Question

Use the figure below to find m< GBA. show all work

Answer 1

Let's begin by listing out the information given to us:

$\begin{gathered} m\angle ABC=102^{\circ} \\ m\angle BCA=44^{\circ}(opposite\text{ }angles) \end{gathered}$

The sum of angles in a triangle (Triangle ABC) is 180 degrees

$\begin{gathered} \angle ABC+\angle BCA+\angle CAB=180^{\circ} \\ \angle ABC=102^{\circ},\angle BCA=44^{\circ} \\ 102+44+\angle CAB=180 \\ \angle CAB=180-(102+44)=180-146 \\ \angle CAB=34^{\circ} \end{gathered}$

To solve for angle GBA, we will use the Trapezium GBCA

The sum of angles in a trapezium add up to 360 degrees. Given by:

$\begin{gathered} \angle AGB+\angle CAG+\angle BCA+\angle CBG=360^{\circ} \\ \angle AGB=90^{\circ},\angle CAG=90^{\circ},\angle BCA=44^{\circ},\angle CBG=\text{?} \\ 90+90+44+\angle CBG=360 \\ \angle CBG=360-(90+90+44) \\ \angle CBG=360-224=136 \\ \angle CBG=136^{\circ} \\ \\ \angle CBG=\angle GBA+\angle ABC \\ \angle CBG=136^{\circ},\angle ABC=102^{\circ},\angle GBA=? \\ 136=\angle GBA+102 \\ \angle GBA=136-102=34 \\ \therefore\angle GBA=34^{\circ} \end{gathered}$