Question

given: $overline{bc}paralleloverline{ef}$, d is the mid - point of $overline{bf}$
prove: $overline{ed}congoverline{cd}$
statements

1. $overline{bc}paralleloverline{ef}$, d is the mid - point of $overline{bf}$
2. $overline{bd}congoverline{df}$
3. $angle bdccongangle edf$
4. $angle ccongangle e$
5. $\triangle cbdcong\triangle efd$
6. $overline{ed}congoverline{cd}$

reasons
1.
2.
3.
4.
5.
6.

Explanation:

Step1: Given information

1. $\overline{BC}\parallel\overline{EF}$, D is the mid - point of $\overline{BF}$ (Given)

Step2: Definition of mid - point

1. $\overline{BD}\cong\overline{DF}$ (Definition of mid - point: A mid - point divides a line segment into two congruent segments)

Step3: Vertical angles are congruent

1. $\angle BDC\cong\angle EDF$ (Vertical angles are congruent)

Step4: Alternate interior angles

1. $\angle C\cong\angle E$ (Since $\overline{BC}\parallel\overline{EF}$, alternate interior angles are congruent)

Step5: Angle - Side - Angle (ASA) congruence

1. $\triangle CBD\cong\triangle EFD$ (By ASA congruence criterion: $\angle C\cong\angle E$, $\overline{BD}\cong\overline{DF}$, $\angle BDC\cong\angle EDF$)

Step6: Corresponding parts of congruent triangles

1. $\overline{ED}\cong\overline{CD}$ (Corresponding parts of congruent triangles are congruent, CPCTC)

Answer:

1. Given
2. Definition of mid - point
3. Vertical angles are congruent
4. Alternate interior angles
5. ASA congruence
6. CPCTC