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. (4 points) 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 by the definition of a perpendicular bisector. ΔBAD is congruent to ΔBCD by the _______1________. Line segment AB is congruent to Line segment BC because _______2________. Consequently, ΔABC is isosceles by definition of an isosceles triangle. 1. corresponding parts of congruent triangles are congruent (CPCTC) 2. the definition of a perpendicular bisector 1. the definition of a perpendicular bisector 2. the definition of congruent angles 1. the definition of congruent angles 2. the definition of a perpendicular bisector 1. Angle-Side-Angle (ASA) Postulate 2. corresponding parts of congruent triangles are congruent (CPCTC)

Answer 1

Answer: 1. Angle-Side-Angle (ASA) Postulate

2. Corresponding parts of congruent triangles are congruent (CPCTC)

Step-by-step explanation:

Given : ABC is a triangle,

In which, ∠BAC ≅ ∠BCA

To prove : Δ ABC is an isosceles triangle,

Proof:

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

∠BAC ≅ ∠BCA                   ( Given )

Since, ∠BDA ≅ ∠BDC       ( Right angles )

AD ≅ DC               (By the definition of a perpendicular bisector. )

Thus, By ASA postulate of congruence,

Δ BAD ≅ Δ BCD

⇒ AB ≅ BC ( By CPCTC )

⇒ ΔABC is isosceles ( by definition of an isosceles triangle )