Geometry "Math Geometry 2024" - Public Release

The equation shown represents a circle in the xy-plane.

${(x + 4)}^{2} + {(y - 3)}^{2} = 36$

Which statement includes the correct center and the correct radius for this circle?

A.

The center is located at $coordinates 4 comma negative 3,$ and the radius is $6 .$
B.

The center is located at $coordinates 4 comma negative 3,$ and the radius is $36 .$
C.

The center is located at $coordinates negative 4 comma 3,$ and the radius is $6 .$
D.

The center is located at $coordinates negative 4 comma 3,$ and the radius is $36 .$

Correcti1

opposite signs on coordinatesi2

opposite signs on coordinates and forgot square root of radiusi4

forget square root of radius.

Given: Parallelogram PQRS with the measure of angle P equals 90 degrees

Prove: Parallelogram PQRS is a rectangle.

An incomplete proof is shown in the table.

Select the correct reasons to complete the proof.

Drag and drop a reason into each box.

Definition of a right angle Subtraction Property of Equality Opposite angles of a parallelogram are congruent. Consecutive angles of a parallelogram are supplementary.

Step	Statement	Reason
1	Parallelogram PQRS with the measure of angle P equals 90 degrees	Given
2	angle P and angle Q are supplementary.
3	the measure of angle P plus the measure of angle Q, equals 180 degrees	Definition of supplementary angles
4	90 degrees plus the measure of angle Q, equals 180 degrees	Substitution
5	the measure of angle Q equals 90 degrees
6	angle P is congruent to angle R and angle Q is congruent to angle S
7	the measure of angle P equals the measure of angle R and the measure of angle Q equals the measure of angle S	Congruent angles have the same measure.
8	the measure of angle R equals the measure of angle S, which equals 90 degrees	Substitution
9	angle P, angle Q, angle R, and angle S are right angles.
10	PQRS is a rectangle.	Definition of a rectangle

In the figure, angle K is congruent to angle R.

Which angles have a cosine that is equal to $\sin (L) ?$

Select all that apply.

A.

angle J
B.

angle K
C.

angle L
D.

angle Q
E.

angle R
F.

angle S

In the xy-plane, triangle R S T is first reflected across the x-axis, then reflected across the y-axis, and finally rotated $180 degrees$ about the origin to obtain the image $Δ R^{'} S^{'} T^{'} .$

Complete each sentence with the best description of the relationship between $Δ R S T$ and $Δ R^{'} S^{'} T^{'} .$

Select from the drop-down menus to correctly complete the statements.

The length of each side of

Δ R^{'} S^{'} T^{'}

is the length of the corresponding side of

Δ R S T .

The area of

Δ R^{'} S^{'} T^{'}

is the area of

Δ R S T .

The location of each vertex of

Δ R^{'} S^{'} T^{'}

the same as the location of the corresponding vertex of

Δ R S T .

An incomplete proof is shown.

Given: Angle P is congruent to angle J, angle Q is congruent to angle K

Prove: segment K L equals 2.8

The incomplete proof is shown in the table.

Drag and drop a reason into each box to complete the proof.

AA similarity SAS similarity Corresponding sides of similar triangles are proportional. The measures of corresponding angles of similar triangles are equal.

Step	Statement	Reason
1	angle P is congruent to angle J, angle Q is congruent to angle K, P R equals 5, Q R equals 7, and J L equals 2.	Given
2	triangle P Q R is similar to triangle J K L
3	$\frac{K L}{Q R} = \frac{J L}{P R}$
4	$\frac{K L}{7} = \frac{2}{5}$	Substitution
5	$K L = \frac{14}{5} = 2.8$	Multiplication property of equality

In the figure shown, segment Q P is congruent to segment Q R.

What is the value of $x ?$

Enter your answer in the space provided.

$x =$

Two triangles are shown.

