Geometry Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 6

Question 101Question →

In the provided technical drawing, two horizontal parallel lines are intersected by a diagonal transversal line. The measures of two same-side interior angles are given by the expressions $(8x - 14)^\circ$ and $(6x + 26)^\circ$ . What is the numerical value of $x$ ?

Show answer & explanation

Answer: 12

Answer

12

The two angles described are same-side interior angles. Because the lines are parallel, these angles are supplementary and their sum is exactly 180 degrees. Setting up the equation

(8x - 14) + (6x + 26) = 180

and solving for

x

correctly yields exactly 12.

Step-by-Step Solution

1

Identify the relationship between the two given angles.

The angles are supplementary.

When two parallel lines are cut by a transversal, same-side interior angles are supplementary, meaning their sum is exactly 180 degrees.

2

Set up an algebraic equation.

(8x - 14) + (6x + 26) = 180

The sum of the measures of the two supplementary angles must equal 180.

3

Combine the like terms on the left side of the equation.

14x + 12 = 180

Adding the variable terms (

8x + 6x = 14x

) and the constant terms (

-14 + 26 = 12

) simplifies the equation.

4

Isolate the variable term.

14x = 168

Subtracting 12 from both sides of the equation leaves only the variable term on the left side.

5

Solve for x.

x = 12

Dividing both sides by 14 gives the final numerical value of x.

Key Concept

Same-side interior angles formed by parallel lines and a transversal are supplementary.

Question 102Question →

Read the scenario below and provide the missing numeric value.

Fill in the blanks below

An architect is designing a courtyard in the shape of a right triangle. The two perpendicular walkways that form the right angle measure

9

meters and

12

meters in length. The length of the diagonal path connecting their endpoints is meters.

Show answer & explanation

Answer

The diagonal path is

15

meters long.

By applying the Pythagorean theorem (

a^2 + b^2 = c^2

) with legs of

9

and

12

, we calculate

9^2 + 12^2 = 81 + 144 = 225

. The square root of

225

is

15

, which is the exact length of the diagonal path.

Step-by-Step Solution

1

Identify the known sides of the right triangle.

The two perpendicular walkways represent the legs of the triangle (

a

and

b

), measuring

9

and

12

.

To accurately apply the Pythagorean theorem, the given measurements must be identified as either legs or the hypotenuse.

2

Apply the Pythagorean theorem (

a^2 + b^2 = c^2

).

9^2 + 12^2 = c^2

The theorem is used to find the hypotenuse (

c

) when the lengths of both legs are known.

3

Square the lengths of both legs.

81 + 144 = c^2

Following mathematical order of operations by evaluating the exponents first.

4

Add the squared values and solve for

c

.

225 = c^2

, which means

c = \sqrt{225} = 15

.

Taking the square root of the sum yields the final straight-line length of the hypotenuse.

Key Concept

Using the Pythagorean theorem (

a^2 + b^2 = c^2

) to calculate the length of the hypotenuse when both legs are known.

Question 103Question →

A flat circular roof of a silo has a circumference of $44$ meters. A sealant must be applied to the entire top surface. If one bucket of sealant covers exactly $30$ square meters, what is the minimum number of buckets that must be purchased to seal the roof? (Use $\pi = \frac{22}{7}$ )

Show answer & explanation

Answer: 6

Answer

6

To find the correct number of buckets, we first derived the radius (

7

meters) from the given circumference (

44

meters). Next, we calculated the total area (

154

square meters). Finally, dividing the total area by the coverage per bucket (

30

square meters) gives a value of

5.133

. Because we cannot purchase a fraction of a bucket and need to cover the entire roof, we must round up to

6

full buckets.

Step-by-Step Solution

1

Set up the circumference formula

C = 2\pi r

using the given values to solve for the radius.

44 = 2 \times \frac{22}{7} \times r

, which simplifies to

44 = \frac{44}{7} \times r

. Multiplying both sides by

\frac{7}{44}

gives

r = 7

meters.

The radius is required to calculate the surface area of the circular roof.

2

Calculate the area of the circular roof using the formula

A = \pi r^2

.

