the provided diagram of triangle abc will help you to prove that the base angles of an isosceles triangle are congruent. the first step is to draw auxiliary line ao. what must be true about $ \overline{ao} $ in order to complete the proof using the asa (angle side angle) triangle congruency theorem? classify each statement as needed or not needed to complete the proof. $ \overline{ao} $ bisects $ \angle bac $ $ \overline{ao} $ is perpendicular to $ \overline{bc} $ point o is the midpoint of $ \overline{bc} $ $ \overline{ao} $ is an altitude.

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Accepted Answer

# Explanation:
## Step1: Analyze the ASA (Angle - Side - Angle) requirement
For two triangles $\triangle ABO$ and $\triangle ACO$ in isosceles triangle $ABC$ ($AB = AC$), we need two angles and the included side to be congruent.
We know $AB=AC$. If $\overline{AO}$ bisects $\angle BAC$, then $\angle BAO=\angle CAO$. Also, $AO = AO$ (common side). So, by ASA ($\angle BAO=\angle CAO$, $AO = AO$, $AB = AC$), $\triangle ABO\cong\triangle ACO$, and then $\angle B=\angle C$.
## Step2: Analyze other statements
- If $\overline{AO}$ is perpendicular to $\overline{BC}$ (or $\overline{AO}$ is an altitude), we get right - angles, but for ASA,  we don't need right - angles. We need the angle bisector condition for the non - right angles.
- If point $O$ is the mid - point of $\overline{BC}$ ($BO = CO$), we would be using SSS (Side - Side - Side) congruence ($AB = AC$, $BO=CO$, $AO = AO$) not ASA.

# Answer:
$\overline{AO}$ bisects $\angle BAC$: Needed
$\overline{AO}$ is perpendicular to $\overline{BC}$: Not needed
Point $O$ is the midpoint of $\overline{BC}$: Not needed
$\overline{AO}$ is an altitude: Not needed

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze the ASA (Angle - Side - Angle) requirement

Step2: Analyze other statements

Answer: