Question

geometry worksheet
congruent triangles #3
complete the proofs.

1. given: (overline{ae} cong overline{cb}), (overline{ab} cong overline{cd}),

and b is the midpoint of (overline{ed})
prove: (\triangle aeb cong \triangle cbd)
(hint: draw the information on the picture as you know it.)
(\begin{array}{c|c}\text{statements} & \text{reasons} \\ hline 1. overline{ae} cong overline{cb}, overline{ab} cong overline{cd}, \\ quad \text{and b is the midpoint of } overline{ed} & 1. \\ 2. overline{eb} cong overline{db} & 2. \\ 3. \triangle aeb cong \triangle cbd & 3. end{array})
diagram: two triangles, aeb and cbd, with e---b---d on a line, a above eb, c above bd

Explanation:

Step1: Identify Reason for Statement 1

The first statement lists the given information: \(\overline{AE} \cong \overline{CB}\), \(\overline{AB} \cong \overline{CD}\), and \(B\) is the midpoint of \(\overline{ED}\). So the reason is "Given".

Step2: Identify Reason for Statement 2

Since \(B\) is the midpoint of \(\overline{ED}\), by the definition of a midpoint, a midpoint divides a segment into two congruent segments. So \(\overline{EB} \cong \overline{DB}\) because of the "Definition of Midpoint".

Step3: Identify Reason for Statement 3

We have three sides: \(\overline{AE} \cong \overline{CB}\) (given), \(\overline{AB} \cong \overline{CD}\) (given), and \(\overline{EB} \cong \overline{DB}\) (from step 2). By the Side - Side - Side (SSS) Congruence Postulate, if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. So \(\triangle AEB \cong \triangle CBD\) by "SSS Congruence Postulate".

Answer:

1. Reason for Statement 1: Given
2. Reason for Statement 2: Definition of Midpoint
3. Reason for Statement 3: SSS Congruence Postulate