Question

given that abcd is a parallelogram, prove $\angle b \cong \angle d$.
move the options to the spaces to complete the proof.

statements	reasons
	definition of a parallelogram
$m\angle a + m\angle b = 180^\circ$ and $m\angle a + m\angle d = 180^\circ$
$m\angle a + m\angle b = m\angle a + m\angle d$
$m\angle b = m\angle d$
$\angle b \cong \angle d$	angles with the same measure are congruent

options:
$\overline{ab} \cong \overline{dc}$ and $\overline{bc} \cong \overline{ad}$
$\overline{ab} \parallel \overline{dc}$ and $\overline{bc} \parallel \overline{ad}$
alternate interior angles theorem
corresponding angles theorem
same side interior angles theorem
substitution
multiplicative property of equality
subtraction property of equality

Explanation:

Step1: Fill first missing statement

By definition of a parallelogram, opposite sides are parallel. So the statement is $\overline{AB} \parallel \overline{DC}$ and $\overline{BC} \parallel \overline{AD}$.

Step2: Fill reason for angle sum

Same-side interior angles are supplementary for parallel lines.
Reason: Same Side Interior Angles Theorem

Step3: Fill reason for equating sums

Set the two equal angle sums equal via substitution.
Reason: Substitution

Step4: Fill reason for angle equality

Subtract $m\angle A$ from both sides using subtraction property.
Reason: Subtraction Property of Equality

Answer:

Statements	Reasons
$\overline{AB} \parallel \overline{DC}$ and $\overline{BC} \parallel \overline{AD}$	Definition of a parallelogram
$m\angle A + m\angle B = 180^\circ$ and $m\angle A + m\angle D = 180^\circ$	Same Side Interior Angles Theorem
$m\angle A + m\angle B = m\angle A + m\angle D$	Substitution
$m\angle B = m\angle D$	Subtraction Property of Equality
$\angle B \cong \angle D$	Angles with the same measure are congruent