Answer:
To prove that △ABD is congruent to △CBD, we need to use the given information that BD bisects angle ∠ABC and ∠BAD is congruent to ∠BCD.
Here is the proof:
Statements:
1. BD bisects angle ∠ABC
2. ∠BAD is congruent to ∠BCD
3. AB is congruent to BC (Given)
4. BD is congruent to BD (Common side)
5. △ABD is congruent to △CBD (ASA congruence postulate)
6.△ABD is congruent to △CBD (CPCTC)
Step-by-step explanation:
In statement 1, it is given that BD bisects angle ∠ABC, which means it divides the angle into two congruent angles.
In statement 2, it is given that ∠BAD is congruent to ∠BCD, which means the angles on each side of BD are congruent.
In statement 3, it is given that AB is congruent to BC, which is the common side between the two triangles.
In statement 4, BD is congruent to BD, which is a reflexive property.
Using the ASA (Angle-Side-Angle) congruence postulate, we can conclude that triangle ABD is congruent to triangle CBD in statement 5. The congruent angles are ∠BAD and ∠BCD, the congruent side is BD, and AB is congruent to BC.
Finally, in statement 6, we use the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to state that △ABD is congruent to △CBD. This means that all corresponding parts of the congruent triangles are congruent.
Therefore, △ABD is congruent to △CBD.
