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?
2, 5, 4, 1, 3, 6
two of these
4, 3, 5, 1, 2, 6
5, 4, 3, 1, 2, 6

Explanation:

Step1: Start with given info

We start with the given facts. First, we state that BD bisects ∠ABC (4) as it is a given condition which is relevant for creating equal - angled parts.

Step2: State the isosceles - triangle given

Then, we state that △ABC is isosceles with vertex angle B (3) as it is the main property of the triangle we are working with.

Step3: Use angle - bisector definition

Next, we use the definition of an angle bisector to get ∠DBA=∠DBC (5) since we already stated that BD is an angle bisector.

Step4: Use isosceles - triangle property

After that, we use the definition of an isosceles triangle to get AB = CB (1) as it is a key property of the isosceles triangle △ABC.

Step5: Use reflexive property

Then, we use the reflexive property of congruence to get BD = BD (2) which is a common side for △ADB and △CDB.

Step6: Prove congruence

Finally, we use the Side - Angle - Side (SAS) congruence criterion to prove that △ADB≅△CDB (6) as we have two sides and the included angle congruent.

Answer:

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