Question

In ΔABC shown below, ∠BAC is congruent to ∠BCA: Triangle ABC, where angles A and C are congruent Given: Base ∠BAC and ∠ACB are congruent. Prove: ΔABC is an isosceles triangle. When completed (fill in the blanks), the following paragraph proves that Line segment AB is congruent to Line segment BC making ΔABC 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 _______1________. Line segment AD is congruent to Line segment DC by _______2________. ΔBAD is congruent to ΔBCD by the Angle-Side-Angle (ASA) Postulate. Line segment AB is congruent to Line segment BC because corresponding parts of congruent triangles are congruent (CPCTC). Consequently, ΔABC is isosceles by definition of an isosceles triangle.

A.)1. the definition of congruent angles
2. Angle-Side-Angle (ASA) Postulate

B.) 1. the definition of congruent angles
2. the definition of a perpendicular bisector

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

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

Answer 1

1. the definition of congruent angles

2. the definition of a perpendicular bisector

Explanation:

Answer 2

Explanation:

Consider the triangle ABC, Construct a perpendicular bisector from point B to Line segment AC and label it as D.

Thus, From ΔBAD and ΔBCD, we have

∠BDA=∠BDC( definition of congruent angles)

AD=CD(definition of perpendicular bisector)

∠BAD=∠BCD (given)

Thus, by ASA rule,

ΔBAD ≅ ΔBCD.

Therefore, by CPCTC, AB=BC and consequently, ΔABC is isosceles by definition of an isosceles triangle.

Thus,option B is correct that is 1. the definition of congruent angles

2. the definition of a perpendicular bisector