Question

given: △abc; $overline{ca}congoverline{cb}$; $overline{ad}$ and $overline{be}$ are altitudes of △abc. prove: $overline{ad}congoverline{be}$ 1. △abc; $overline{ca}congoverline{cb}$; $overline{ad}$ and $overline{be}$ are altitudes of △abc. 1. given 2. definition of an altitude 3. all right angles are congruent. 4. ∠abc≅∠bac

Explanation:

Step1: State given information

Given $\triangle ABC$ with $\overline{CA}\cong\overline{CB}$ and $\overline{AD},\overline{BE}$ as altitudes.

Step2: Define right - angles

Since $\overline{AD}$ and $\overline{BE}$ are altitudes, $\angle ADB = \angle BEC = 90^{\circ}$.

Step3: Use right - angle congruence

All right - angles are congruent, so $\angle ADB\cong\angle BEC$.

Step4: Identify base - angles of isosceles triangle

In $\triangle ABC$, because $\overline{CA}\cong\overline{CB}$, $\angle ABC\cong\angle BAC$ (base - angles of an isosceles triangle).

Step5: Apply reflexive property

$\overline{AB}$ is congruent to itself, i.e., $\overline{AB}\cong\overline{BA}$.

Step6: Prove triangle congruence

By AAS congruence criterion (two angles and a non - included side), $\triangle ADB\cong\triangle BEC$.

Step7: Use CPCTC

Since the triangles are congruent, corresponding parts are congruent, so $\overline{AD}\cong\overline{BE}$.

Answer:

1. $\triangle ABC; \overline{CA}\cong\overline{CB}; \overline{AD}$ and $\overline{BE}$ are altitudes of $\triangle ABC$. - Given
2. $\angle ADB = \angle BEC=90^{\circ}$ - Definition of an altitude
3. $\angle ADB\cong\angle BEC$ - All right - angles are congruent
4. $\angle ABC\cong\angle BAC$ - Base - angles of an isosceles triangle are congruent ($\triangle ABC$ is isosceles since $\overline{CA}\cong\overline{CB}$)
5. $\overline{AB}\cong\overline{BA}$ - Reflexive property of congruence
6. $\triangle ADB\cong\triangle BEC$ - AAS (Angle - Angle - Side) congruence criterion ($\angle ADB\cong\angle BEC$, $\angle BAD\cong\angle ABE$, $\overline{AB}\cong\overline{BA}$)
7. $\overline{AD}\cong\overline{BE}$ - Corresponding parts of congruent triangles are congruent (CPCTC)