Geometry Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 2

Match each polygon description based on its angle properties to the correct geometric shape.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Match the

540^\circ

sum to Pentagon, the

120^\circ

interior angle to Regular Hexagon, the

1080^\circ

sum to Octagon, and the equal interior/exterior sums to Quadrilateral.

Each match correctly applies the sum of interior angles formula,

S = (n-2) \times 180^\circ

, or relies on the fact that the sum of exterior angles for any convex polygon is exactly

360^\circ

. Solving for

n

reveals the specific polygon.

Step-by-Step Solution

1

Find the number of sides for a polygon with a

540^\circ

interior angle sum.

n = 5

We apply the formula

(n-2) \times 180 = 540

, which simplifies to

n-2 = 3

.

2

Determine the number of sides for a regular polygon with a

120^\circ

interior angle.

n = 6

The exterior angle is

180^\circ - 120^\circ = 60^\circ

. The number of sides is

360^\circ / 60^\circ = 6

.

3

Find the number of sides for a polygon with a

1080^\circ

interior angle sum.

n = 8

Applying the formula gives

(n-2) \times 180 = 1080

, which simplifies to

n-2 = 6

.

4

Identify the polygon where the interior angle sum equals the exterior angle sum.

n = 4

Since the exterior sum is always

360^\circ

, we set

(n-2) \times 180 = 360

, yielding

n-2 = 2

.

Key Concept

Polygon Interior and Exterior Angle Formulas

Estimated Time:1m 30s

Question 22Question →

A circular stage has a radius of $5$ meters. Which of the following represents the total area of the stage, in square meters?

10\pi

5\pi

25\pi

100\pi

Show answer & explanation

Answer: $25\pi$

Answer

The area of the stage is

25\pi

square meters.

The correct answer is found by using the formula for the area of a circle,

A = \pi r^2

. Substituting the radius of 5 meters into the formula gives

A = \pi (5)^2 = 25\pi

.

Step-by-Step Solution

1

Identify the correct formula for the area of a circle.

A = \pi r^2

The question asks for the total area of a circular surface.

2

Substitute the given radius into the area formula.

A = \pi (5)^2

The problem states that the radius of the stage is 5 meters.

3

Calculate the final area.

A = 25\pi

Squaring 5 yields 25, resulting in an area of

25\pi

.

Key Concept

Area of a Circle

Estimated Time:45s

Question 23Question →

A rectangular community garden has an area of $120$ square feet. If the length of the garden is $15$ feet, what is the width of the garden, in feet?

Show answer & explanation

Answer: 8

Answer

8

The area of a rectangle is calculated by multiplying its length by its width (

A = l \times w

). To find the missing width when the area and length are known, you divide the total area by the length (

w = A \div l

). Dividing

120

by

15

results in exactly

8

feet.

Step-by-Step Solution

1

Identify the given values from the problem

Area = 120, Length = 15

To know which values to substitute into the formula for the area of a rectangle.

2

Apply the area formula for a rectangle

120 = 15 \times w

The formula relates the total area to the length and width: Area = length \times width.

3

Solve for the unknown width

w = 8

Dividing the area by the length isolates the width variable, giving the final dimension.

Key Concept

Calculating a missing dimension of a rectangle given its area and one other dimension.

Estimated Time:45s

Question 24Question →

Two horizontal parallel lines, $L_1$ and $L_2$ , are drawn with $L_1$ above $L_2$ . A point $P$ is located between the two lines. Point $A$ is on $L_1$ and point $B$ is on $L_2$ , such that $P$ is positioned to the right of both $A$ and $B$ . Segment $AP$ and segment $BP$ form a zigzag pattern. The interior angle at $A$ , below $L_1$ and opening towards $P$ , measures $(5x - 4)^\circ$ . The interior angle at $B$ , above $L_2$ and opening towards $P$ , measures $(3x + 16)^\circ$ . The angle $\angle APB$ , opening towards the left, measures $(11x - 42)^\circ$ . What is the measure of the complement of the smaller of the two angles at $A$ and $B$ ?

44^\circ

52^\circ

20^\circ

110^\circ

Show answer & explanation

Answer: $20^\circ$

Answer

The correct answer is

20^\circ

.

