Question

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)

Answer 1

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 2

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)