Question

5
given: $overline{bc}\parallel\overline{ef}$. $d$ is the midpoint of $overline{bf}$
prove: $overline{ed}\cong\overline{cd}$

statements

reasons

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1. $overline{bc}\parallel\overline{ef}$

1. given

2.

2.

3.

3.

4.

4.

5.

5.

6.

6.

7.

7.

Explanation:

Step1: State given parallel lines

$\overline{BC} \parallel \overline{EF}$

Step2: State given midpoint

$D$ is the midpoint of $\overline{BF}$, so $\overline{BD} \cong \overline{FD}$

Step3: Identify alternate interior angles

$\angle B \cong \angle F$ (Alternate Interior Angles Theorem, since $\overline{BC} \parallel \overline{EF}$)

Step4: Identify vertical angles

$\angle BDC \cong \angle FDE$ (Vertical Angles Congruence Theorem)

Step5: Prove triangle congruence

$\triangle BDC \cong \triangle FDE$ (ASA Congruence Postulate: $\overline{BD} \cong \overline{FD}$, $\angle B \cong \angle F$, $\angle BDC \cong \angle FDE$)

Step6: Corresponding parts of congruent triangles

$\overline{ED} \cong \overline{CD}$ (CPCTC: Corresponding Parts of Congruent Triangles are Congruent)

Answer:

The completed two-column proof is:

Statements	Reasons
2. $D$ is the midpoint of $\overline{BF}$	2. Given
3. $\overline{BD} \cong \overline{FD}$	3. Definition of a Midpoint
4. $\angle B \cong \angle F$	4. Alternate Interior Angles Theorem
5. $\angle BDC \cong \angle FDE$	5. Vertical Angles Congruence Theorem
6. $\triangle BDC \cong \triangle FDE$	6. ASA Congruence Postulate
7. $\overline{ED} \cong \overline{CD}$	7. CPCTC