A = \frac{22}{7} \times (7)^2 = \frac{22}{7} \times 49 = 154

square meters.

The total area determines the total amount of sealant that is needed.

3

Determine the number of buckets needed by dividing the total area by the coverage per bucket.

154 \div 30 = 5

with a remainder of

4

(or

5.133...

). Since partial buckets cannot be purchased, you must round up to

6

buckets.

To seal the entire surface area without leaving any sections exposed, you must purchase enough full buckets to cover the total area.

Key Concept

Calculating the area of a circle from its circumference and applying the result to a real-world scenario requiring discrete quantities.

Question 104Question →

A canvas tarp is cut into the shape of a right triangle. The area of the tarp is $60$ square feet, and its shortest leg measures $8$ feet. What is the perimeter of the tarp in feet?

23

40

46

60

Show answer & explanation

Answer: $40$

Answer

40

To find the perimeter, all three sides of the right triangle are needed. Using the given area (

60

) and shortest leg (

8

), the length of the other leg is found with the area formula

A = \frac{1}{2}bh

, yielding

15

feet. Next, the Pythagorean theorem (

8^2 + 15^2 = c^2

) reveals the hypotenuse is

17

feet. Summing the three sides (

8 + 15 + 17

) gives the correct perimeter of

40

feet.

Step-by-Step Solution

1

Find the length of the other leg using the area formula for a triangle,

A = \frac{1}{2}bh

.

60 = \frac{1}{2}(8)b \Rightarrow 60 = 4b \Rightarrow b = 15

feet.

The perimeter requires all three side lengths, so the missing leg must be calculated first.

2

Use the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the hypotenuse.

8^2 + 15^2 = c^2 \Rightarrow 64 + 225 = c^2 \Rightarrow 289 = c^2 \Rightarrow c = 17

feet.

The hypotenuse is the third side needed to find the complete outer boundary of the right triangle.

3

Calculate the perimeter by summing the three side lengths.

P = 8 + 15 + 17 = 40

feet.

The perimeter of any polygon is the total length of its outer boundary.

Key Concept

Calculating missing dimensions of a right triangle using area and the Pythagorean theorem to find the perimeter.

Estimated Time:1m 0s

Question 105Question →

An aviation engineer is evaluating the intersection of two flight paths on a radar display. The flight paths are represented by two straight intersecting lines. Two adjacent angles formed by this intersection have measures of $(8x)^\circ$ and $(2x + 10)^\circ$ . What is the measure of the smaller of these two adjacent angles?

26^\circ

28^\circ

34^\circ

44^\circ

Show answer & explanation

Answer: $44^\circ$

Answer

The smaller angle measures

44^\circ

.

Adjacent angles on a straight line form a linear pair and are supplementary, meaning their sum is 180 degrees. Setting up the equation 8x + 2x + 10 = 180 simplifies to 10x + 10 = 180. Subtracting 10 from both sides leaves 10x = 170, and dividing by 10 yields x = 17. Substituting x back into the expressions gives angle measures of 8(17) = 136 degrees and 2(17) + 10 = 44 degrees. The question asks for the smaller angle, which is 44 degrees.

Step-by-Step Solution

1

Recognize that adjacent angles formed by intersecting lines create a linear pair.

The angles are supplementary, meaning their sum is

180^\circ

.

This geometric rule provides the basis for setting up the equation.

2

Set up an equation representing the sum of the two angles.

8x + (2x + 10) = 180

This translates the geometric relationship into an algebraic expression.

3

Combine the like terms on the left side of the equation.

10x + 10 = 180

Simplifying the equation is the first step in isolating the variable x.

4

Solve for x by subtracting 10 from both sides and then dividing by 10.

10x = 170 \Rightarrow x = 17

These standard algebraic operations isolate x and determine its exact value.

5

Substitute x = 17 back into both original expressions to find the angle measures.

8(17) = 136^\circ

and

2(17) + 10 = 34 + 10 = 44^\circ

The question asks for a specific angle measure, not the variable x.

6

Identify the smaller of the two calculated angles.

