Question

proof complete the proof.
given: b is the mid - point of $overline{ae}$. b is the mid - point of $overline{cd}$.
prove: $\triangle abdcong\triangle ebc$
statements reasons

1. b is the mid - point of $overline{ae}$. 1. given
2. $angle b$ 2. definition of midpoint
3. b is the mid - point of $overline{cd}$. 3. definition of midpoint
4. 4. definition of midpoint
5. $angle abdcongangle ebc$ 5.
6. $\triangle abdcong\triangle ebc$ 6.

Explanation:

Step1: Identify equal segments from mid - point

Since $B$ is the mid - point of $\overline{AE}$, by the definition of a mid - point, $AB = EB$.

Step2: Identify other equal segments

As $B$ is the mid - point of $\overline{CD}$, by the definition of a mid - point, $DB = CB$.

Step3: Identify equal angles

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

Step4: Prove triangle congruence

We have two pairs of equal sides ($AB = EB$ and $DB = CB$) and one pair of equal angles ($\angle ABD\cong\angle EBC$). By the Side - Angle - Side (SAS) congruence criterion, $\triangle ABD\cong\triangle EBC$.

Answer:

1. Statements: $AB = EB$

• Reasons: Definition of mid - point

1. Statements: $DB=CB$

• Reasons: Definition of mid - point

1. Statements: $\angle ABD\cong\angle EBC$

• Reasons: Vertical angles are congruent

1. Statements: $\triangle ABD\cong\triangle EBC$

• Reasons: Side - Angle - Side (SAS) congruence criterion