Answer
Proof for the problem:
1. $\angle 1$ and $\angle 2$ are right angles (1. Given)
2. $\angle 1\cong\angle 2$ (2. Two corresponding angles are both $90^o$)
3. $\vec{AB}$ bisects $\angle ABD$ (3. Given)
4. $\angle BAC\cong\angle BAD$ (4. The bisector of an angle separates it into 2 congruent angles)
5. $\overline{AB}\cong\overline{AB}$ (5. Identity)
6. $\triangle ABC\cong\triangle ABD$ (6. ASA)
Work Step by Step
1) First, we see that $\angle 1$ and $\angle 2$ are both right angles.
Therefore, $\angle 1\cong\angle 2$
2) It is also given that $\vec{AB}$ bisects $\angle ABD$.
So, $\angle BAC\cong\angle BAD$
3) Furthermore, by Identity, we see that $\overline{AB}\cong\overline{AB}$
Now we see that 2 angles and the included side of $\triangle ABC$ are congruent with 2 corresponding angles and the included side of $\triangle ABD$.
So we would use ASA to prove triangles congruent.
Now we would construct a proof for the problem:
1. $\angle 1$ and $\angle 2$ are right angles (1. Given)
2. $\angle 1\cong\angle 2$ (2. Two corresponding angles are both $90^o$)
3. $\vec{AB}$ bisects $\angle ABD$ (3. Given)
4. $\angle BAC\cong\angle BAD$ (4. The bisector of an angle separates it into 2 congruent angles)
5. $\overline{AB}\cong\overline{AB}$ (5. Identity)
6. $\triangle ABC\cong\triangle ABD$ (6. ASA)