Based on the properties of parallel lines intersected by transversals, an auxiliary parallel line drawn through the vertex of the zigzag shows that the middle angle equals the sum of the two outer angles. Setting up the equation

(5x - 4) + (3x + 16) = 11x - 42

yields

x = 18

. Substituting this value back in gives angles of

86^\circ

and

70^\circ

. The smaller angle is

70^\circ

, and its complement is found by subtracting it from

90^\circ

, which gives

20^\circ

.

Step-by-Step Solution

1

Apply geometric properties to set up an equation.

(5x - 4) + (3x + 16) = 11x - 42

By drawing an auxiliary line through point P parallel to L1 and L2, the angle at P is split into two alternate interior angles. Therefore, the angle opening left equals the sum of the two angles opening right.

2

Combine like terms on the left side of the equation.

8x + 12 = 11x - 42

Combining the

x

terms (

5x + 3x = 8x

) and the constants (

-4 + 16 = 12

) simplifies the equation.

3

Solve for x.

x = 18

Subtract

8x

from both sides to get

12 = 3x - 42

. Add

42

to both sides to get

54 = 3x

. Divide by

3

to find

x = 18

.

4

Substitute x back into the expressions to find the measures of the angles at A and B.

Angle at A =

86^\circ

, Angle at B =

70^\circ

Angle at A:

5(18) - 4 = 90 - 4 = 86^\circ

. Angle at B:

3(18) + 16 = 54 + 16 = 70^\circ

.

5

Identify the smaller angle and calculate its complement.

20^\circ

The smaller angle is

70^\circ

. The complement of an angle is

90^\circ

minus the angle, so

90^\circ - 70^\circ = 20^\circ

.

Key Concept

Parallel Lines and Transversals (Zigzag Theorem)

Question 25Question →

A newly constructed rectangular parking lot is $40$ yards long and $25$ yards wide. What is the total perimeter of the parking lot, in yards?

65

105

130

1,000

Show answer & explanation

Answer: $130$

Answer

The total perimeter of the parking lot is

130

yards.

To find the perimeter of a rectangle, you must add the lengths of all four sides. Using the formula

P = 2(l + w)

, we substitute the given length (

40

) and width (

25

). Adding those gives

65

, and multiplying by

2

results in a total perimeter of

130

yards.

Step-by-Step Solution

1

Identify the shape and the given dimensions.

Shape: Rectangle. Length (

l

) =

40

yards, Width (

w

) =

25

yards.

Understanding the shape and dimensions is necessary to select the correct formula.

2

Recall the formula for the perimeter of a rectangle.

P = 2l + 2w

or

P = 2(l + w)

Perimeter is the total distance around the outside of the shape.

3

Substitute the dimensions into the formula and solve.

P = 2(40 + 25) = 2(65) = 130

Calculating the expression yields the final perimeter in yards.

Key Concept

Perimeter of Rectangles

Alternative Method

Instead of using the formula

P = 2(l + w)

, you can simply add all four sides together:

40 + 40 + 25 + 25 = 130

.

Estimated Time:45s

Question 26Question →

Solve the following word problem by applying properties of right triangles.

Fill in the blanks below

A 25-foot ladder is leaning against a vertical wall, with its base placed 7 feet away from the bottom of the wall. If the top of the ladder slides exactly 4 feet straight down the wall, the base of the ladder will slide feet further away from the wall.

Show answer & explanation

Answer

8

By applying the Pythagorean theorem to the initial position, we establish that the ladder reached a height of 24 feet (

25^2 - 7^2 = 24^2

). When it slips 4 feet down, the new height becomes 20 feet. Using the Pythagorean theorem again for the new position with a constant hypotenuse of 25, the new base distance is 15 feet (

25^2 - 20^2 = 15^2

). The ladder started 7 feet away, so it moved an additional 8 feet (

15 - 7 = 8

).

Step-by-Step Solution

1

Find the initial height of the top of the ladder on the wall using the Pythagorean theorem.

a^2 + 7^2 = 25^2

, which simplifies to

a^2 + 49 = 625

. Solving for

a^2

gives

576

, so

a = 24

feet.

The ladder, the wall, and the ground form a right triangle where the ladder is the hypotenuse.

2

Calculate the new height of the top of the ladder after it slides down the wall.

24 - 4 = 20

feet.

The top of the ladder slid exactly 4 feet downwards, reducing the vertical leg of our right triangle.

