Proofs with Congruent Triangles
Completion requirements
B. Proofs with Congruent Triangles
The congruence rules you saw in the last section can be used in proofs. Information required for the proof will often be given in the form of a diagram and/or text. Sometimes you will need to draw additional information to complete a proof.
Example 1
Given: BC bisects \(\angle\)ABD
\(\angle\)ACB = \(\angle\)DCB
Prove: \(\triangle\)ABC 
\(\triangle\)DBC
If you can show that there is enough information to satisfy one of the congruence conditions, you will have proven that the two triangles are congruent.
Proof:
| Statement | Justification |
|---|---|
| \(\angle\)ABC = \(\angle\)DBC | BC bisects \(\angle\)ABD (given). |
| \(\angle\)ACB = \(\angle\)DCB | Given. |
| BC = BC |
They are the same line. (This is sometimes called the reflexive property of equality.) |
| \(\triangle\)ABC \(\triangle\)DBC |
ASA (The previous three statements show that two angles and the contained side are equal, which is the ASA congruence.) |