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 property that perpendicular lines intersect to form right angles, $\angle FBC = \angle FED=90^{\circ}$. So, $\triangle CBF$ and $\triangle DEF$ are right - triangles by the definition of a right - triangle.

Step2: Use mid - point property

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

Step3: Prove right - triangle congruence

We know that $\overline{FB}\cong\overline{FE}$. In right - triangles $\triangle CBF$ and $\triangle DEF$, we have $CF = DF$ and $FB = FE$. By the Hypotenuse - Leg (HL) congruence criterion for right - triangles, $\triangle CBF\cong\triangle DEF$.

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 Isosceles Triangle Theorem (if two angles of a triangle are equal, then the sides opposite those angles are equal), $CA = DA$. So, $\triangle CAD$ is isosceles.

Answer:

$\triangle CAD$ is isosceles.