Question

proof: complete the proof.
given: b is the mid - point of \\(\overline{ae}). b is the mid - point of \\(\overline{cd}).
prove: \\(\triangle abd\cong\triangle ebc\\)
statements reasons

1. b is the mid - point of \\(\overline{ae}). 1. definition of mid - point
2. 2. definition of mid - point
3. b is the mid - point of \\(\overline{cd}). 3. definition of mid - point
4. 4.
5. \\(\angle abd\cong\angle ebc\\) 5.
6. \\(\triangle abd\cong\triangle ebc\\) 6.

Explanation:

Step1: State given information

Given that B is the mid - point of $\overline{AE}$ and $\overline{CD}$

Step2: Use mid - point property

By the definition of a mid - point, we get congruent line segments.

Step3: Restate given

Re - state the given mid - point information for the other line segment.

Step4: Identify vertical angles

Vertical angles $\angle ABD$ and $\angle EBC$ are congruent.

Step5: Apply SAS criterion

With two pairs of congruent sides and the included congruent angles, we can apply SAS.

Step6: Conclude triangle congruence

We conclude that $\triangle ABD\cong\triangle EBC$ by the SAS congruence postulate.

Answer:

1. Given
2. If a point is the mid - point of a line segment, it divides the line segment into two congruent segments. So, $\overline{AB}\cong\overline{BE}$ and $\overline{DB}\cong\overline{BC}$
3. Given
4. Vertical angles are congruent. So, $\angle ABD\cong\angle EBC$
5. Side - Angle - Side (SAS) congruence criterion. Since $\overline{AB}\cong\overline{BE}$, $\angle ABD\cong\angle EBC$, and $\overline{DB}\cong\overline{BC}$, then $\triangle ABD\cong\triangle EBC$
6. $\triangle ABD\cong\triangle EBC$ (by SAS)