Which statement describes the additional information needed to prove $Q R equals 18?$

KEY i1

Three pairs of congruent sides are not provided i2

Angles $J$ and $R$ are not “included” between the two pairs of congruent angles. i3

Three pairs of congruent sides are not provided

Starting with $\bar{P Q}$ (not shown) a square will be constructed using a compass and straightedge so that $\bar{P Q}$ is one of its sides.

Which other constructions must be completed during the construction of the square?

Select all that apply.

In the following figure, triangle D E F is mapped onto triangle A B C by a dilation with center $N$ .

If $E N equals 16$ and $B N equals 20$ , what is the scale factor of the dilation?

Enter your answer as a fraction in the spaces provided.

fraction bar

The square pyramid shown in the following figure will be sliced by a plane.

Determine whether each of the following two-dimensional shapes is a possible cross-section formed when the pyramid is sliced by the plane.

Move each two-dimensional shape into the correct side of the table.

Given: triangle X Y Z with point Y on line k, and line k is parallel to segment X Z.

Which statement will most likely be used to prove that the measure of angle X, plus the measure of angle X Y Z, plus the measure of angle Z, equals 180 degrees?

Key (Alternate interior angles) i2

Not necessarily true so it would not be used in a proof i3

Not necessarily true so it would not be used in a proof i4

Not true since

Triangle $A B C$ is shown in the following $x y$ ‑plane with A. with coordinates 2 comma 3 B with coordinates 20 comma 11, and C with coordinates 16 comma 3.

What is the area, in square units, of triangle A B C?

Enter your answer in the space provided.

square units

Given: lines $k$ and $n$ intersect to form angles $1,2,3$ and $4 .$

Prove: angle 2 is congruent to angle 4

An incomplete proof is shown.

Step	Statement	Reason
1	Lines $k$ and $n$ intersect to form angles $1,2,3$ and $4 .$	Given
2	Angles $1$ and $2$ form a linear pair. Angles $1$ and $4$ form a linear pair.	Definition of a linear pair
3	Angles $1$ and $2$ are supplementary. Angles $1$ and $4$ are supplementary.	Angles that form a linear pair are supplementary.
4	the measure of angle 1 plus the measure of angle 2, equals 180 degrees; the measure of angle 1 plus the measure of angle 4, equals 180 degrees.	The sum of the measures of the angles is $180 degrees.$
5	the measure of angle 1 plus the measure of angle 2, equals the measure of angle 1 plus the measure of angle 4	Transitive Property
6	the measure of angle 2 equals the measure of angle 4	$?$
7	angle 2 is congruent to angle 4	Angles that have the same measure are congruent.

Which reason for step $6$ correctly completes the proof?

Correct i1

Misunderstood the reason i2

Misunderstood the reason i4

Misunderstood the reason

Line $ℓ$ in the xy-plane passes through the point $with coordinates negative 4 comma 5$ and is perpendicular to the line with the equation $y = \frac{1}{2} x + 5$ .

What is the $y$ -intercept of line $ℓ$ ?

Select one answer.

A.

$negative 3$
B.

$5$
C.

$7$
D.

$13$

In right $Δ X Y Z,$ the length of the hypotenuse $\bar{Y Z}$ is $85$ inches and $\tan Z = \frac{3}{4} .$

What are the lengths, in inches, of the legs $\bar{X Y}$ and $\bar{X Z} ?$

Enter your answers in the spaces provided.

$X Y =$ inches

$X Z =$ inches

A can is in the shape of a right circular cylinder with an inner diameter of $7.5$ centimeters and an inner height of $12.5$ centimeters. The can is placed on its circular base, and $440$ milliliters of juice is poured into the can.

Given that $1$ milliliter is equivalent to $1$ cubic centimeter, what is the height of the juice in the can to the nearest tenth of a centimeter?

Enter your answer in the space provided.

The coordinates of the vertices of quadrilateral $P Q R S$ are shown in the xy-plane.