44^\circ

The prompt specifically requests the measure of the smaller angle.

Key Concept

Linear pairs of angles form a straight line and are supplementary, meaning they add up to 180 degrees.

Estimated Time:1m 15s

Question 106Question →

A rectangular garden has a perimeter of $56$ meters. The length of the garden is $4$ meters greater than its width. If a $3$ -meter wide paved path is built entirely inside the garden along all four of its borders, what is the area, in square meters, of the remaining unpaved section of the garden?

Show answer & explanation

Answer: 60

Answer

The area of the remaining unpaved section of the garden is 60 square meters.

The original dimensions of the garden are found to be 12 meters and 16 meters based on the given perimeter and side relationships. The inner unpaved section has dimensions of 12 - 6 = 6 meters and 16 - 6 = 10 meters because the 3-meter path cuts into both sides of each dimension. Multiplying these inner dimensions yields a final area of 60 square meters.

Step-by-Step Solution

1

Use the perimeter formula to find the outer length and width of the garden.

Width = 12 meters, Length = 16 meters.

The relationship between length and width (L = W + 4) and the total perimeter formula

2L + 2W = 56

allows us to find the exact dimensions.

2

Calculate the dimensions of the inner unpaved area by subtracting the path's width.

Inner width = 6 meters, Inner length = 10 meters.

The 3-meter path runs along all four borders, meaning the path's width must be subtracted twice (3 + 3 = 6 meters) from both the outer length and the outer width.

3

Calculate the final area of the inner unpaved section.

60 square meters.

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

10 \times 6

).

Key Concept

Solving for unknown dimensions using perimeter, and determining the area of a smaller inscribed rectangle by correctly subtracting border widths.

Question 107Question →

A radar system on a ship sweeps a $45^\circ$ sector of the surrounding water. If the total area scanned by this single sweep is $32\pi$ square nautical miles, what is the maximum detection range (radius) of the radar in nautical miles?

2

32

16

128

Show answer & explanation

Answer: 16

Answer

The maximum detection range (radius) of the radar is 16 nautical miles.

The area of a sector is found using the formula

A = \frac{\theta}{360} \pi r^2

. Substituting the given values gives

32\pi = \frac{45}{360} \pi r^2

. Simplifying the fraction

\frac{45}{360}

to

\frac{1}{8}

and dividing both sides by

\pi

yields

32 = \frac{1}{8} r^2

. Multiplying both sides by 8 isolates

r^2

, resulting in

256 = r^2

. Taking the square root gives a radius of 16.

Step-by-Step Solution

1

Identify the mathematical formula for the area of a circle's sector.

Area = (θ / 360) * π * r^2

This formula relates the known area and central angle to the unknown radius.

2

Substitute the known values from the word problem into the formula.

32π = (45 / 360) * π * r^2

The scanned area is 32π and the sweep angle is 45 degrees.

3

Simplify the fraction and divide both sides by π.

32 = (1/8) * r^2

The fraction 45/360 simplifies to 1/8, and removing π from both sides simplifies the algebraic equation.

4

Multiply both sides of the equation by 8 to isolate r^2.

256 = r^2

Applying the correct inverse operation (multiplying by the reciprocal of 1/8) isolates the squared variable.

5

Take the square root of both sides to solve for r.

r = 16

Since a physical distance must be positive, the principal square root gives the correct radius.

Key Concept

Calculating the radius of a circle given the area of a sector.

Question 108Question →

A rectangular swimming pool is twice as long as it is wide. It is surrounded by a tiled deck that has a uniform width of $2$ meters. If the area of the deck alone is $136$ square meters, what is the perimeter of the swimming pool in meters?

50

60

132

200

Show answer & explanation

Answer: 60

Answer

60

The correct answer accurately models the outer dimensions of the pool plus deck as

(w + 4)

and

(2w + 4)

. Subtracting the inner pool area

(2w^2)

leaves the deck area equation

12w + 16 = 136

. Solving this yields a width of

10

meters and a length of

20

meters, giving a perimeter of

60

meters.

Step-by-Step Solution

1

Define variables for the dimensions of the pool and the outer boundaries including the deck.

