Given: segment BD bisects angle ABC; AB=BC. Prove: triangle ABD congruent to triangle CBD

Can you prove it by side angle side?

AB = BC

BD = BD it is a shared side

Angle ABD = Angle CBD

We have 2 sides and the angle between those 2 sides.

I drew a picture.

To prove that triangle ABD is congruent to triangle CBD, we can use the angle-side-angle (ASA) congruence criterion. We will show that triangle ABD and triangle CBD have congruent angles and a congruent side.

1. Given: Segment BD bisects angle ABC
AB = BC

2. To prove: Triangle ABD is congruent to Triangle CBD

Proof:

Step 1: Angle ABD is congruent to angle CBD
- By the definition of angle bisector, the angle bisector cuts the angle into two congruent angles.
- Therefore, angle ABD is congruent to angle CBD.

Step 2: Side AB is congruent to side BC
- Given that AB = BC.

Step 3: Angle B is congruent to itself (Reflexive property)

By the ASA congruence criterion, we have shown that triangle ABD is congruent to triangle CBD.

To prove the congruence of triangles ABD and CBD, we need to show that they have three congruent parts: two sides and the included angle. Here's how we can approach the proof:

1. Draw a clear diagram: Sketch triangle ABC, where AB = BC. Mark point D on side AC. Segment BD should bisect angle ABC such that angles ABD and CBD are congruent.

2. State the given information: In this case, we're given that segment BD bisects angle ABC and AB = BC.

3. Identify the parts to prove congruent: To prove that triangles ABD and CBD are congruent, we need to show that the following parts are congruent:
a. AB = CB (Common side)
b. BD = BD (Common side)
c. angle ABD = angle CBD (Included angle)

4. Prove congruence of sides:
a. AB = CB: Since AB = BC (given), this side is already congruent.

b. BD = BD: Both segments are the same segment, so they are congruent by the reflexive property of congruence.

5. Prove congruence of angles:
a. angle ABD = angle CBD: Since segment BD bisects angle ABC (given), angles ABD and CBD are congruent.

6. Conclude using the Side-Angle-Side (SAS) congruence criterion: By proving that AB = CB, BD = BD, and angle ABD = angle CBD, we have shown that triangles ABD and CBD are congruent according to the SAS criterion.

Therefore, triangle ABD is congruent to triangle CBD.