Question

In the diagram $\overleftrightarrow{AB}$bisects $\angle DHF,\overleftrightarrow{AB}\perp \overline{GH},\overleftrightarrow{AB}\parallel \overleftrightarrow{CD}$, and $m\angle AHD={\mathbf{60}}^{\circ }$. Find the measure of each angle.

$\angle FHG$

Short answer

The angle $\angle FHG$ is${\mathbf{30}}^{\circ }$

Step-by-step solution

Step 1. Observe the diagram.

Here, $\angle AHD={60}^{\circ },\overleftrightarrow{AB}\perp \overline{GH},\overleftrightarrow{AB}\parallel \overleftrightarrow{CD}$and $\overleftrightarrow{AB}$bisects $\angle DHF$.

Step 2. Show the calculation.

From the figure,

$\overleftrightarrow{AB}$bisects $\angle DHF$

$$\begin{gathered} \angle AHD=\angle AHF={60}^{\circ } \\ \angle AHG=\angle AHF+\angle FHG \\ {90}^{\circ }={60}^{\circ }+\angle FHG \\ \angle FHG={30}^{\circ } \end{gathered}$$

Hence the angle$\angle FHG={30}^{\circ }$

Step 3. State the conclusion.

Therefore, $\angle FHG={30}^{\circ }$.