Question

proof # 1
given: ( overline{ab} cong overline{ac}, overline{bc} perp overline{ad} )
prove: ( \triangle bda cong \triangle cda )
( overline{ab} cong overline{ac} ) sas ( cong ) theorem ( overline{ad} cong overline{ad} ) hl ( cong ) theorem
definition of ( perp ) lines definition of midpoint given ( overline{bc} perp overline{ad} )
reflexive property ( angle bda ) and ( angle cda ) are right angles ( \triangle bda cong \triangle cda )

Explanation:

Step1: State given information

1. $\overline{AB}\cong\overline{AC}$ - Given
2. $\overline{BC}\perp\overline{AD}$ - Given

Step2: Use definition of perpendicular lines

Since $\overline{BC}\perp\overline{AD}$, by the Definition of $\perp$ lines, $\angle BDA$ and $\angle CDA$ are right angles.

Step3: Use reflexive property

$\overline{AD}\cong\overline{AD}$ - Reflexive Property (a segment is congruent to itself)

Step4: Apply HL (Hypotenuse - Leg) congruence theorem

We have two right - angled triangles ($\triangle BDA$ and $\triangle CDA$) where the hypotenuse ($\overline{AB}\cong\overline{AC}$) and one leg ($\overline{AD}\cong\overline{AD}$) are congruent. So, by the $HL\cong$ Theorem.

Step5: Conclude triangle congruence

$\triangle BDA\cong\triangle CDA$

Answer:

1. $\overline{AB}\cong\overline{AC}$ - Given
2. $\overline{BC}\perp\overline{AD}$ - Given
3. $\angle BDA$ and $\angle CDA$ are right angles - Definition of $\perp$ lines
4. $\overline{AD}\cong\overline{AD}$ - Reflexive Property
5. $\triangle BDA\cong\triangle CDA$ - $HL\cong$ Theorem