3

Find the new distance of the base of the ladder from the wall.

b^2 + 20^2 = 25^2

, which simplifies to

b^2 + 400 = 625

. Solving for

b^2

gives

225

, so

b = 15

feet.

The ladder's overall length does not change, creating a new right triangle with a height of 20 and a hypotenuse of 25.

4

Calculate the difference between the new base distance and the original base distance.

15 - 7 = 8

feet.

The question specifically asks how much further the base slid away from the wall, not the final total distance.

Key Concept

Applying the Pythagorean theorem iteratively to linked right triangles that share a common hypotenuse length

Question 27Question →

An architect is designing a room in the shape of a convex pentagon. The measures of the interior angles of the room are defined algebraically as $(x + 20)^\circ$ , $(x + 10)^\circ$ , $(x + 5)^\circ$ , $(x - 5)^\circ$ , and $(2x)^\circ$ . What is the measure of the smallest interior angle of this room?

10^\circ

50^\circ

80^\circ

90^\circ

Show answer & explanation

Answer: $80^\circ$

Answer

The smallest interior angle measures

80^\circ

.

A pentagon has 5 sides, so the sum of its interior angles is

(5 - 2) \times 180^\circ = 540^\circ

. Setting the sum of the algebraic expressions equal to

540^\circ

yields

6x + 30 = 540

. Subtracting

30

leaves

6x = 510

, so

x = 85

. Substituting

85

into the smallest expression,

(x - 5)^\circ

, gives

85 - 5 = 80^\circ

.

Step-by-Step Solution

1

Determine the sum of the interior angles for a pentagon.

Using the formula

(n - 2) \times 180^\circ

for a 5-sided polygon (

n = 5

), the sum is

(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

.

Before solving for

x

, the total sum that the algebraic expressions add up to must be known.

2

Set up an equation by adding all the given angle expressions.

(x + 20) + (x + 10) + (x + 5) + (x - 5) + 2x = 540

The sum of the individual interior angles must equal the total calculated sum of the polygon's angles.

3

Combine like terms and solve for

x

.

6x + 30 = 540 \Rightarrow 6x = 510 \Rightarrow x = 85

Simplifying the algebraic equation isolates the variable

x

.

4

Substitute

x = 85

back into each expression to find all angle measures and identify the smallest one.

The angles are

105^\circ

,

95^\circ

,

90^\circ

,

80^\circ

, and

170^\circ

. The smallest of these is

80^\circ

.

The variable

x

alone is not the answer; it must be substituted back into the expressions to find the actual angle measures.

Key Concept

The sum of the interior angles of a convex polygon is

(n-2) \times 180^\circ

. This geometric principle can be combined with algebraic equations to solve for unknown angle measures.

Question 28Question →

A civil engineer is analyzing the intersection of two straight structural beams, Beam 1 and Beam 2. Angle $X$ and angle $Y$ form a linear pair along Beam 1. Angle $Y$ and angle $Z$ are vertical angles formed by the intersection of the two beams. The measure of angle $X$ is given by the expression $(3x - 10)^\circ$ and the measure of angle $Z$ is represented by $(2x + 15)^\circ$ . What is the measure of angle $Y$ in degrees?

49

65

85

89

Show answer & explanation

Answer: 85

Answer

85

The correct answer is found by recognizing that since angle X and angle Y form a linear pair, they sum to 180 degrees. Because angle Y and angle Z are vertical angles, they are equal. Substituting Z for Y means angle X and angle Z must also sum to 180 degrees. Setting up the equation

(3x - 10) + (2x + 15) = 180

yields

x = 35

. Substituting this back into the expression for angle Z (which is the same measure as angle Y) gives

2(35) + 15 = 85^\circ

.

Step-by-Step Solution

1

Identify the relationship between angle X and angle Z.

Since angle X and angle Y form a linear pair, they are supplementary (

X + Y = 180^\circ

). Angle Y and angle Z are vertical angles, so they are equal (

Y = Z

). Therefore, angle X and angle Z must also be supplementary (

X + Z = 180^\circ

).

Establishing the geometric relationship allows us to set up an algebraic equation based on the given expressions.

2

Set up and solve the algebraic equation for x.

(3x - 10) + (2x + 15) = 180 \rightarrow 5x + 5 = 180 \rightarrow 5x = 175 \rightarrow x = 35

.

