Question

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

Explanation:

Step1: Define right - angled triangles

Since $\overline{AD}\perp\overline{AF}$ and $\overline{FC}\perp\overline{CH}$, $\triangle DAF$ and $\triangle HCF$ are right - angled triangles. So, $\angle A = 90^{\circ}$ and $\angle C=90^{\circ}$, and we can write $\angle A\cong\angle C$ because all right angles are congruent.

Step2: Use vertical angles property

Given that $\angle1$ and $\angle2$ are vertical angles. By the vertical angles theorem, vertical angles are congruent, so $\angle1\cong\angle2$.

Step3: Apply the angle - angle (AA) similarity criterion

In $\triangle DAF$ and $\triangle HCF$, we have $\angle A\cong\angle C$ and $\angle1\cong\angle2$. By the AA similarity criterion, $\triangle DAF\sim\triangle HCF$.

Step4: Prove the congruence of the desired angles

Since $\triangle DAF\sim\triangle HCF$, corresponding angles of similar triangles are congruent. So, $\angle D\cong\angle H$.

Answer:

1. $\angle A = 90^{\circ},\angle C = 90^{\circ}$; 3. All right angles are congruent.