Question

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 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.

AO bisects BAC

AO is perpendicular to BC

Point O is the midpoint of BC

AO is an altitude.

answer choices are:
Needed
Not needed

Answer 1

I did this:

1. Needed

2. Needed

3. Not Needed

4. Needed

Explanation:

Answer 2

AO bisects BAC - Needed

AO is perpendicular to BC - Needed

Point O is the midpoint of BC - Not needed

AO is an altitude - Not needed

Explanation:

In ΔAOB,

• <BOA = 90° (since AO is perpendicular to BC )

• <BAO =
  `(1)/(2)`<BAC (since AO bisects BAC)

In ΔAOC,

• <COA = 90° (since AO is perpendicular to BC )

• <CAO =
  `(1)/(2)`<BAC (since AO bisects BAC)

In ΔAOB and ΔAOC,

• <BOA = <COA (since both are 90°)
• <BAO = CAO (since AO bisects BAC)
• AO=AO

So, we can conclude ΔAOB ≅ ΔAOC (ASA property) and hence the base angles <ABO and < ACO of the isosceles triangle ABC are also congruent.