Question

given: $overline{fb}congoverline{fe}$, $overline{fb}perpoverline{ac}$, $overline{fe}perpoverline{ad}$, f is the mid - point of $overline{cd}$. prove: $\triangle cad$ is isosceles. 1. $overline{fb}perpoverline{ac}$, $overline{fe}perpoverline{ad}$ 2. perpendicular lines intersect to form right angles. 3. $\triangle def$ and $\triangle cbf$ are right triangles. 3. definition of a right triangle

Explanation:

Step1: Identify right - angled triangles

Since $\overline{FB}\perp\overline{AC}$ and $\overline{FE}\perp\overline{AD}$, by the definition that perpendicular lines intersect to form right angles, $\angle FBC = \angle FED=90^{\circ}$. So, $\triangle CBF$ and $\triangle DEF$ are right - triangles.

Step2: Use mid - point property

Given that $F$ is the mid - point of $\overline{CD}$, we have $CF = DF$. Also, given $\overline{FB}\cong\overline{FE}$.

Step3: Prove right - triangle congruence

By the Hypotenuse - Leg (HL) congruence criterion in right - triangles $\triangle CBF$ and $\triangle DEF$ (where the hypotenuse and one leg are equal: $CF = DF$ and $FB = FE$), we have $\triangle CBF\cong\triangle DEF$.

Step4: Get equal angles

Since $\triangle CBF\cong\triangle DEF$, corresponding angles are equal. So, $\angle C=\angle D$.

Step5: Prove isosceles triangle

In $\triangle CAD$, if $\angle C=\angle D$, then by the definition of an isosceles triangle (a triangle with two equal angles is isosceles), $\triangle CAD$ is isosceles.

Answer:

We have proven that $\triangle CAD$ is isosceles.