CBD=180$.

Therefore, it can be obtained that:

$\begin{array}{c}\angle ABC+\angle CBD=180\\ x+\left(\frac{2}{3}x-5\right)=180\\ x+\frac{2}{3}x-5=180\\ x+\frac{2}{3}x=180+5\\ \frac{3x+2x}{3}=185\\ \frac{5x}{3}=185\\ x=\frac{185\times 3}{5}\\ x=37\times 3\\ x=111\end{array}$

­- Find the measure of .

Substitute 111 for $x$in $m\angle ABC=x°$.

Therefore, it can be obtained that:

$m\angle ABC=111°$

Therefore, the measure of $\angle ABC$ is $111°$.

­- Find the measure of .

Substitute 111 for $x$in $m\angle CBD=\left(\frac{2}{3}x-5\right)°$

Therefore, it can be obtained that:

$\begin{array}{c}m\angle CBD=\left(\frac{2}{3}\left(111\right)-5\right)°\\ =\left(2\times 37-5\right)°\\ =\left(74-5\right)°\\ =\left(69\right)°\\ =69°\end{array}$

Therefore, the measure of $\angle CBD$ is $69°$.