Praxis Math Quiz: Apply Shape Properties

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Apply Shape PropertiesQuestion 1 of 20

A shape is formed by joining an equilateral triangle to one side of a square. The two shapes share a common side. If the perimeter of the entire composite shape is 60 inches, what is the length of the common side?

10 inches

12 inches

15 inches

20 inches

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Praxis Math Quiz

Praxis Math Quiz: Apply Shape Properties

Practice Apply Shape Properties in Praxis Math 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 Apply Shape Properties, giving you a quick way to practice the rules, question types, and explanations that matter most for Praxis Math.

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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.

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Question 1

10 inches
12 inches (correct answer)
15 inches
20 inches

Explanation: Let the side length of the square be

s

. Since the equilateral triangle is joined to one side, its side length is also

s

. The perimeter of the composite shape consists of the sides that are on the exterior. The square has 4 sides and the triangle has 3 sides. One side is shared, so it is not on the perimeter. Thus, the perimeter is composed of 3 sides from the square and 2 sides from the triangle. The total number of sides forming the perimeter is

3 + 2 = 5

. Since all side lengths are equal to

s

, the perimeter is

5s

. We are given that the perimeter is 60 inches. So,

5s = 60

, which means

s = 12

inches.

Question 2

Refer to the figure. Trapezoid $WXYZ$ has $\overline{WX}\parallel\overline{ZY}$ , $WX=6$ , $ZY=14$ , and legs $WZ=XY=5$ . Point $M$ is the midpoint of $\overline{WZ}$ and point $N$ is the midpoint of $\overline{XY}$ . What is the length of $\overline{MN}$ ?

$8$
$10$ (correct answer)
$11$
$12$

Explanation: The segment connecting midpoints of the legs of a trapezoid (the midsegment) is parallel to the bases and has length equal to the average of the bases:

MN=\frac{WX+ZY}{2}=\frac{6+14}{2}=10

. (A) 8 is the difference 14−6. (C) 11 is average including one leg. (D) 12 adds the legs' average. The leg lengths are irrelevant distractors.

Question 3

A triangle has two sides of length 7 cm and 12 cm. If the length of the third side, $x$ , is an integer, what is the difference between the maximum and minimum possible values of $x$ ?

11
12 (correct answer)
13
14

Explanation: According to the Triangle Inequality Theorem, the length of the third side

x

must be greater than the difference of the other two sides and less than their sum. So,

12 - 7 < x < 12 + 7

, which simplifies to

5 < x < 19

. Since

x

must be an integer, the minimum possible value for

x

is 6 and the maximum possible value is 18. The difference between the maximum and minimum values is

18 - 6 = 12

Question 4

In rectangle ABCD, the diagonals intersect at point E. If the measure of angle BDC is 40°, what is the measure of angle AEB?

40°
50°
80°
100° (correct answer)

Explanation: In rectangle ABCD, opposite sides are parallel, so AB || DC. When diagonal BD intersects these parallel sides, alternate interior angles are equal. Therefore, angle ABD = angle BDC = 40°. The diagonals of a rectangle bisect each other, so AE = BE, making triangle ABE isosceles. In an isosceles triangle, the base angles are equal, so angle BAE = angle ABE = 40°. Using the triangle angle sum: angle AEB = 180° - 40° - 40° = 100°.

Question 5

In isosceles triangle ABC, side AB is congruent to side AC. If the measure of angle B is 75°, what is the measure of the exterior angle at vertex A?

30°
105°
150° (correct answer)
165°

Explanation: In an isosceles triangle, the angles opposite the congruent sides are equal. Since AB = AC, angle C must be equal to angle B. So,

m\angle C = 75°

. The sum of angles in a triangle is 180°, so

m\angle A + m\angle B + m\angle C = 180°

. This means

m\angle A + 75° + 75° = 180°

, which simplifies to

m\angle A + 150° = 180°

. Therefore, the interior angle at vertex A is

m\angle A = 30°

. The exterior angle at a vertex is supplementary to the interior angle. The exterior angle at vertex A is

180° - 30° = 150°

. Alternatively, the exterior angle at A is equal to the sum of the remote interior angles B and C, which is

75° + 75° = 150°

Question 6

In parallelogram ABCD, the diagonals intersect at point E. If AE = $2x + 3$ and EC = $4x - 9$ , what is the length of the diagonal AC?

