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Proving a Quadrilateral Is a Parallelogram: Practice

12 multiple-choice questions, progressively harder.

Level 2 · Intermediate 0 / 12 answered

Question 1 of 12

1

Convex $ABCD$ has side lengths in centimeters.
$\begin{gathered} AB=3x-2,\\ CD=13,\\ BC=x+1,\\ AD=6. \end{gathered}$
Which value makes both opposite-side pairs congruent?

Answer choices for question 1 A $x=4$ B $x=6$ C $x=5$ D No value
2

Convex $ABCD$ has these side lengths in the same unit.
$\begin{gathered} AB=2x+1,\\ CD=9,\\ BC=x+3,\\ AD=8. \end{gathered}$
Which value of $x$ makes both opposite-side pairs congruent?

Answer choices for question 2 A No value B $x=4$ C $x=5$ D $x=\frac92$
3

Diagonals $AC$ and $BD$ of convex $ABCD$ meet at $E$ . Their parts have lengths in the same unit.
$\begin{gathered} AE=2x+3,\\ EC=x+8,\\ BE=y+1,\\ ED=3y-7. \end{gathered}$
Which values make both diagonals bisect each other?

Answer choices for question 3 A
$x=4$

$y=5$
B
$x=5$

$y=4$
C
$x=5$

$y=2$
D
$x=11$

$y=4$
4

In convex $ABCD$ , both opposite-angle pairs are congruent. Angle $A$ measures $(2x+10)^\circ$ and angle $C$ measures $(4x-30)^\circ$ . What is the measure of angle $B$ ?

Answer choices for question 4 A $20^\circ$ B $50^\circ$ C $310^\circ$ D $130^\circ$
5

Convex $ABCD$ has two side equalities.
$\begin{gathered} AB=CD,\\ AD=BC. \end{gathered}$
After drawing diagonal $BD$ , which ordered triangle correspondence is guaranteed by these matches and SSS?

Answer choices for question 5 A $\triangle ABD\cong\triangle CDB$ B $\triangle ABD\cong\triangle CBD$ C $\triangle ABD\cong\triangle BCD$ D $\triangle ABD\cong\triangle DCB$
6

Diagonals of convex $ABCD$ bisect each other at $E$ . Which reason proves $\triangle AEB\cong\triangle CED$ ?

Answer choices for question 6 A SSS, using three given side equalities B ASA, using the fact that the opposite sides are already parallel C SSA, using two diagonal lengths and one angle D SAS, using two half-diagonal equalities and vertical angles
7

In convex $ABCD$ , angles $A,C$ both measure $p$ , and angles $B,D$ both measure $q$ . Which equation turns this evidence into supplementary consecutive angles?

Answer choices for question 7 A $p=q$ B $p+q=360^\circ$ C $p+q=180^\circ$ D $p-q=180^\circ$
8

In the shown convex $ABCD$ , the givens include a parallel pair and a congruent pair.
$\begin{gathered} AB\parallel CD,\\ AD=BC. \end{gathered}$
Which fact proves that this particular quadrilateral is not a parallelogram?

Answer choices for question 8 A $AD=BC$ B $AB\parallel CD$ C $AB\ne CD$ D The drawing is wider than it is tall.
9

Diagonals of convex $ABCD$ meet at $E$ . The parts have lengths in the same unit.
$\begin{gathered} AE=6,\\ EC=6,\\ BE=5,\\ ED=3. \end{gathered}$
Which conclusion is certain?

Answer choices for question 9 A It is a parallelogram because one diagonal is bisected. B It is not a parallelogram. C It is a parallelogram because all diagonal parts are positive. D It may be a parallelogram if the drawing is rotated.
10

Convex $ABCD$ has this evidence about one opposite-side pair.
$\begin{gathered} AB\parallel CD,\\ AB=CD. \end{gathered}$
With diagonal $AC$ , which included-angle equality supports SAS for triangles $BAC$ and $DCA$ ?

Answer choices for question 10 A $\angle BAC\cong\angle DCA$ B $\angle ABC\cong\angle CDA$ C $\angle BCA\cong\angle DAC$ D $\angle BAD\cong\angle DCB$
11

In convex $ABCD$ , diagonals meet at $E$ . The parts have lengths in centimeters.
$\begin{gathered} AE=2,\\ BE=2,\\ EC=3,\\ ED=3. \end{gathered}$
Which conclusion is justified?

Answer choices for question 11 A It is a parallelogram because there are two pairs of equal parts. B It is a parallelogram because the whole diagonals are congruent. C It must have four congruent sides. D It is not a parallelogram because a diagonal is not bisected.
12

The diagonals of convex $ABCD$ bisect each other at $E$ . SAS gives two congruences.
$\begin{gathered} \triangle AEB\cong\triangle CED,\\ \triangle BEC\cong\triangle DEA. \end{gathered}$
Which conclusion completes a valid side-based proof of a parallelogram?

Answer choices for question 12 A $AC=BD$ only B
$AB=CD$

$BC=AD$
C $AB=BC$ only D
$AE=BE$

$EC=ED$

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