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 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.

corresponding parts of congruent triangles are congruent (CPCTC)

the definition of a perpendicular bisector

the definition of a perpendicular bisector

the definition of congruent angles

the definition of congruent angles

the definition of a perpendicular bisector

Angle-Side-Angle (ASA) Postulate

corresponding parts of congruent triangles are congruent (CPCTC)

Blank 1: ASA Postulate
Blank 2: CPCTC

Based on the info. your given and I drew (screenshot). Blank 1 is ASA Post. and Blank 2 is CPCTC because of ASA Post.

i am so sorry!
I meant to draw line segment AD is congruent to AC.... However Line segment AB is congruent to BC....

Answer Link

pinquancaro pinquancaro · Answer 2 · 2018-10-17T01:42:11-04:00

Refer to the image attached.

Given: \(\angle BAC\) and \(\angle ACB\) are congruent.

To Prove: \(\Delta\)ABC is an isosceles triangle.

Construction: Construct a perpendicular bisector from point B to Line segment AC.

Consider triangle ABD and BDC,

\(\angle BAD = \angle BCD\) (given)

\(\angle ADB = \angle BDC = 90^\circ\)

(By the definition of a perpendicular bisector)

\(AD=DC\) (By the definition of a perpendicular bisector)

Therefore, \(\Delta ABD \cong \Delta BDC\) by Angle Side Angle(ASA) Postulate.

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