Question

given: \\( \angle a \cong \angle c, \angle d \cong \angle b, \overline{ad} \cong \overline{cb}, \overline{ae} \cong \overline{ce}, \overline{ac} \\) bisects \\( \overline{bd} \\).
prove: \\( \triangle aed \cong \triangle ceb \\)
proof

statements	reasons
2 ?	2. vertical angles are \\( \cong \\).
3. \\( \overline{ad} \cong \overline{cb}; \overline{ae} \cong \overline{ce} \\)	3. ?
4. ?	4. given
5. ?	5. def. of segment bisector
6. \\( \triangle aed \cong \triangle ceb \\)	6. ?

Explanation:

Step1: State the given angles

The problem statement directly gives $\angle A\cong\angle C$ and $\angle D\cong\angle B$. So the reason for statement 1 is "Given".

Step2: Identify vertical angles

Vertical angles are formed when two lines intersect. In the figure, $\angle AED$ and $\angle CEB$ are vertical angles. By the property of vertical angles, $\angle AED\cong\angle CEB$.

Step3: State the given segments

The problem statement directly gives $\overline{AD}\cong\overline{CB}$ and $\overline{AE}\cong\overline{CE}$. So the reason for statement 3 is "Given".

Step4: Restate the given about the bisector

The problem statement gives that $\overline{AC}$ bisects $\overline{BD}$. So this is statement 4.

Step5: Use the definition of segment bisector

If a segment bisector divides a segment into two congruent parts, since $\overline{AC}$ bisects $\overline{BD}$, then $\overline{DE}\cong\overline{BE}$ (by the definition of segment bisector).

Step6: Apply the congruence theorem

We have two pairs of congruent angles ($\angle A\cong\angle C$, $\angle D\cong\angle B$) and a pair of non - included congruent sides ($\overline{AD}\cong\overline{CB}$). By the AAS (Angle - Angle - Side) Congruence Theorem, $\triangle AED\cong\triangle CEB$.

Answer:

1. Given
2. $\angle AED\cong\angle CEB$
3. Given
4. $\overline{AC}$ bisects $\overline{BD}$
5. $\overline{DE}\cong\overline{BE}$
6. $AAS$ (Angle - Angle - Side) Congruence Theorem