Finding the numerical value of the variable is necessary to determine the specific angle measures.

3

Substitute the value of x back into the expression for angle Z (which equals angle Y).

Angle

Y = \text{Angle } Z = 2(35) + 15 = 70 + 15 = 85^\circ

.

The question specifically asks for the measure of angle Y in degrees.

Key Concept

Applying algebraic equations to solve for unknown values in supplementary and vertical angle relationships.

Question 29Question →

A competitive swimmer is walking around the outer edge of a rectangular pool to warm up. If the pool has a length of $50$ meters and a width of $25$ meters, what is the total perimeter of the pool, in meters?

Show answer & explanation

Answer: 150

Answer

The total perimeter of the pool is 150 meters.

The perimeter of a rectangle is found by adding the lengths of all four sides. Using the formula

P = 2l + 2w

, we calculate

2(50) + 2(25) = 150

meters.

Step-by-Step Solution

1

Identify the shape and the goal.

The shape is a rectangle and we need to find its perimeter.

Understanding what needs to be calculated determines the correct formula.

2

Recall the perimeter formula for a rectangle.

P = 2l + 2w

or

P = 2(l + w)

The perimeter is the total distance around the outside of the shape.

3

Substitute the given dimensions into the formula.

P = 2(50) + 2(25)

The length (

l

) is

50

and the width (

w

) is

25

.

4

Perform the arithmetic operations.

100 + 50 = 150

Multiplying the terms and adding them together yields the final perimeter.

Key Concept

Perimeter of Rectangles

Question 30Question →

A rectangular metallic sheet has a length of $28$ inches and a width of $15$ inches. A triangular metallic sheet has the exact same area. If the base of the triangular sheet is $24$ inches, what is its height, in inches?

Show answer & explanation

Answer: 35

Answer

35

The correct height is found by first determining the area of the rectangle (

28 \times 15 = 420

). Since the triangle shares this exact area and has a given base of

24

, solving the triangle area formula

420 = \frac{1}{2}(24)h

yields

12h = 420

, which simplifies to

h = 35

.

Step-by-Step Solution

1

Calculate the area of the rectangular sheet.

Area = 420 square inches

The area of the rectangle is required to find the area of the triangle, as the problem explicitly states their areas are equal.

2

Set up the area equation for the triangular sheet.

420 = 0.5 * 24 * h

Substituting the known area and base into the triangle area formula allows us to solve for the unknown height.

3

Solve the algebraic equation for the height (h).

h = 35

Simplifying 420 = 12h by dividing both sides by 12 isolates the height variable.

Key Concept

Calculating and equating the area of different 2D figures using standard area formulas.

Estimated Time:1m 0s

Question 31Question →

A convex pentagon has three interior angles that are equal in measure. If the remaining two interior angles are supplementary to each other, what is the measure of one of the three equal angles?

60^\circ

120^\circ

150^\circ

180^\circ

Show answer & explanation

Answer: $120^\circ$

Answer

The measure of one of the three equal angles is

120^\circ

.

A pentagon has 5 sides, so the total sum of its interior angles is

(5 - 2) \times 180^\circ = 540^\circ

. The problem states that three angles are equal (we can call each

x

) and the other two are supplementary. Supplementary angles add up to

180^\circ

. Setting up the algebraic equation gives

3x + 180^\circ = 540^\circ

. Subtracting

180^\circ

from both sides yields

3x = 360^\circ

. Dividing by 3, we find that

x = 120^\circ

.

Step-by-Step Solution

1

Calculate the total sum of the interior angles of a pentagon.

The sum is

(5 - 2) \times 180^\circ = 540^\circ

.

The formula for the sum of interior angles of any convex polygon is

(n - 2) \times 180^\circ

, where

n

is the number of sides.

2

Set up an algebraic equation representing the sum of the five interior angles.

3x + 180^\circ = 540^\circ

Three angles are equal (let them be

x

), and the remaining two angles are supplementary, meaning their combined sum is exactly

180^\circ

.

3

Solve the equation for

x

.

3x = 360^\circ \Rightarrow x = 120^\circ

Subtracting

180^\circ

from both sides isolates the

3x

term. Dividing both sides by 3 gives the value of a single equal angle.

