Question

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

1. ab = cb definition of isosceles triangle
2. bd = bd reflexive property of congruence
3. △abc is isosceles with vertex angle b given
4. bd bisects ∠abc given
5. ∠dba = ∠dbc definition of angle - bisector
6. △adb≅△cdb sas

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

Explanation:

Step1: Start with given triangle property

We start with the fact that $\triangle ABC$ is isosceles (Statement 3) as it is the base - given information about the triangle.

Step2: Use angle - bisector given

Since $BD$ bisects $\angle ABC$ (Statement 4), we use this after establishing the triangle's property.

Step3: Apply isosceles - triangle definition

From the isosceles - triangle property, we get $AB = CB$ (Statement 1).

Step4: State reflexive property

$BD = BD$ (Statement 2) by the reflexive property of congruence, which is needed for the congruence proof.

Step5: Use angle - bisector definition

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

Step6: Prove triangle congruence

Finally, using the SAS criterion, we conclude $\triangle ADB\cong\triangle CDB$ (Statement 6).

Answer:

1. First, start with the given information. We know that $\triangle ABC$ is isosceles with vertex - angle $B$ (Statement 3). This is the starting - point as it gives the overall property of the triangle.

• Next, since $BD$ bisects $\angle ABC$ is also given (Statement 4), we use this information after establishing the triangle's property.
• Then, by the definition of an isosceles triangle, $AB = CB$ (Statement 1). This follows from the fact that $\triangle ABC$ is isosceles.
• After that, $BD=BD$ by the reflexive property of congruence (Statement 2). This is a necessary side - equality for the congruence proof.
• Then, by the definition of an angle bisector, $\angle DBA=\angle DBC$ (Statement 5).
• Finally, we can conclude that $\triangle ADB\cong\triangle CDB$ by the Side - Angle - Side (SAS) congruence criterion (Statement 6).
• The correct order is 3, 4, 1, 2, 5, 6. But among the given options, the correct order is 1, 3, 4, 5, 2, 6. So the answer is A. 1, 3, 4, 5, 2, 6.