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

corresponding parts of congruent triangles are congruent (CPCTC)

B1. corresponding parts of congruent triangles are congruent (CPCTC)

Angle-Side-Angle (ASA) Postulate

C1. the definition of a perpendicular bisector

Angle-Side-Angle (ASA) Postulate

D1. corresponding parts of congruent triangles are congruent (CPCTC)

the definition of a perpendicular bisector

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.