Key Concept

The sum of interior angles of a polygon and the geometric property of supplementary angles.

Question 32Question →

Match each geometric angle scenario to its final calculated angle measure.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The geometric relationships determine the equations: the supplementary pair resolves to

140^\circ

, the complementary pair resolves to

60^\circ

, the alternate interior angles evaluate to

95^\circ

, and the vertical angles evaluate to

90^\circ

.

Each correct match correctly identifies the angle relationship, translates it into an accurate algebraic equation, solves for the variable, and completes the final substitution to find the requested angle measure.

Step-by-Step Solution

1

Identify the core geometric relationship for each scenario.

Determine if the angles sum to

180^\circ

(supplementary), sum to

90^\circ

(complementary), or are mathematically equivalent (alternate interior, vertical).

Recognizing the geometric property is necessary to set up the correct algebraic equation.

2

Translate the text into algebraic equations and solve for the unknown variable.

The variables are calculated as

x = 40

,

x = 60

,

x = 20

, and

y = 20

for the four respective scenarios.

Isolating the variable is an intermediate step required to find the specific angle measurements.

3

Calculate the final requested angle measure using the solved variable.

Substitute the variables back into the expressions or apply the logic to find the specific 'larger angle' requested.

The question asks for the final angle measure, not just the isolated variable.

Key Concept

Lines and Angles

Question 33Question →

In a convex pentagon, two of the interior angles are equal in measure. The other three interior angles are also equal to each other, and each of these three is $40^\circ$ larger than the measure of one of the first two angles. What is the measure of the largest exterior angle of this pentagon?

132^\circ

96^\circ

80^\circ

48^\circ

Show answer & explanation

Answer: $96^\circ$

Answer

The largest exterior angle is

96^\circ

.

The correct answer is

96^\circ

because setting up the equation for a pentagon's sum of interior angles,

2x + 3(x + 40) = 540

, gives

x = 84^\circ

as the smallest interior angle. The exterior angle is supplementary to the interior angle, so

180^\circ - 84^\circ = 96^\circ

.

Step-by-Step Solution

1

Determine the sum of the interior angles of a pentagon.

Using the formula

(n - 2) \times 180^\circ

, for

n = 5

, the sum is

(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

.

You must know the total sum of the interior angles before you can solve for any individual unknown angle.

2

Set up an algebraic equation to represent the interior angles.

Let the two smaller equal angles be

x

. The three larger equal angles are

(x + 40^\circ)

. The equation is

2x + 3(x + 40^\circ) = 540^\circ

.

Translating the word problem into algebra allows us to solve for the specific angle measures.

3

Solve the equation for

x

.

Distribute the 3 to get

2x + 3x + 120^\circ = 540^\circ

. Combine like terms:

5x + 120^\circ = 540^\circ

. Subtract 120 from both sides:

5x = 420^\circ

. Divide by 5:

x = 84^\circ

.

Solving for

x

provides the measure of the smallest interior angles.

4

Determine the largest exterior angle.

An exterior angle is supplementary to its adjacent interior angle. The largest exterior angle corresponds to the smallest interior angle. The largest exterior angle is

180^\circ - 84^\circ = 96^\circ

.

Because exterior and interior angles on a straight line add up to 180 degrees, the smallest interior angle creates the largest exterior angle.

Key Concept

Quadrilaterals and Polygons

Question 34Question →

A rectangular floor is $18$ feet long and $12$ feet wide. A square rug with a side length of $9$ feet is placed on the floor. What is the area of the floor that is NOT covered by the rug, in square feet?

180

24

135

297

Show answer & explanation

Answer: $135$

Answer

The area of the floor not covered by the rug is

135

square feet.

The correct answer is found by calculating the total area of the floor (

18 \times 12 = 216

sq ft) and subtracting the area of the square rug (

9 \times 9 = 81

sq ft). This gives an uncovered area of

216 - 81 = 135

square feet.

Step-by-Step Solution

1

Calculate the total area of the rectangular floor.

\text{Area of floor} = 18 \times 12 = 216 \text{ square feet}

The area of a rectangle is found by multiplying its length by its width.

2

Calculate the area of the square rug.

\text{Area of rug} = 9 \times 9 = 81 \text{ square feet}

The area of a square is found by squaring its side length.

3

Subtract the area of the rug from the total area of the floor.

