Geometry Quiz: Theorems About Parallelograms

7 questions · exam conditions

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Theorems About ParallelogramsQuestion 1 of 7

Triangle $JKL$ has medians $JM$ , $KN$ , and $LO$ that intersect at centroid $G$ . If $JM = 21$ and $KN = 18$ , and the distance from $K$ to $G$ is $12$ , what is the distance from $G$ to the midpoint $M$ of side $KL$ ?

6

units using centroid division properties of medians

14

units using centroid division properties of medians

9

units using centroid division properties of medians

7

units using centroid division properties of medians

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Geometry Quiz

Geometry Quiz: Theorems About Parallelograms

Practice Theorems About Parallelograms in Geometry with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Theorems About Parallelograms, giving you a quick way to practice the rules, question types, and explanations that matter most for Geometry.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$6$ units using centroid division properties of medians
$14$ units using centroid division properties of medians
$9$ units using centroid division properties of medians
$7$ units using centroid division properties of medians (correct answer)

Explanation: When you encounter problems involving medians and centroids, remember that the centroid divides each median in a specific 2:1 ratio, with the longer segment extending from the vertex to the centroid. The centroid

G

divides every median so that the distance from any vertex to

G

is twice the distance from

G

to the opposite midpoint. Since the distance from

K

G

is 12, and

G

divides median

KN

in a 2:1 ratio, the distance from

G

to midpoint

N

must be

12 ÷ 2 = 6

. We can verify this using the given information: if

KG = 12

and

GN = 6

, then the total length

KN = 12 + 6 = 18

, which matches the given median length. Now, for median

JM

with length 21, the centroid divides it the same way. The distance from

J

G

\frac{2}{3} × 21 = 14

, and the distance from

G

to midpoint

M

\frac{1}{3} × 21 = 7

. Choice A gives 6 units, which incorrectly applies the distance from

G

N

instead of finding the distance to

M

. Choice B gives 14 units, which represents the distance from

J

G

, not from

G

M

. Choice C gives 9 units, which appears to incorrectly calculate half of the median length

JM

. Study tip: Always remember the centroid's 2:1 division rule: vertex to centroid is

\frac{2}{3}

of the median length, while centroid to midpoint is

\frac{1}{3}

of the median length.

Question 2

Triangle $ABC$ has vertices $A$ , $B$ , and $C$ . The centroid $G$ is located at coordinates $(4, 3)$ . If vertex $A$ is at $(1, 6)$ and vertex $B$ is at $(5, 2)$ , what are the coordinates of vertex $C$ ?

$(6, 1)$ using the centroid formula for coordinate geometry (correct answer)
$(6, 4)$ using the centroid formula for coordinate geometry
$(10, 1)$ using the centroid formula for coordinate geometry
$(12, 9)$ using the centroid formula for coordinate geometry

Explanation: The centroid is located at

\left(\frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3}\right)

. Given

G(4,3)

A(1,6)

, and

B(5,2)

: For x-coordinate:

\frac{1 + 5 + x_C}{3} = 4

, so

6 + x_C = 12

, thus

x_C = 6

. For y-coordinate:

\frac{6 + 2 + y_C}{3} = 3

, so

8 + y_C = 9

, thus

y_C = 1

. Therefore

C = (6,1)

. Choice B uses incorrect y-calculation. Choice C doubles the x-coordinate. Choice D uses the sum instead of solving for the unknown coordinate.

Question 3

In triangle $XYZ$ , $\angle X = 65°$ and $\angle Y = 45°$ . If triangle $XYZ$ is reflected across the perpendicular bisector of side $XZ$ to form triangle $X'Y'Z'$ , what is the measure of $\angle Y'X'Z'$ in the reflected triangle?

$45°$ because reflections preserve all angle measures exactly
$115°$ because the reflection changes the angle orientation significantly
$70°$ because reflections preserve all angle measures exactly
$65°$ because reflections preserve all angle measures exactly (correct answer)

Explanation: When you encounter reflection problems in geometry, remember that reflections are rigid transformations that preserve all distances, angles, and shapes exactly. The reflected figure is congruent to the original, just in a new position. First, let's find all angles in the original triangle. Since angles in a triangle sum to

180°

, we have

\angle Z = 180° - 65° - 45° = 70°

. When triangle

XYZ

is reflected across the perpendicular bisector of side

XZ

, the triangle flips over that line. Points

X

and

Z

land on themselves (since they're on the perpendicular bisector), while point

Y

moves to position

Y'

on the opposite side. In the reflected triangle

X'Y'Z'

