Question

In ∆ABC shown below, ∡BAC is congruent to ∡BCA. Given: Base ∡BAC and ∡ACB are congruent.

Prove: ∆ABC is an isosceles triangle.

Construct a perpendicular bisector from point B to line segment AC.
Label the point of intersection between this perpendicular bisector and line segment AC as point D.
m∡BDA and m∡BDC is 90° by the definition of a perpendicular bisector.
∡BDA is congruent to ∡BDC by the definition of congruent angles. Line segment AD is congruent to line segment DC by _______1________.
∆BAD is congruent to ∆BCD by the _______2________. Line segment AB is congruent to line segment BC because corresponding parts of congruent triangles are congruent (CPCTC).
Consequently, ∆ABC is isosceles by definition of an isosceles triangle.

options are:
a)1. Angle-Side-Angle (ASA) Postulate
2. corresponding parts of congruent triangles are congruent (CPCTC)

b) 1. corresponding parts of congruent triangles are congruent (CPCTC)
2. Angle-Side-Angle (ASA) Postulate

c) 1. the definition of a perpendicular bisector
2. Angle-Side-Angle (ASA) Postulate

d) 1. corresponding parts of congruent triangles are congruent (CPCTC)
2. the definition of a perpendicular bisector

Answer 1

Refer to the image attached.

Given:
`\angle BAC` and
`\angle BCA` are congruent.

To Prove:
`\Delta ABC` is an isosceles triangle.

Construction: Construct a perpendicular bisector from point B to line segment AC. Label the point of intersection between this perpendicular bisector and line segment AC as point D.

Proof:

Consider
`\Delta BDA, \Delta BDC`

`\angle BDA = \angle BDC= 90^\circ`

(By the definition of perpendicular bisector)

`AD=DC` (By the definition of perpendicular bisector)

So, Line segment AD is congruent to DC by the definition of perpendicular bisector.

`\angle BAC` =
`\angle BCA` (given)

So,
`\Delta BDA \cong \Delta BDC` by ASA congruence postulate.

∆BAD is congruent to ∆BCD by the ASA congruence Postulate.

Line segment AB is congruent to line segment BC because corresponding parts of congruent triangles are congruent (CPCTC).

So, Option C is the correct answer.