Question

complete the proof.
given: angle 1 congruent to angle 2, angle 3 congruent to angle 4
prove: abcd is a parallelogram

1. angle 1 congruent to angle 2, angle 3 congruent to angle 4
2. segment bc is parallel to segment ad
3. segment ab is parallel to segment dc
4. abcd is a parallelogram

choose
choose

given
converse of the alternate interior angles theorem (case 1)
adjacent angles are supplementary
alternate exterior angles theorem (case 2)
alternate interior angles theorem (case 1)
two pairs of opposite sides are congruent
diagonal bisects each other
alternate angles are congruent
two pairs of opposite sides are parallel
converse of the alternate exterior angles theorem (case 2)

Explanation:

Step1: State given congruences

$\angle 1 \cong \angle 2$, $\angle 3 \cong \angle 4$ (given)

Step2: Prove $AB \parallel DC$

$\angle 3 \cong \angle 4$ are alternate interior angles, so by alternate interior angles converse, $AB \parallel DC$.

Step3: Prove $AD \parallel BC$

$\angle 1 \cong \angle 2$ are alternate interior angles, so by alternate interior angles converse, $AD \parallel BC$.

Step4: Conclude ABCD is parallelogram

A quadrilateral with both pairs of opposite sides parallel is a parallelogram.

Answer:

1. $\boldsymbol{\text{given}}$
2. $\boldsymbol{\text{alternate interior angles (converse, Step 2)}}$
3. $\boldsymbol{\text{alternate interior angles (converse, Step 3)}}$
4. $\boldsymbol{\text{two pairs of opposite sides are parallel}}$