216 - 81 = 135 \text{ square feet}

The uncovered area is the difference between the total floor area and the area covered by the rug.

Key Concept

Calculating and subtracting areas of 2D figures

Estimated Time:1m 0s

Question 35Question →

A graphic designer creates a rectangular product label that is $5.5$ centimeters long and $4.0$ centimeters wide. What is the area of the label, in square centimeters?

19

9.5

22

220

Show answer & explanation

Answer: $22$

Answer

The correct area of the label is

22

square centimeters.

The area of a rectangle is found by multiplying its length by its width. The length of the label is

5.5

cm and the width is

4.0

cm. Multiplying these together gives

5.5 \times 4.0 = 22

square centimeters.

Step-by-Step Solution

1

Identify the shape and the goal from the word problem.

The shape is a rectangle and the goal is to find its area.

Identifying the shape ensures the correct mathematical formula is applied.

2

Set up the area formula for a rectangle.

\text{Area} = \text{length} \times \text{width}

This is the standard geometric formula for the area of a rectangle.

3

Substitute the given values into the formula and multiply.

\text{Area} = 5.5 \times 4.0 = 22

Multiplying the given dimensions provides the final area in square centimeters.

Key Concept

The area of a rectangle is found by multiplying its length by its width (

A = l \times w

).

Question 36Question →

Match each angle relationship term with its correct geometric definition.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Complementary Angles match with the sum of

90^\circ

, Supplementary Angles match with the sum of

180^\circ

, Vertical Angles match with opposite angles of intersecting lines, and Adjacent Angles match with angles sharing a common vertex and side.

The correct matches directly apply the standard mathematical definitions for these fundamental angle relationships. Correctly distinguishing between these terms is essential for solving complex geometry problems.

Step-by-Step Solution

1

Identify the definition of Complementary Angles.

Matches with 'Two angles whose measures add up to

90^\circ

'.

This is a fundamental geometric property for complementary pairs.

2

Identify the definition of Supplementary Angles.

Matches with 'Two angles whose measures add up to

180^\circ

'.

Supplementary pairs form a straight angle or line when combined adjacently.

3

Identify the definition of Vertical Angles.

Matches with 'Opposite angles formed by the intersection of two lines, which are always equal in measure'.

The intersection of two lines inherently creates these opposite, congruent angle pairs.

4

Identify the definition of Adjacent Angles.

Matches with 'Two angles that share a common vertex and side, but have no interior points in common'.

This physical layout defines adjacency in geometry without dictating a specific angle sum.

Key Concept

Basic Angle Relationships and Definitions

Question 37Question →

Match each geometric description to the correct polygon or quadrilateral it uniquely describes.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The geometric descriptions correspond perfectly to the Dodecagon, Octagon, Rectangle, and Nonagon based on their algebraic angle properties and specific diagonal traits.

Each provided description translates to a specific algebraic or geometric property that uniquely identifies a single polygon. Setting up and solving the respective equations for

n

(number of sides) or applying logical diagnostic properties of quadrilaterals produces the correct matching pairs.

Step-by-Step Solution

1

Analyze the polygon where the interior angle sum is five times the exterior angle sum.

180(n-2) = 5(360) \Rightarrow 180(n-2) = 1800 \Rightarrow n-2 = 10 \Rightarrow n=12

.

Applying the universal constant for exterior angle sums (

360^\circ

) allows solving algebraically for the

12

-sided dodecagon.

2

Solve for the polygon where the number of diagonals equals

2.5

times the number of vertices (

n

).

\frac{n(n-3)}{2} = 2.5n \Rightarrow n-3 = 5 \Rightarrow n=8

.

Equating the formula for the total number of diagonals to

2.5n

allows us to simplify and identify the

8

-sided octagon.

3

Deduce the quadrilateral with congruent, bisecting, non-perpendicular diagonals.

The shape is uniquely identified as a rectangle (excluding a square).

Bisecting diagonals prove it is a parallelogram, congruent diagonals prove it is a rectangle, and non-perpendicularity proves it is not a square.

4

Calculate the sides of the polygon where the interior angle is

100^\circ

larger than the exterior angle.

I + E = 180^\circ

and

I - E = 100^\circ

. Solving the system gives

E = 40^\circ

. Since

n = \frac{360}{E}

,

n = \frac{360}{40} = 9

.

