Question

given: $overline{ad}perpoverline{af}$, $overline{fc}perpoverline{ch}$, $angle1$ and $angle2$ are vertical angles. prove: $angle dcongangle h$

Explanation:

Step1: Recall vertical angle property

Vertical angles $\angle1$ and $\angle2$ are congruent by the vertical - angle theorem.

Step2: Restate given perpendiculars

The given perpendiculars $\overline{AD}\perp\overline{AF}$ and $\overline{FC}\perp\overline{CH}$ mean $\angle DAF = 90^{\circ}$ and $\angle HCF=90^{\circ}$.

Step3: Use AA similarity for right - triangles

In $\triangle DAF$ and $\triangle HCF$, with $\angle A\cong\angle C$ and $\angle DAF=\angle HCF = 90^{\circ}$, by AA similarity for right - triangles, $\angle D\cong\angle H$.

Answer:

1. Reason: Vertical angles are congruent.
2. Statement: $\overline{AD}\perp\overline{AF},\overline{FC}\perp\overline{CH}$ (already given in the overall problem - likely this is just to restate for the flow of the proof).
3. Statement: $\angle D\cong\angle H$

Reason: In right - triangles $\triangle DAF$ and $\triangle HCF$, since $\angle A\cong\angle C$ (from step 3) and $\angle DAF=\angle HCF = 90^{\circ}$ (from the perpendicular given statements), by the Angle - Angle (AA) similarity criterion for right - triangles, the third pair of angles must be congruent. In right - triangles, if two non - right angles are congruent, the remaining non - right angles are also congruent. So $\angle D\cong\angle H$.