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Parallelograms: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

Which proof route establishes the opposite-side property from the definition of a parallelogram without circular reasoning?

Answer choices for question 1 A Assume opposite sides match, then invoke SSS across a diagonal B Assume a half-turn, then use it to prove diagonal bisection, then prove the half-turn C Use equal full diagonals, then SSS D Parallel pairs, then alternate angles, then ASA across a diagonal, then corresponding sides
2

In parallelogram $ABCD$ , lengths in centimeters satisfy
$\begin{gathered} AB=2x+3,\\ CD=21. \end{gathered}$
Its diagonals meet at $E$ , with
$\begin{gathered} AE=y+6,\\ EC=3y-2. \end{gathered}$
Which values follow?

Answer choices for question 2 A
$x=9$

$y=2$
B
$x=9$

$y=4$
C
$x=12$

$y=4$
D
$x=21$

$y=8$
3

In parallelogram $ABCD$ , diagonals meet at $O$ . Point $P$ lies on $AB$ and $Q$ lies on $CD$ , with $AP=CQ$ . Why does the half-turn about $O$ send $P$ to $Q$ ?

Answer choices for question 3 A Every point of side $AB$ is fixed by a half-turn B It sends $P$ to the midpoint of $CD$ regardless of $AP$ C It sends ray $AB$ to ray $CD$ and preserves distance from the mapped endpoint D A reflection in $AC$ always exchanges $P$ and $Q$
4

Opposite angles of a parallelogram measure $(3t+6)^\circ$ and $(5t-34)^\circ$ . What is the measure of either of the other two angles?

Answer choices for question 4 A $114^\circ$ B $66^\circ$ C $20^\circ$ D $57^\circ$
5

Diagonals of parallelogram $ABCD$ meet at $E$ . The ratio $AE:BE$ is $5:3$ , and $AC$ is $30$ cm. How long is $BD$ ?

Answer choices for question 5 A $9$ cm B $50$ cm C $18$ cm D $15$ cm
6

In parallelogram $ABCD$ , diagonal $AC$ makes $\angle BAC$ measure $20^\circ$ and $\angle CAD$ measure $45^\circ$ . Which conclusion is guaranteed?

Answer choices for question 6 A $m\angle CAD=20^\circ$ B $m\angle ABC=65^\circ$ C $m\angle BCD=115^\circ$ D $m\angle DCA=20^\circ$
7

Which additional description of a parallelogram guarantees that it is a general member with no reflection axes?

Answer choices for question 7 A It has no right angle and has unequal adjacent sides B Its adjacent sides are unequal C It has an acute angle D Its diagonals intersect
8

Why cannot a reflection exchange both pairs of opposite vertices of a convex parallelogram at once?

Answer choices for question 8 A A reflection cannot exchange any two vertices B Its axis would need to be perpendicular to both crossing diagonals C The diagonals would have to have unequal lengths D A reflection always fixes every point of the shape
9

A proof of diagonal bisection in parallelogram $ABCD$ uses these established facts, where $E$ is the diagonal intersection.
$\begin{gathered} AB=CD,\\ \angle BAE\cong\angle DCE,\\ \angle ABE\cong\angle CDE. \end{gathered}$
Which conclusion is justified by exactly these facts at this stage of the proof?

Answer choices for question 9 A $\triangle ABE\cong\triangle DCE$ by ASA B $\triangle ABE\cong\triangle CDE$ by ASA C $\triangle ABE\cong\triangle CDE$ by SSS D The triangles are similar only because no side pair is known
10

Point $P$ lies on side $AD$ of parallelogram $ABCD$ , and its half-turn image about the diagonal intersection is $Q$ . Which description must hold for $Q$ ?

Answer choices for question 10 A $Q$ lies on $CB$ , with $CQ=AP$ B $Q$ lies on $CD$ , with $CQ=AP$ C $Q$ lies on $CB$ , with $BQ=AP$ D $Q$ is always the midpoint of $CB$
11

A parallelogram has diagonal lengths $12$ cm and $8$ cm. Which statement describes what its half-turn proof needs?

Answer choices for question 11 A All four half-diagonal lengths must equal $5$ cm B The diagonals must be perpendicular C Each diagonal must be an angle bisector D Each diagonal has equal halves, even though the two diagonals differ
12

For a given parallelogram, which conclusion follows directly from an $80^\circ$ angle without any information about side lengths or diagonals?

Answer choices for question 12 A It has a reflection axis through that vertex B Its diagonals meet at $90^\circ$ C Its two neighboring angles are $100^\circ$ D Its adjacent sides are congruent

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