Recognizing that interior and exterior angles form a linear pair lets us set up a system of equations. Finding the exterior angle easily leads to the

9

-sided nonagon.

Key Concept

Advanced Polygon Analysis and Quadrilateral Properties

Question 38Question →

A painter is marking a new parallelogram-shaped crosswalk on a street. The base of the parallelogram is $45$ feet long, and the perpendicular height across the street is $20$ feet. What is the total area of the crosswalk, in square feet?

Show answer & explanation

Answer: 900

Answer

900

The area of a parallelogram is calculated by multiplying its base by its perpendicular height. Given a base of

45

feet and a height of

20

feet, the area is

45 \times 20 = 900

square feet.

Step-by-Step Solution

1

Identify the relevant formula

A = b \times h

The area of a parallelogram is the product of its base and perpendicular height.

2

Substitute the given values into the formula

A = 45 \times 20

The base (

b

) is given as

45

feet and the height (

h

) is

20

feet.

3

Calculate the final area

900

Multiplying

45

by

20

gives

900

.

Key Concept

Area of a Parallelogram

Estimated Time:30s

Question 39Question →

In a geometric diagram, angle $\alpha$ and angle $\beta$ are complementary. Furthermore, angle $\beta$ and angle $\gamma$ are supplementary. If the measure of angle $\alpha$ is represented by the expression $(2x - 5)^\circ$ and the measure of angle $\gamma$ is represented by $(5x - 2)^\circ$ , what is the measure of angle $\beta$ ?

37^\circ

97^\circ

135^\circ

153^\circ

Show answer & explanation

Answer: $37^\circ$

Answer

37^\circ

The correct answer is found by recognizing that angle

\beta

can be expressed in two ways:

90^\circ - \alpha

and

180^\circ - \gamma

. Setting

90 - (2x - 5)

equal to

180 - (5x - 2)

allows us to solve for

x = 29

. Substituting

x

back into the first expression yields

95 - 2(29) = 37^\circ

.

Step-by-Step Solution

1

Express angle beta in terms of x using the complementary relationship.

\beta = 90 - (2x - 5) = 95 - 2x

Since alpha and beta are complementary, their sum is 90 degrees.

2

Express angle beta in terms of x using the supplementary relationship.

\beta = 180 - (5x - 2) = 182 - 5x

Since beta and gamma are supplementary, their sum is 180 degrees.

3

Set the two expressions for beta equal to each other and solve for x.

95 - 2x = 182 - 5x \Rightarrow 3x = 87 \Rightarrow x = 29

Both expressions represent the same angle, so they must be algebraically equivalent.

4

Substitute the value of x back into either expression to find the measure of angle beta.

\beta = 95 - 2(29) = 95 - 58 = 37^\circ

This provides the final numerical measure of the target angle.

Key Concept

Algebraic application of complementary and supplementary angle relationships

Estimated Time:2m 0s

Question 40Question →

An artist is framing a rectangular painting that measures $10$ inches in length and $8$ inches in width. The frame has a uniform width of $2$ inches on all sides. What is the total area of the frame itself, in square inches?

Show answer & explanation

Answer: 88

Answer

88

To find the area of the frame, subtract the area of the painting from the total area of the painting with the frame included. Because the frame is 2 inches wide on all sides, the overall dimensions increase by 4 inches in both directions. The outer rectangle is 14 inches by 12 inches (168 square inches). The inner painting is 10 inches by 8 inches (80 square inches). Subtracting the inner area from the outer area gives 168 - 80 = 88 square inches.

Step-by-Step Solution

1

Determine the outer dimensions of the framed painting.

Outer length = 14 inches, Outer width = 12 inches

The frame adds 2 inches to both the left and right sides (total of 4 inches to length), and 2 inches to both the top and bottom (total of 4 inches to width).

2

Calculate the total combined area of the painting and the frame.

168 square inches

Area of a rectangle is length multiplied by width (14 * 12).

3

Calculate the area of the inner painting.

80 square inches

The painting dimensions are 10 by 8 (10 * 8).

4

Subtract the inner area from the total area to find the area of the frame.

88 square inches

The frame is the region between the outer and inner rectangles.

Key Concept

The area of a border or frame is found by calculating the total outer area and subtracting the inner area.

Geometry

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