Question

Please help! Due in 10 minutes! 100 Points!

Options for 3) BAD is congruent to BCD, ABD is congruent to CBD
Options for 5) ASA, SAS

Answer 1

3. m 4BAD = m 4DAC

5. ASA axiom of congruence

For every isosceles triangle.

Answer 2

3. m ∡BAD ≅ m ∡DAC

5. ASA axiom of congruence

For every isosceles triangle.

Explanation:

In ΔABC

Statement (Reason in bracket)

1. AB ≅ BC (Given)

2. Construct BD as the angle bisector of ∡ ABC. where point D is on AC (We can construct a bisector for any angle)

3. m ∡BAD ≅ m ∡DAC (Definition of angle bisector)

4. BD ≅ BD(Reflexive property)

5. Δ DAB ≅ ΔDCB (By SAS axiom)

6. ∡A≅∡C (Corresponding parts of congruent triangles are congruent.)

The bisector of the vertex angle of an isosceles triangle bisects the base of the triangle equally. This is because the angle bisector divides the vertex angle into two angles with equal measures, and the base angles are already congruent.

So, we prove the conclusion true for every isosceles triangle.

Note:

Definition of an angle bisector of an isosceles triangle:

An angle bisector of an isosceles triangle is a line segment that divides one of the angles of the triangle into two angles with equal measures.