Given: line AB is congruent to line AC, Angle BAD is congruent to angle CAD.

Prove: line AD bisects BC

Picture: An upside down triangle divided inhalf to form two triangles Angle BAD and angle CAD. They share a common side of AD. and point D bisects line BC. and Angle c and Angle B are corresponding and congruent.

what i have so far
given info
Line DA is congruent to Line DA by Reflexive Prop or EQuality
angle C is congruent to angle B by the Corresponding parts of congruent triangles are congruen. CPCTC.
but i do not know where to go from that..

To prove that line AD bisects line BC, you can use the congruence of segments and angles in the given triangle. Here's a step-by-step explanation of how to complete the proof:

1. Given: Line AB is congruent to line AC (AB ≅ AC) and angle BAD is congruent to angle CAD (BAD ≅ CAD).
2. Draw the triangle ABC with line AD intersecting BC at point D.
3. We need to prove that line AD bisects BC, meaning that BD ≅ DC.
4. To prove that, we can use the Triangle Congruence Postulate (SSS, SAS, ASA) along with properties of congruent angles (CPCTC).

Proof:

Step 1: AB ≅ AC (Given)
Step 2: Angle BAD ≅ Angle CAD (Given)

Step 3: BD ≅ DC (To be proven)

Step 4: Consider triangle BDA and triangle CDA
Step 5: AD ≅ AD (Reflexive Property of Congruence)

Step 6: AB ≅ AC (Given)
Step 7: Angle BAD ≅ Angle CAD (Given)
Step 8: Angle BDA ≅ Angle CDA (Corresponding Parts of Congruent Triangles are Congruent - CPCTC)

Step 9: By SAS (Side-Angle-Side) congruence, triangle BDA ≅ triangle CDA
Step 10: BD ≅ CD and AD ≅ AD (Corresponding parts of congruent triangles are congruent - CPCTC)

Step 11: Therefore, line AD bisects line BC (BD ≅ CD)

By proving that the triangles BDA and CDA are congruent, you have shown that line AD not only intersects line BC but also bisects it by dividing it into two equal segments, BD and DC.