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: State given information

Given $\overline{FB}\cong\overline{FE},\overline{FB}\perp\overline{AC},\overline{FE}\perp\overline{AD}$, F is mid - point of $\overline{CD}$

Step2: Define right angles

Since $\overline{FB}\perp\overline{AC}$ and $\overline{FE}\perp\overline{AD}$, $\angle FBC = \angle FED = 90^{\circ}$

Step3: Identify right - triangles

$\triangle DEF$ and $\triangle CBF$ are right - triangles by definition of right - triangle

Step4: Use mid - point property

$CF = DF$ as F is the mid - point of $\overline{CD}$

Step5: Recall given side equality

$FB = FE$ (Given)

Step6: Prove triangle congruence

By HL congruence criterion (Hypotenuse - Leg), $\triangle CBF\cong\triangle DEF$

Step7: Find equal angles

Corresponding parts of congruent triangles are equal, so $\angle C=\angle D$

Step8: Prove isosceles triangle

Since $\angle C=\angle D$ in $\triangle CAD$, $\triangle CAD$ is isosceles.

Answer:

1. Given
2. $\angle FBC = \angle FED=90^{\circ}$
3. Given
4. $CF = DF$ (Since F is the mid - point of CD)
5. $FB = FE$ (Given)
6. $\triangle CBF\cong\triangle DEF$ (HL congruence criterion, Hypotenuse - Leg, as $FB = FE$ and $CF = DF$)
7. $\angle C=\angle D$ (Corresponding parts of congruent triangles are equal)
8. $\triangle CAD$ is isosceles (If two angles of a triangle are equal, the triangle is isosceles)