Question

1. sat/act in quadrilateral abcd, ∠a ≅ ∠c. which additional statement can be used to show that abcd is a parallelogram? a ( mangle a + mangle c = 180 ) b ( overline{bd} ) bisects ( overline{ac} ) c ( mangle b + mangle d = 180 ) d ( angle b cong angle d )

Explanation:

Step1: Analyze given condition

We know $\angle A \cong \angle C$, so $m\angle A = m\angle C$.

Step2: Evaluate Option A

Substitute $m\angle A = m\angle C$ into $m\angle A + m\angle C = 180$:
$2m\angle A = 180 \implies m\angle A = 90$. This only tells us $\angle A$ and $\angle C$ are right angles, not that $ABCD$ is a parallelogram.

Step3: Evaluate Option B

A diagonal bisecting another diagonal is not sufficient to prove a quadrilateral is a parallelogram (the reverse is true, but this one-way condition does not guarantee the shape is a parallelogram).

Step4: Evaluate Option C

$m\angle B + m\angle D = 180$ does not pair with $\angle A \cong \angle C$ to prove opposite sides are parallel or all conditions for a parallelogram.

Step5: Evaluate Option D

If $\angle B \cong \angle D$, then we have both pairs of opposite angles congruent. For any quadrilateral, the sum of interior angles is $360^\circ$:
$m\angle A + m\angle B + m\angle C + m\angle D = 360^\circ$
Substitute $m\angle A = m\angle C$ and $m\angle B = m\angle D$:
$2m\angle A + 2m\angle B = 360^\circ \implies m\angle A + m\angle B = 180^\circ$
This means $\overline{AD} \parallel \overline{BC}$. Similarly, $m\angle A + m\angle D = 180^\circ$, so $\overline{AB} \parallel \overline{CD}$. Thus, $ABCD$ is a parallelogram.

Answer:

D. $\angle B \cong \angle D$