Let width =

w

and length =

2w

. The outer dimensions, which add the 2-meter deck to both sides, are width =

w + 4

and length =

2w + 4

.

This establishes algebraic expressions to represent the areas.

2

Set up an equation for the deck area by subtracting the pool's area from the total outer area.

(w + 4)(2w + 4) - (w)(2w) = 136

The area of the deck alone is given as 136 square meters.

3

Expand the expressions and solve the equation for

w

.

(2w^2 + 12w + 16) - 2w^2 = 136 \rightarrow 12w + 16 = 136 \rightarrow 12w = 120 \rightarrow w = 10

meters.

Simplifying isolates the variable

w

to find the width of the pool.

4

Calculate the pool's length and then its perimeter.

Length =

2(10) = 20

meters. Perimeter =

2(10 + 20) = 60

meters.

The question asks for the perimeter of the swimming pool.

Key Concept

Solving for unknown geometric dimensions using the area of a surrounding path, followed by perimeter calculation.

Question 109Question →

Match each angle relationship description with the correct calculated value for its variable.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches are:

x = 20

,

y = 10

,

z = 25

, and

w = 30

.

Each relationship dictates a specific equation setup. Vertical and corresponding angles require setting the expressions equal to each other. Complementary angles require their sum to equal

90^\circ

, while same-side interior angles require their sum to equal

180^\circ

.

Step-by-Step Solution

1

Determine the relationship and setup the equation for the vertical angles.

4x - 12 = 2x + 28

Vertical angles are congruent (equal in measure).

2

Solve the equation for x.

x = 20

Subtract

2x

from both sides (

2x - 12 = 28

), add 12 to both sides (

2x = 40

), and divide by 2.

3

Determine the relationship and setup the equation for the complementary angles.

(4y + 10) + (2y + 20) = 90

Complementary angles have a sum of

90^\circ

.

4

Solve the equation for y.

y = 10

Combine like terms (

6y + 30 = 90

), subtract 30 (

6y = 60

), and divide by 6.

5

Determine the relationship and setup the equation for the same-side interior angles.

(3z + 10) + (z + 70) = 180

Same-side interior angles between parallel lines are supplementary (sum to

180^\circ

).

6

Solve the equation for z.

z = 25

Combine like terms (

4z + 80 = 180

), subtract 80 (

4z = 100

), and divide by 4.

7

Determine the relationship and setup the equation for the corresponding angles.

5w - 10 = 3w + 50

Corresponding angles between parallel lines are congruent (equal in measure).

8

Solve the equation for w.

w = 30

Subtract

3w

from both sides (

2w - 10 = 50

), add 10 (

2w = 60

), and divide by 2.

Key Concept

Geometric angle relationships (vertical, complementary, supplementary, corresponding, same-side interior) can be modeled and solved using linear algebraic equations.

Question 110Question →

Match each description of a 3D solid to its corresponding numerical value for volume or total surface area. (Note: All answers are expressed in terms of $\pi$ .)

Click a left item, then click its matching right item

Show answer & explanation

Answer

The volume of the cylinder matches

48\pi

, the volume of the sphere matches

36\pi

, the total surface area of the cylinder matches

30\pi

, and the surface area of the sphere matches

64\pi

.

Each correct numerical value is found by identifying the specified 3D solid, recalling its corresponding volume or surface area formula, and accurately substituting the given dimensions into the formula.

Step-by-Step Solution

1

Calculate the volume of the cylinder.

V = \pi(2^2)(12) = 48\pi

Apply the cylinder volume formula

V = \pi r^2 h

.

2

Calculate the volume of the sphere.

Radius is

3

.

V = \frac{4}{3}\pi(3^3) = 36\pi

Convert diameter to radius, then apply the sphere volume formula

V = \frac{4}{3}\pi r^3

.

3

Calculate the total surface area of the closed cylinder.

SA = 2\pi(3^2) + 2\pi(3)(2) = 30\pi

Apply the cylinder total surface area formula

SA = 2\pi r^2 + 2\pi rh

.

4

Calculate the surface area of the sphere.