, vertex

X'

corresponds to original vertex

X

. Since reflections preserve all angle measures,

\angle Y'X'Z' = \angle YXZ = 65°

. Choice A is incorrect because it gives

45°

, which was the measure of

\angle Y

in the original triangle, not

\angle X

. Choice B is wrong because

115°

doesn't correspond to any angle in this triangle, and the reasoning about "orientation change" is false—reflections don't change angle measures. Choice C incorrectly gives

70°

, which was

\angle Z

in the original triangle. Choice D correctly identifies that

\angle Y'X'Z' = 65°

because reflections preserve angle measures exactly. Study tip: Remember that all rigid transformations (reflections, rotations, translations) preserve angle measures. When solving reflection problems, identify corresponding vertices carefully, then apply the preservation property.

Question 4

In triangle $ABC$ , point $M$ is the midpoint of side $AB$ and point $N$ is the midpoint of side $AC$ . If $BC = 18$ and $\angle BAC = 50°$ , what can be concluded about the relationship between segment $MN$ and side $BC$ ?

$MN$ is parallel to $BC$ and $MN = 9$ , with $\angle AMN = 50°$ (correct answer)
$MN$ is parallel to $BC$ and $MN = 18$ , with $\angle AMN = 50°$
$MN$ is perpendicular to $BC$ and $MN = 9$ , with $\angle AMN = 40°$
$MN$ is parallel to $BC$ and $MN = 9$ , with $\angle AMN = 130°$

Explanation: By the midpoint theorem, the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. Therefore,

MN \parallel BC

and

MN = \frac{1}{2} \cdot 18 = 9

. Since

MN \parallel BC

, corresponding angles are equal, so

\angle AMN = \angle BAC = 50°

. Choice B incorrectly states

MN = 18

(should be half). Choice C incorrectly claims perpendicularity and wrong angle. Choice D incorrectly calculates the angle as supplementary.

Question 5

In triangle ABC, point D lies on side BC such that AD bisects ‚à†BAC. If AB = 12 and AC = 15, what is the ratio BD:DC?

4:5 by the angle bisector theorem and proportional segments (correct answer)
5:4 by the angle bisector theorem and proportional segments
3:4 by the angle bisector theorem and proportional segments
4:3 by the angle bisector theorem and proportional segments

Explanation: By the angle bisector theorem, when AD bisects ‚à†BAC, the ratio BD:DC = AB:AC = 12:15 = 4:5. Choice B reverses the ratio. Choices C and D use incorrect proportions that don't match the given side lengths.

Question 6

In triangle $PQR$ , the three medians intersect at point $G$ . If median $PS$ has length $15$ , where $S$ is the midpoint of $QR$ , what is the distance from $G$ to vertex $P$ ?

$5$ units from $P$ along the median to the centroid
$7.5$ units from $P$ along the median to the centroid
$10$ units from $P$ along the median to the centroid (correct answer)
$12$ units from $P$ along the median to the centroid

Explanation: The medians of a triangle meet at the centroid, which divides each median in a 2:1 ratio, with the longer segment being from the vertex to the centroid. Since median

PS = 15

, the distance from

P

to centroid

G

\frac{2}{3} \cdot 15 = 10

units. Choice A gives

\frac{1}{3}

of the median length. Choice B gives half the median length. Choice D gives

\frac{4}{5}

of the median length, incorrectly applying a different ratio.

Question 7

Triangle $DEF$ is isosceles with $DE = DF$ . If the measure of the exterior angle at vertex $E$ is $110°$ , what is the measure of $\angle EDF$ ?

$35°$ because the base angles sum to $70°$
$40°$ because each base angle measures $70°$ (correct answer)
$70°$ because the exterior angle equals the base angle
$110°$ because vertical angles are congruent to exterior angles

Explanation: The exterior angle at

E

110°

, so the interior angle at

E

180° - 110° = 70°

. Since triangle

DEF

is isosceles with

DE = DF

, the base angles

\angle DEF

and

\angle DFE

are congruent, each measuring

70°

. Using the triangle angle sum theorem:

\angle EDF + 70° + 70° = 180°

, so

\angle EDF = 40°

. Choice A incorrectly divides the base angle sum by 2. Choice C confuses the exterior angle with a base angle. Choice D incorrectly applies the exterior angle relationship.