Question

Given: angle A is congruent to angle C



and



segment B D is drawn such that angle A B D is congruent to angle C B D





Prove: triangle A B C is isosceles

A diagram shows triangle A B C with segment B D.
Question
Move options to the blanks to complete the proof.

Response area with 5 blank spaces
In triangle A B C, it is given that angle A is congruent to angle C and angle A B D is congruent to angle C B D.


By the reflexive property,
Blank space 4 empty
is congruent to
Blank space 6 empty
. Then, triangle A B D is congruent to triangle C B D by
Blank space 7 empty
.



Therefore,
Blank space 8 empty
is congruent to
Blank space 9 empty
because corresponding parts of congruent triangles are congruent.



By definition, triangle A B C is isosceles.

Answer options with 8 options.

Answer 1

Let's fill in the blanks with appropriate answers to complete the proof for triangle ABC being isosceles.

1. **Blank space 4:** \( \overline{BD} \)
2. **Blank space 6:** \( \overline{BD} \)
3. **Blank space 7:** the Side-Angle-Side (SAS) Postulate
4. **Blank space 8:** \( \overline{AB} \)
5. **Blank space 9:** \( \overline{CB} \)

The completed proof is as follows:

In triangle ABC, it is given that angle A is congruent to angle C and angle ABD is congruent to angle CBD.

By the reflexive property, \( \overline{BD} \) is congruent to \( \overline{BD} \). Then, triangle ABD is congruent to triangle CBD by the Side-Angle-Side (SAS) Postulate.

Therefore, \( \overline{AB} \) is congruent to \( \overline{CB} \) because corresponding parts of congruent triangles are congruent.

By definition, triangle ABC is isosceles.