SA = 4\pi(4^2) = 64\pi

Apply the sphere surface area formula

SA = 4\pi r^2

.

Key Concept

Applying appropriate volume and surface area formulas for cylinders and spheres.

Question 111Question →

A landscaping company is designing a parallelogram-shaped flower bed that will cover exactly $288$ square feet. If the design requires the base of the flower bed to be twice its height, what must the length of the base be in feet?

Show answer & explanation

Answer: 24

Answer

24

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

A = b \times h

). By substituting

2h

for the base, the equation becomes

288 = 2h^2

. Solving for

h

yields

12

. The base is twice the height, so

2 \times 12 = 24

.

Step-by-Step Solution

1

Set up the area formula

A = b \times h

To relate the given area to the dimensions of the parallelogram.

2

Express the base in terms of height

b = 2h

The problem states the base is twice the height.

3

Substitute the expressions into the area equation

288 = (2h) \times h

To create an equation with only one variable.

4

Solve for the height (

h

)

2h^2 = 288 \Rightarrow h^2 = 144 \Rightarrow h = 12

To find the numerical value of the height.

5

Calculate the base (

b

)

b = 2 \times 12 = 24

The question asks for the length of the base, not the height.

Key Concept

Area of a parallelogram and algebraic substitution

Estimated Time:1m 0s

Question 112Question →

A transversal line cuts across two parallel lines, forming a pair of same-side interior angles. If the measures of these angles are $(7x - 5)^\circ$ and $(3x + 25)^\circ$ , what is the measure of the larger angle?

43^\circ

46^\circ

107^\circ

117^\circ

Show answer & explanation

Answer: $107^\circ$

Answer

The larger angle measures

107^\circ

.

Same-side interior angles are supplementary, so setting their sum equal to

180^\circ

gives

10x + 20 = 180

, solving to

x = 16

. Substituting this value back into the original expressions yields angle measures of

107^\circ

and

73^\circ

. The larger of the two is

107^\circ

.

Step-by-Step Solution

1

Set up an equation based on the geometric relationship.

(7x - 5) + (3x + 25) = 180

Same-side interior angles formed by parallel lines and a transversal are supplementary, meaning they add up to 180 degrees.

2

Combine like terms to simplify the left side of the equation.

10x + 20 = 180

Combine the variable terms (

7x + 3x = 10x

) and the constant terms (

-5 + 25 = 20

).

3

Solve the linear equation for

x

.

10x = 160 \Rightarrow x = 16

Subtract 20 from both sides, then divide by 10 to isolate the variable

x

.

4

Substitute

x = 16

back into the original algebraic expressions to find the measures of both angles.

7(16) - 5 = 112 - 5 = 107^\circ

and

3(16) + 25 = 48 + 25 = 73^\circ

The exact measures must be calculated in order to determine which angle is larger.

5

Compare the two angles and identify the larger one.

107^\circ

Between

107^\circ

and

73^\circ

,

107^\circ

is the larger value.

Key Concept

Same-side (consecutive) interior angles formed by a transversal intersecting parallel lines are supplementary.

Estimated Time:1m 30s

Question 113Question →

A cylindrical water tank has a volume of $250\pi$ cubic feet and a height of $10$ feet. The exterior curved side (lateral surface) of the tank needs a new coat of paint, but the flat top and bottom bases will not be painted. What is the total area, in square feet, that needs to be painted?

150\pi

200\pi

100\pi

250\pi

Show answer & explanation

Answer: $100\pi$

Answer

The total area that needs to be painted is

100\pi

square feet.

The area to be painted is the lateral surface area of the cylinder, calculated with the formula

2\pi r h

. To find the unknown radius, use the given volume:

250\pi = \pi r^2(10)

. Solving this yields

r^2 = 25

, so

r = 5

. Substituting

r = 5

and

h = 10

into the lateral surface area formula gives

2\pi(5)(10) = 100\pi

square feet.

Step-by-Step Solution

1

Set up the volume formula for a cylinder to solve for the radius.

V = \pi r^2 h \Rightarrow 250\pi = \pi r^2 (10)

The volume and height are given, so the radius is the only unknown variable.

