Question

given that abcd is a parallelogram. prove ∠b≅∠d
move the options to the spaces to complete the proof.
statements reasons
abcd is a parallelogram given
definition of a parallelogram
m∠a + m∠b = 180° and m∠a + m∠d = 180°
m∠a + m∠b = m∠a + m∠d
m∠b = m∠d
∠b≅∠d angles with the same measure are congruent
ab≅dc and bc≅ad
ab || dc and bc || ad
alternate interior angles theorem
corresponding angles theorem
same - side interior angles theorem
substitution
multiplicative property of equality
subtraction property of equality

Explanation:

Step1: Recall parallelogram property

$\overline{AB}\parallel\overline{DC}$ and $\overline{BC}\parallel\overline{AD}$ (Definition of a parallelogram)

Step2: Use same - side interior angles

$m\angle A + m\angle B=180^{\circ}$ and $m\angle A + m\angle D = 180^{\circ}$ (Same - Side Interior Angles Theorem)

Step3: Set up equality

$m\angle A + m\angle B=m\angle A + m\angle D$ (Substitution)

Step4: Solve for angle measures

$m\angle B=m\angle D$ (Subtraction Property of Equality)

Step5: Prove angle congruence

$\angle B\cong\angle D$ (Angles with the same measure are congruent)

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