6
15
24
30 (correct answer)

Explanation: A property of parallelograms is that their diagonals bisect each other. This means that segment AE must be equal in length to segment EC. Set up the equation:

2x + 3 = 4x - 9

. Subtract

2x

from both sides to get

3 = 2x - 9

. Add 9 to both sides to get

12 = 2x

. Solve for

x

to find

x = 6

. The question asks for the length of the entire diagonal AC. First, find the length of AE by substituting

x=6

AE = 2(6) + 3 = 12 + 3 = 15

. Since the diagonals bisect each other,

AC = 2 \times AE = 2 \times 15 = 30

Question 7

In parallelogram EFGH, the measure of angle E is $(3x + 10)°$ and the measure of angle G is $(5x - 30)°$ . What is the measure of angle F?

70°
90°
110° (correct answer)
120°

Explanation: In a parallelogram, opposite angles are equal in measure. Therefore,

m\angle E = m\angle G

. We can set up the equation:

3x + 10 = 5x - 30

. To solve for

x

, subtract

3x

from both sides:

10 = 2x - 30

. Add 30 to both sides:

40 = 2x

. Thus,

x = 20

. Now, find the measure of angle E:

m\angle E = 3(20) + 10 = 60 + 10 = 70°

. In a parallelogram, consecutive angles are supplementary (add up to 180°). Therefore,

m\angle E + m\angle F = 180°

. Substituting the value for angle E:

70° + m\angle F = 180°

. Solving for angle F gives

m\angle F = 110°

Question 8

A quadrilateral has diagonals that are congruent, but they do not bisect each other. What is the most specific name for this quadrilateral?

Rectangle
Rhombus
Parallelogram
Isosceles Trapezoid (correct answer)

Explanation: Let's analyze the properties. In a parallelogram, diagonals bisect each other. Since the diagonals do not bisect each other, it cannot be a parallelogram, and therefore cannot be a rectangle, rhombus, or square. An isosceles trapezoid has congruent diagonals. In a general isosceles trapezoid, the diagonals do not bisect each other. Therefore, the most specific name for a quadrilateral with these properties is an isosceles trapezoid.

Question 9

Each interior angle of a regular polygon measures 150°. How many diagonals does this polygon have?

35
44
54 (correct answer)
65

Explanation: First, find the number of sides,

n

. The formula for the measure of an interior angle of a regular n-gon is

\frac{(n-2) \times 180}{n}

. Set this equal to 150:

\frac{(n-2) \times 180}{n} = 150

. Multiply both sides by

n

180(n-2) = 150n

. Distribute:

180n - 360 = 150n

. Subtract

150n

from both sides:

30n - 360 = 0

. Add 360 to both sides:

30n = 360

. Divide by 30:

n = 12

. The polygon is a dodecagon. The formula for the number of diagonals in an n-gon is

D = \frac{n(n-3)}{2}

. Substitute

n=12

D = \frac{12(12-3)}{2} = \frac{12(9)}{2} = \frac{108}{2} = 54

Question 10

A quadrilateral has diagonals that are perpendicular bisectors of each other. If the quadrilateral is not a square, which statement must be true?

All four interior angles are congruent.
All four sides are congruent. (correct answer)
The diagonals are congruent in length.
Adjacent angles are complementary.

Explanation: A quadrilateral with diagonals that are perpendicular bisectors of each other is a rhombus. A square is a special type of rhombus. If the quadrilateral is not a square, it is a non-square rhombus. A key property of all rhombuses is that all four sides are congruent. Therefore, this statement must be true. Distractor A is true for a square, but not a non-square rhombus. Distractor C is a property of rectangles and squares, not all rhombuses. Distractor D is incorrect; adjacent angles in a rhombus are supplementary, not complementary.

Question 11

According to the inclusive definition, a trapezoid is a quadrilateral with at least one pair of parallel sides. Using this definition, which of the following statements is true?

No rhombus is a trapezoid.
All rectangles are trapezoids. (correct answer)
A kite with two right angles is a trapezoid.
A trapezoid can never be a parallelogram.

