Question

Find the measure of each numbered angle.

Short answer

The angles are:

$\begin{array}{l}\text{m}\angle \mathbf{5}={\mathbf{90}}^{\circ }\\ \text{m}\angle \mathbf{6}={\mathbf{54}}^{\circ }\\ \text{m}\angle \mathbf{7}={\mathbf{36}}^{\circ }\\ \text{m}\angle \mathbf{8}={\mathbf{54}}^{\circ }\end{array}$

Step-by-step solution

- Consider the diagram. 

Here, $AB\parallel CD$.

- Show the calculation.

Since, $AB\parallel CD$,

Here, $m\angle 5=m\angle F=90°$, since they are corresponding angles, where FE is the transversal and AB is parallel to CD.

$m\angle 5+m\angle 6+m\angle GEC=180°$ …… [linear angles]

That implies,

$\begin{array}{c}m\angle 6+{90}^{\circ }+{36}^{\circ }={180}^{\circ }\\ m\angle 6+{126}^{\circ }={180}^{\circ }\end{array}$

Subtract $126°$ from each side, then,

$\begin{array}{c}m\angle 6={180}^{\circ }-{126}^{\circ }\\ m\angle 6={54}^{\circ }\end{array}$

Now, $m\angle 7=m\angle 6=36°$, since they are vertically opposite angles, where GE is the transversal and AB is parallel to CD.

And $m\angle 8=m\angle GEC=54°$, since they are vertically opposite angles, where GE is the transversal and AB is parallel to CD.

- State the conclusion.

Therefore,

$\begin{array}{l}m\angle 5={90}^{\circ }\\ m\angle 6={54}^{\circ }\\ m\angle 7={36}^{\circ }\\ m\angle 8={54}^{\circ }\end{array}$