Question

complete the following proof.
given: $overline{db}$ bisects $overline{ac}$
$angle 1 \cong \angle 2$
prove: $abcd$ is a parallelogram

1. segment $db$ bisects segment $ac$, angle 1 congruent to angle 2 choose
2. segment $ae$ is congruent to segment $ec$ choose

choose
given
definition of bisect (line bisect)
cpctc
vertical angles theorem
diagonals bisect each other in parallelogram
asa
definition of bisect (line segment)

1. angle $dea$ congruent to angle $bec$
2. triangle $dea$ congruent to triangle $bec$
3. segment $de$ congruent to segment $be$
4. segment $ac$ bisects segment $db$
5. $abcd$ is a parallelogram

Explanation:

Step1: State given premises

Given: $\overline{DB}$ bisects $\overline{AC}$, $\angle 1 \cong \angle 2$

Step2: Apply bisector definition (first)

$\overline{AE} \cong \overline{EC}$ (definition of bisect (first))

Step3: Use vertical angles theorem

$\angle DEA \cong \angle BEC$ (vertical angles theorem)

Step4: Apply ASA congruence

$\triangle DEA \cong \triangle BEC$ (ASA: $\angle 1 \cong \angle 2$, $\overline{AE} \cong \overline{EC}$, $\angle DEA \cong \angle BEC$)

Step5: Use CPCTC

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

Step6: Apply bisector definition (second)

$\overline{AC}$ bisects $\overline{DB}$ (definition of bisect (second))

Step7: Use parallelogram diagonal theorem

ABCD is a parallelogram (diagonals bisect each other in parallelograms)

Answer:

1. given
2. definition of bisect (first)
3. vertical angles theorem
4. ASA
5. CPCTC
6. definition of bisect (second)
7. diagonals bisect each other in parallelograms