2

Solve for the radius (

r

).

250\pi = 10\pi r^2 \Rightarrow r^2 = 25 \Rightarrow r = 5

feet

Dividing both sides by

10\pi

isolates

r^2

, and taking the square root gives the radius.

3

Identify the correct formula for the area to be painted.

Lateral Surface Area =

2\pi r h

The problem specifies painting only the exterior curved side (lateral surface), excluding the top and bottom bases.

4

Calculate the lateral surface area.

2\pi(5)(10) = 100\pi

square feet

Substituting

r = 5

and

h = 10

into the formula yields the final painted area.

Key Concept

Solving for an unknown dimension using the volume formula, then calculating the lateral surface area of a cylinder.

Question 114Question →

A city planner is reviewing a design where two straight, parallel avenues are intersected by a diagonal rail line. The measure of one obtuse angle at the first intersection is $(12x - 20)^\circ$ , and the measure of its corresponding angle at the second intersection is $(8x + 40)^\circ$ .

What is the measure of the acute angle formed by the intersection of the rail line and the avenues?

76^\circ

80^\circ

20^\circ

64^\circ

Show answer & explanation

Answer: $20^\circ$

Answer

The measure of the acute angle is

20^\circ

.

Because the avenues are parallel, their corresponding angles are equal. Setting the expressions equal gives

12x - 20 = 8x + 40

. Subtracting

8x

and adding

20

to both sides results in

4x = 60

, which simplifies to

x = 15

. Substituting

15

back into either expression yields the obtuse angle measure:

12(15) - 20 = 160^\circ

. The question asks for the adjacent acute angle, which forms a linear pair with the obtuse angle. Subtracting from

180^\circ

gives

180^\circ - 160^\circ = 20^\circ

.

Step-by-Step Solution

1

Set the expressions for the two corresponding angles equal to each other.

12x - 20 = 8x + 40

When two parallel lines are intersected by a transversal, corresponding angles are equal in measure.

2

Solve the algebraic equation for

x

.

4x = 60 \Rightarrow x = 15

Subtracting

8x

from both sides and adding

20

to both sides isolates the variable

x

.

3

Substitute

x = 15

back into either expression to find the measure of the obtuse angle.

12(15) - 20 = 180 - 20 = 160^\circ

The expressions represent the obtuse corresponding angles, so plugging in the value of

x

gives the actual angle measure.

4

Subtract the obtuse angle from

180^\circ

to find the measure of the adjacent acute angle.

180^\circ - 160^\circ = 20^\circ

Angles on a straight line form a linear pair and are supplementary, meaning their sum must be

180^\circ

.

Key Concept

Corresponding angles on parallel lines are equal, and adjacent angles on a straight line are supplementary (sum to

180^\circ

).

Question 115Question →

A circular metal plate has a total area of $100\pi \text{ cm}^2$ . A smaller circular hole is drilled completely out of its exact center. If the area of the remaining metal plate is $64\pi \text{ cm}^2$ , what is the radius of the drilled hole?

6 \text{ cm}

12 \text{ cm}

18 \text{ cm}

36 \text{ cm}

Show answer & explanation

Answer: $6 \text{ cm}$

Answer

The radius of the drilled hole is

6 \text{ cm}

.

The area of the removed hole is the difference between the original total area and the remaining area, which is

100\pi - 64\pi = 36\pi \text{ cm}^2

. Using the circle area formula (

A = \pi r^2

), setting

\pi r^2 = 36\pi

yields

r^2 = 36

. Taking the square root of both sides confirms the radius is

6 \text{ cm}

.

Step-by-Step Solution

1

Find the area of the drilled hole by subtracting the remaining area from the total area.

100\pi - 64\pi = 36\pi \text{ cm}^2

.

The drilled hole accounts for the area that was removed from the total plate.

2

Set up the area formula for the drilled hole to solve for the unknown radius.

A = \pi r^2 \rightarrow 36\pi = \pi r^2

.

The area of a circle is calculated using the formula

A = \pi r^2

.

