Question

given that △abc is isosceles with vertex angle b, and bd bisects ∠abc, prove that △adb≅△cdb.
statement\treason

1. ab = cb\tdefinition of isosceles triangle
2. bd = bd\treflexive property of congruence
3. △abc is isosceles with vertex angle b\tgiven
4. bd bisects ∠abc\tgiven
5. ∠dba = ∠dbc\tdefinition of angle - bisector
6. △adb≅△cdb\tsas

the proof shown is written in an incorrect order. what is a possible order that the statements and reasons could be written in?
4, 5, 1, 3, 2, 5
2, 4, 3, 5, 6, 1
3, 1, 4, 5, 2, 6
two of these

Explanation:

Step1: Start with given info

First, state that $\triangle ABC$ is isosceles with vertex - angle $B$ (Statement 3) as it is given.

Step2: Use isosceles - triangle def.

Then, by the definition of an isosceles triangle, $AB = CB$ (Statement 1).

Step3: State angle - bisector given

Next, state that $BD$ bisects $\angle ABC$ (Statement 4) as it is given.

Step4: Use angle - bisector def.

Then, by the definition of an angle bisector, $\angle DBA=\angle DBC$ (Statement 5).

Step5: Use reflexive property

After that, $BD = BD$ by the reflexive property of congruence (Statement 2).

Step6: Prove triangles congruent

Finally, conclude that $\triangle ADB\cong\triangle CDB$ by the Side - Angle - Side (SAS) congruence criterion (Statement 6).

Answer:

C. 3, 1, 4, 5, 2, 6