Question

proof complete the two - column proof by dragging the missing statements and reasons.
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.?

keyboard help
$\overline{de} \cong \overline{be}$
given (2nd use)
$\angle aed \cong \angle ceb$
$\overline{ac}$ bisects $\overline{bd}$.
def. of congruent triangles
given (1st use)

Explanation:

Step1: Fill Reason for Statement 1

The first statement \(\angle A\cong\angle C\), \(\angle D\cong\angle B\) is given (first use), so the reason is "Given (first use)".

Step2: Fill Statement for Reason 2

The reason is "Vertical angles are \(\cong\)", so the statement is \(\angle AED\cong\angle CEB\) (since \(\angle AED\) and \(\angle CEB\) are vertical angles).

Step3: Fill Reason for Statement 3

The statement \(\overline{AD}\cong\overline{CB}\), \(\overline{AE}\cong\overline{CE}\) is given (and used), so the reason is "Given (and used)".

Step4: Fill Statement for Reason 4

The reason is "Given", and from the given we know \(AC\) bisects \(BD\), so the statement is \(\overline{AC}\) bisects \(\overline{BD}\).

Step5: Fill Statement for Reason 5

The reason is "Def. of segment bisector", which means \(E\) is the midpoint of \(BD\), so \(\overline{DE}\cong\overline{BE}\).

Step6: Fill Reason for Statement 6

To prove \(\triangle AED\cong\triangle CEB\), we can use the congruence criteria. We have angles and sides: \(\angle A\cong\angle C\), \(\angle D\cong\angle B\), \(\overline{AD}\cong\overline{CB}\) (or other combinations), so the reason is "Def. of congruent triangles" (or ASA/AAS congruence, but based on the options, "Def. of congruent triangles" fits).

Answer:

Statements	Reasons
2. \(\angle AED\cong\angle CEB\)	2. Vertical angles are \(\cong\)
3. \(\overline{AD}\cong\overline{CB}\), \(\overline{AE}\cong\overline{CE}\)	3. Given (and used)
4. \(\overline{AC}\) bisects \(\overline{BD}\)	4. Given
5. \(\overline{DE}\cong\overline{BE}\)	5. Def. of segment bisector
6. \(\triangle AED\cong\triangle CEB\)	6. Def. of congruent triangles