Question

Given: $\overline{AB}\perp \overline{BF};\overline{HD}\perp \overline{BF}$and $\overline{GF}\perp \overline{BF}$.

$\angle A\cong \angle G$

Which numbered angles must be congruent?

Short answer

The congruent angles are $\angle 1\cong \angle 2,\angle 3\cong \angle 4$and $\angle 4\cong \angle 6$.

Step-by-step solution

Step 1. Apply definition of vertically opposite angles.

Vertically opposite angles are the opposite angles formed due to intersection of two lines.

Consider $\Delta HDC\text{and}\Delta ABC$,

$\angle 1\text{and}\angle 2$are vertically opposite angles, therefore, $\angle 1\cong \angle 2$.

Step 2. Description of step.

$\overline{HD}\perp \overline{BF}$and $\overline{FG}\perp \overline{BF}$implies that $\overline{HD}\left|\right|\overline{FG}$, that is, if two lines are perpendicular to same line then they are parallel to each other.

When two lines are intersected by a transversal then the angles formed on the opposite sides of a transversal with respect to two lines are congruent. Therefore,

$\angle 4\cong \angle 6$

Step 3. Apply the definition of angle bisector.

Angle bisector is a line which divides an angle into two equal parts.

From the figure, it can be observed that $\overline{HD}$bisect $\angle CHE$. Therefore,

$\angle 3\cong \angle 4$