3

Solve the equation for the radius

r

.

r^2 = 36

, so

r = 6 \text{ cm}

.

Dividing both sides by

\pi

leaves

r^2 = 36

, and taking the square root of both sides gives the radius.

Key Concept

Perimeter and Area of 2D Figures

Estimated Time:1m 0s

Question 116Question →

During the alignment of a satellite's solar panels, sensors record various angular positions as algebraic expressions. Pair each geometric relationship described below with the exact numerical value of its unknown variable.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The linear pair matches to

x = 35

. The right angle matches to

y = 18

. The vertical angles match to

z = 16

. The corresponding angles match to

w = 9

.

Each relationship dictates a specific mathematical equation based on geometric definitions: linear pairs sum to 180, right angle components sum to 90, vertical angles are equal, and corresponding angles are equal. Solving these multi-term algebraic equations yields the correct numerical values.

Step-by-Step Solution

1

Set up and solve an equation for the linear pair.

3x + 10 + 2x - 5 = 180 \Rightarrow 5x = 175 \Rightarrow x = 35

.

Angles in a linear pair are supplementary and their measures sum to 180 degrees.

2

Set up and solve an equation for the adjacent angles forming a right angle.

4y - 12 + y + 12 = 90 \Rightarrow 5y = 90 \Rightarrow y = 18

.

Angles that form a right angle are complementary and their measures sum to 90 degrees.

3

Set up and solve an equation for the vertical angles.

6z - 18 = 4z + 14 \Rightarrow 2z = 32 \Rightarrow z = 16

.

Vertical angles are congruent, meaning their measures are equal.

4

Set up and solve an equation for the corresponding angles.

7w - 3 = 4w + 24 \Rightarrow 3w = 27 \Rightarrow w = 9

.

Corresponding angles formed by parallel lines and a transversal are congruent.

Key Concept

Using geometric angle relationships (supplementary, complementary, vertical, and corresponding) to write and solve algebraic equations.

Question 117Question →

Match each geometric term related to circles with its correct mathematical definition.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Chord matches with a line segment having endpoints on the circle. Secant matches with a straight line intersecting at two points. Tangent matches with a straight line touching at exactly one point. Diameter matches with a line segment passing through the center.

Each geometric term possesses a precise mathematical definition based on whether it is a finite segment or an infinite line, and how many distinct points it shares with the circle. A chord is a finite segment with two points on the circle, a diameter is a specific chord that passes through the center, a secant is an infinite line intersecting the circle at two points, and a tangent is an infinite line intersecting at exactly one point.

Step-by-Step Solution

1

Differentiate between lines and line segments in the definitions.

Identify that Chord and Diameter are finite line segments, while Secant and Tangent are straight lines extending infinitely.

This fundamental distinction splits the matching options into two distinct groups, making the problem easier to solve.

2

Distinguish between the two line segments (Chord and Diameter).

A diameter must pass through the center point, making it a specific type of chord. Therefore, 'Diameter' matches the definition mentioning the center, and 'Chord' matches the broader segment definition.

Both have endpoints on the circle, but only the diameter requires passing through the center point.

3

Distinguish between the two straight lines (Secant and Tangent).

A secant cuts through the circle at two points, while a tangent only touches the outer boundary at one point.

Counting the exact number of intersection points (two for a secant, one for a tangent) determines the correct match.

Key Concept

Geometric properties and terminology of lines and segments associated with circles

Question 118Question →

Two adjacent angles, $\angle A$ and $\angle B$ , form a straight line. If the measure of $\angle A$ is $(12x - 10)^\circ$ and the measure of $\angle B$ is $(6x + 28)^\circ$ , what is the measure of the larger angle?

52^\circ

58^\circ

98^\circ

118^\circ

Show answer & explanation

Answer: $98^\circ$

Answer

The larger angle measures 98 degrees.

The correct answer is

98^\circ

because the angles are supplementary, meaning their sum is

180^\circ

. Setting up the equation

(12x - 10) + (6x + 28) = 180

simplifies to

18x + 18 = 180

. Solving this gives

x = 9

. Substituting

x = 9

into both expressions reveals the angles are

98^\circ

and

82^\circ

. Comparing the two,

98^\circ

is the larger measure.