Explanation: The inclusive definition of a trapezoid requires at least one pair of parallel sides. A rectangle, by definition, has two pairs of parallel sides. Since two is 'at least one', all rectangles fit the definition of a trapezoid. A: A rhombus has two pairs of parallel sides, so it is a trapezoid by this definition. C: A kite has no parallel sides unless it is also a rhombus. D: A parallelogram has two pairs of parallel sides, so it is always a trapezoid under the inclusive definition.

Question 12

Refer to the figure. In the figure, $\triangle ABC\sim\triangle ADE$ with $D$ on $\overline{AB}$ and $E$ on $\overline{AC}$ . If $AD=4$ , $DB=6$ , and the area of trapezoid $DBCE$ is $84$ , what is the area of $\triangle ADE$ ?

$16$ (correct answer)
$24$
$33.6$
$56$

Explanation: The ratio of similarity is

AD:AB=4:10=2:5

, so the ratio of areas is

4:25

. Let the area of

\triangle ADE=4k

; then area of

\triangle ABC=25k

and area of trapezoid

DBCE=25k-4k=21k=84

, giving

k=4

. Area of

\triangle ADE=4k=16

. (B) 24 uses linear ratio

2/5

of 84−... error. (C) 33.6 uses ratio 2:5 on 84 directly. (D) 56 uses

21k

wrong subtraction.

Question 13

In the figure shown, equilateral triangle $ABC$ has side length $12$ . Point $D$ is on $\overline{BC}$ such that $BD=4$ . What is the length of $\overline{AD}$ ?

$4\sqrt{7}$ (correct answer)
$8\sqrt{2}$
$4\sqrt{3}$
$2\sqrt{37}$

Explanation: Drop a perpendicular from

A

\overline{BC}

meeting at midpoint

M

(since triangle is equilateral), so

BM=6

and

AM=6\sqrt{3}

. Then

DM=BM-BD=6-4=2

, and

AD=\sqrt{AM^2+DM^2}=\sqrt{108+4}=\sqrt{112}=4\sqrt{7}

. Alternatively, use the Law of Cosines in triangle

ABD

AD^2=12^2+4^2-2(12)(4)\cos 60°=144+16-48=112

. (B)

8\sqrt{2}=\sqrt{128}

uses cos 60° = ½ sign wrong. (C)

4\sqrt{3}

is the altitude length mistaken. (D)

2\sqrt{37}=\sqrt{148}

uses

+48

instead of

-48

Question 14

Refer to the figure. In the figure, $ABCD$ is a square with side length $10$ , and semicircles are drawn with $\overline{AB}$ and $\overline{CD}$ as diameters, both opening into the interior of the square. What is the area of the region inside the square but outside both semicircles?

$100-25\pi$ (correct answer)
$100-50\pi$
$50-25\pi$
$100-12.5\pi$

Explanation: Each semicircle has radius

5

, so area

\frac{1}{2}\pi(5)^2=\frac{25\pi}{2}

. The two semicircles meet at the center of the square (both diameters are 10, and the square has side 10, so each semicircle extends 5 units into the square — exactly reaching the center line). The two semicircles do not overlap; they are tangent at the center. Total area of semicircles

=2\cdot\frac{25\pi}{2}=25\pi

. Region outside both

=100-25\pi

. (B) forgets to halve the full circle areas. (C) halves the square area erroneously. (D) halves 25π incorrectly.

Question 15

The side lengths of a triangle are given by $x$ , $x+4$ , and $2x-1$ . Which of the following is a possible value for $x$ ?

1.5
2.0
2.5
3.0 (correct answer)

Explanation: For a valid triangle, all side lengths must be positive, and the sum of the lengths of any two sides must be greater than the length of the third side. First, ensure positive side lengths:

x > 0

x+4 > 0

(true for

x>0

), and

2x-1 > 0

(so

x > 0.5

). The strictest condition is

x > 0.5

. Now, apply the Triangle Inequality Theorem: 1)

x + (x+4) > 2x-1 \implies 2x+4 > 2x-1 \implies 4 > -1

(This is always true). 2)

x + (2x-1) > x+4 \implies 3x-1 > x+4 \implies 2x > 5 \implies x > 2.5

. 3)

(x+4) + (2x-1) > x \implies 3x+3 > x \implies 2x > -3 \implies x > -1.5

(This is redundant). The combined conditions are

x > 0.5

and

x > 2.5

. The stricter condition is

x > 2.5

. Of the given choices, only 3.0 is greater than 2.5.

