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Proving a Quadrilateral Is a Parallelogram: Practice

12 multiple-choice questions, progressively harder.

Level 1 · Foundational 0 / 12 answered

Question 1 of 12

1

Convex quadrilateral $ABCD$ has the following side equalities.
$\begin{gathered} AB=CD,\\ BC=AD. \end{gathered}$
Which conclusion is guaranteed?

Answer choices for question 1 A It has four right angles. B It is a parallelogram. C Its diagonals have equal lengths. D It has four congruent sides.
2

Diagonals $AC$ and $BD$ of convex $ABCD$ meet at $E$ . Given $AE=EC$ , which extra equality completes the diagonal-bisection test?

Answer choices for question 2 A $AE=BE$ B $AC=BD$ C $EC=ED$ D $BE=ED$
3

Which angle evidence alone guarantees that a convex quadrilateral $ABCD$ is a parallelogram?

Answer choices for question 3 A
$\angle A\cong\angle C$

$\angle B\cong\angle D$
B $\angle A\cong\angle C$ only C $\angle A\cong\angle B$ only D $\angle A$ is acute
4

Convex $ABCD$ has $AB\parallel CD$ . Which extra fact completes the test on that same opposite-side pair?

Answer choices for question 4 A $AD=BC$ B $AC=BD$ C $AB=CD$ D $AB=BC$
5

Having at least one congruent opposite-side pair is which kind of condition for a convex quadrilateral to be a parallelogram?

Answer choices for question 5 A Sufficient, but not necessary B Both necessary and sufficient C Necessary, but not sufficient D Neither necessary nor sufficient
6

In convex $ABCD$ , diagonals meet at $E$ , and $AE=EC$ . Nothing else is known. Which conclusion is justified?

Answer choices for question 6 A The evidence does not guarantee a parallelogram. B It must be a parallelogram. C It must have four congruent sides. D It cannot be a parallelogram.
7

In the figure, draw diagonal $AC$ in convex $ABCD$ . Which fact supplies the third side match for an SSS comparison of triangles $ABC$ and $CDA$ ?

Answer choices for question 7 A $AB=BC$ B $AC=BD$ C $\angle A=\angle C$ D $AC=CA$
8

In convex $ABCD$ , diagonals meet at $E$ . If $\angle BAE\cong\angle DCE$ , which parallel pair follows by the alternate interior-angle converse?

Answer choices for question 8 A $AD\parallel BC$ B $AB\parallel CD$ C $AC\parallel BD$ D $AB\parallel AC$
9

The interior angles of convex $ABCD$ , in boundary order, are $71^\circ$ , $109^\circ$ , $71^\circ$ , and $109^\circ$ . Why must it be a parallelogram?

Answer choices for question 9 A Both pairs of opposite angles are congruent. B All four angles are congruent. C One angle is acute. D Its diagonals are known to be perpendicular.
10

A convex quadrilateral has one opposite-side pair parallel. What does this fact alone guarantee?

Answer choices for question 10 A Both opposite-side pairs are parallel B Both opposite-side pairs are congruent C At least one parallel pair, but no parallelogram conclusion D Both diagonals bisect each other
11

Diagonals of convex $ABCD$ meet at $E$ . The two halves of $AC$ are each $4$ cm, and the two halves of $BD$ are each $6$ cm. What follows?

Answer choices for question 11 A It is not a parallelogram because the diagonals differ. B It is a parallelogram even though the diagonal lengths differ. C It is a parallelogram only if all four halves are equal. D There is not enough evidence because no side is marked parallel.
12

Convex $ABCD$ has the following equalities between adjacent sides.
$\begin{gathered} AB=AD,\\ BC=CD. \end{gathered}$
What follows about a parallelogram?

Answer choices for question 12 A It must be a parallelogram by the opposite-side test. B It must have both diagonals bisected. C It cannot be a parallelogram. D These equalities alone do not guarantee one.

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