Step-by-Step Solution

1

Recognize the geometric relationship between the two angles.

Since the angles form a straight line, they are a linear pair and are supplementary. Their sum must equal

180^\circ

.

Adjacent angles on a straight line add up to 180 degrees.

2

Set up the algebraic equation based on the supplementary relationship.

(12x - 10) + (6x + 28) = 180

This translates the geometric concept into a solvable algebraic equation.

3

Combine like terms to simplify the equation.

18x + 18 = 180

Grouping the

x

variables (

12x + 6x

) and constants (

-10 + 28

) simplifies the problem.

4

Solve for

x

.

18x = 162 \Rightarrow x = 9

Subtracting 18 from both sides and then dividing by 18 isolates the variable.

5

Substitute

x = 9

into both original expressions to find the angle measures.

\angle A = 12(9) - 10 = 108 - 10 = 98^\circ

and

\angle B = 6(9) + 28 = 54 + 28 = 82^\circ

.

Finding the value of

x

is not the final step; the exact angle measurements must be calculated.

6

Identify the larger angle.

Since

98^\circ > 82^\circ

, the larger angle is

98^\circ

.

The question specifically asks for the measure of the larger angle.

Key Concept

Adjacent angles forming a straight line are supplementary and sum to 180 degrees.

Question 119Question →

The circumference of a circular stained glass window is $18\pi$ inches. What is the area of the window in square inches?

(Enter only the numerical value that multiplies $\pi$ . For example, if the area is $10\pi$ , enter 10.)

Show answer & explanation

Answer: 81

Answer

81

The correct answer is 81. First, the radius is found by dividing the circumference (

18\pi

) by

2\pi

, which gives a radius of 9 inches. Then, the area is calculated by squaring this radius and multiplying by

\pi

(

9^2 = 81

), resulting in a total area of

81\pi

. Since the prompt asks for just the numerical value multiplying

\pi

, the final response is 81.

Step-by-Step Solution

1

Find the radius of the circular window using the circumference formula

C = 2\pi r

.

18\pi = 2\pi r \Rightarrow r = 9

inches.

The area formula requires the radius, so we must work backward from the given circumference to find it.

2

Calculate the area of the window using the area formula

A = \pi r^2

.

A = \pi(9)^2 = 81\pi

square inches.

Squaring the radius and multiplying by

\pi

yields the total area of the circle.

3

Identify the requested numerical value.

81

The question specifically asks for the numerical coefficient that multiplies

\pi

.

Key Concept

Calculating the area of a circle by first determining the radius from the circumference.

Question 120Question →

Two angles are supplementary. The measure of the larger angle is $24^\circ$ more than three times the measure of the smaller angle. What is the measure, in degrees, of the larger angle?

Show answer & explanation

Answer: 141

Answer

141

Supplementary angles have a total sum of

180^\circ

. By letting the smaller angle equal

x

, the larger angle becomes

3x + 24

. Adding these gives

x + (3x + 24) = 180

. Simplifying this results in

4x = 156

, so

x = 39

. Since the question asks for the larger angle, we must substitute

x = 39

into our expression for the larger angle:

3(39) + 24 = 117 + 24 = 141

. Therefore, the larger angle measures

141^\circ

.

Step-by-Step Solution

1

Assign algebraic expressions to the unknown angles.

Smaller angle =

x

, Larger angle =

3x + 24

This translates the given text relationships into mathematical variables that can be used to set up an equation.

2

Set up an equation based on the geometric property of supplementary angles.

x + (3x + 24) = 180

Supplementary angles, by definition, always add up to exactly 180 degrees.

3

Simplify and solve the linear equation for

x

.

4x = 156

, meaning

x = 39

Isolating

x

provides the measure of the smaller angle.

4

Calculate the measure of the larger angle.

3(39) + 24 = 141

The question specifically asks for the measure of the larger angle, so we must plug

x

back into the original expression.

Key Concept

Solving algebraic equations involving supplementary angles

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