Question 16

The measure of an exterior angle of a triangle is 110°. One of the non-adjacent interior angles measures 40°. Which of the following correctly classifies the triangle?

Scalene and acute
Isosceles and obtuse
Isosceles and acute (correct answer)
Scalene and right

Explanation: The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Let the interior angles be A, B, and C. Let the exterior angle at vertex C be 110°. Then

A + B = 110°

. We are given one of these is 40°, let's say

A = 40°

. Then

40° + B = 110°

, which means

B = 70°

. The interior angle at vertex C is supplementary to the exterior angle, so

C = 180° - 110° = 70°

. The three interior angles are 40°, 70°, and 70°. Since two angles are equal, the triangle is isosceles. Since all angles are less than 90°, the triangle is acute.

Question 17

The lengths of the sides of a triangle are 9, 40, and 41. What is the sum of the measures of the two smallest angles of the triangle?

45°
60°
90° (correct answer)
99°

Explanation: First, determine the type of triangle by checking the Pythagorean theorem. The two shorter sides are 9 and 40, and the longest side is 41. Check if

9^2 + 40^2 = 41^2

81 + 1600 = 1681

, and

41^2 = 1681

. Since the condition is met, the triangle is a right triangle, and the angle opposite the hypotenuse (the side of length 41) is 90°. The sum of all three angles in a triangle is 180°. The two smallest angles are the two acute angles. Their sum is

180° - 90° = 90°

Question 18

Which of the following sets of side lengths could form an isosceles right triangle?

5, 5, 8
6, 8, 10
6, 6, $6\sqrt{2}$ (correct answer)
4, $4\sqrt{3}$ , 8

Explanation: An isosceles right triangle must satisfy two conditions: two sides must be equal (isosceles), and the side lengths must satisfy the Pythagorean theorem

a^2 + b^2 = c^2

(right triangle). A: It is isosceles, but

5^2 + 5^2 = 50 \neq 8^2 = 64

. B: It is a right triangle (

6^2+8^2=100=10^2

), but it is not isosceles. C: It is isosceles with two sides of length 6. Let's check the Pythagorean theorem:

6^2 + 6^2 = 36 + 36 = 72

. The third side squared is

(6\sqrt{2})^2 = 36 \times 2 = 72

. Since

6^2 + 6^2 = (6\sqrt{2})^2

, it is a right triangle. D: This represents the side ratios of a 30-60-90 triangle, which is a right triangle but not isosceles.

Question 19

The sum of the measures of the interior angles of a polygon is 1,080°. If the polygon is regular, what is the measure of one of its exterior angles?

36°
45° (correct answer)
60°
135°

Explanation: The formula for the sum of the interior angles of a polygon with

n

sides is

S = (n-2) \times 180°

. We are given

S = 1080°

. So,

1080 = (n-2) \times 180

. Dividing both sides by 180 gives

6 = n-2

, so

n = 8

. The polygon is an octagon. For any convex polygon, the sum of the exterior angles is 360°. In a regular polygon, all exterior angles are equal. Therefore, the measure of one exterior angle is

360° / n = 360° / 8 = 45°

Question 20

A circle is inscribed within a square such that it is tangent to all four sides of the square. If the area of the circle is $25\pi$ square units, what is the perimeter of the square?

20
40 (correct answer)
50
100

Explanation: The formula for the area of a circle is

A = \pi r^2

. We are given that the area is

25\pi

. So,

25\pi = \pi r^2

, which means

r^2 = 25

, and the radius

r = 5

. For a circle inscribed in a square, the diameter of the circle is equal to the side length of the square. The diameter is twice the radius, so

d = 2r = 2(5) = 10

. Thus, the side length of the square is 10 units. The perimeter of a square is

P = 4s

, where

s

is the side length. Therefore, the perimeter is

4 \times 10 = 40

units.