Part A

Prove that quadrilateral $P Q R S$ is a trapezoid.

Enter your answer and your work or explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Part B

Determine whether quadrilateral $P Q R S$ is an isosceles trapezoid. Show your work or explain your answer.

Enter your answer and your work or explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

Holistic Rubric for 4-Point Reasoning Constructed Response Items
Sample Top Score Response Part A Calculating the slopes: The slope of side $P Q :$ $\frac{10 â 7}{4 â 10} = \frac{3}{â 6} = â \frac{1}{2}$ The slope of side $R S :$ $\frac{5 â 4}{4 â 6} = \frac{1}{â 2}$ The slope of side $P S :$ $\frac{10 â 5}{4 â 4} = \frac{5}{0}$ which is undefined The slope of side $Q R :$ $\frac{7 â 4}{10 â 6} = \frac{3}{4}$ $P Q R S$ is a trapezoid because sides $P Q$ and $R S$ are parallel and sides $P S$ and $Q R$ are not parallel. Part B Calculating the side lengths: $P S = \sqrt{{(10 â 5)}^{2} + {(4 â 4)}^{2}} = 5$ $Q R = \sqrt{{(7 â 4)}^{2} + {(10 â 6)}^{2}} = 5$ Since the side lengths are equal, $P Q R S$ is an isosceles trapezoid. Scoring
Points	Sample Characteristics
4 Points	A four-point response for reasoning items provides evidence of correct, complete, and appropriate mathematical reasoning. The response may: be clear and well developed with logical reasoning communicated by the use of precise and appropriate representations, symbols, drawings, or mathematical vocabulary. demonstrate a thorough understanding of the mathematics. contain minor flaws that do not detract from the correct reasoning or demonstration of a thorough understanding.
3 Points	A three-point response for reasoning items provides evidence of essentially correct, essentially complete, and essentially appropriate mathematical reasoning. The response may: be clear and developed with logical reasoning communicated by the use of essentially precise and appropriate representations, symbols, drawings, or mathematical vocabulary. contain minor flaws that detract from the correct reasoning or demonstration of a thorough understanding.
2 Points	A two-point response for reasoning items provides evidence of partially correct mathematical reasoning. The response may: display an incomplete reasoning process. contain mathematical flaws.
1 Point	A one-point response for reasoning items provides limited evidence of correct mathematical reasoning. The response may: demonstrate the beginning of a valid chain of reasoning. reflect a lack of essential understanding of the underlying mathematical concepts. contain the correct solution, but work is limited or missing. contain errors in the fundamental mathematical procedures or reasoning. contain omissions or irregularities that lead to an inadequate solution.
0 Point	A zero-point response is completely incorrect, incoherent or irrelevant.

A bird flew from a point on the ground directly to the edge of the roof of a building. The height of the building is $40$ feet, and the angle of elevation the bird’s flight path made with the ground is $26 degrees.$

Which expression models the total distance, in feet, the bird flew?

Correcti1

Used cosine instead of sinei3

Used cosine instead of sine and incorrectly solved the equationi4

Incorrectly solved the equation

A cone has a base radius of $3$ centimeters and a height of $5$ centimeters. A student correctly calculates its volume to be $15 pi$ cubic centimeters.

The student thinks that a simpler formula for the volume of a cone is $V = π r h$ because $pi, open parenthesis, 3, close parenthesis, open parenthesis, 5, close parenthesis, equals 15 pi.$

Which statement explains the conditions for which the student’s claim would be true?

Correcti1

The formula does not always work.i3

The revised formula is not affected by the height.i4

The revised formula would still work if the height was $6$ and therefore the product would be $18 .$

In the following figure, a sector of a circle has a central angle of 80 degrees The area of the circle is $9 pi$ square units.

If the sector has an area of $k π$ square units, what is the value of $k ?$

Enter your answer in the space provided.

$k =$

A company is designing a soup can that is in the shape of a right circular cylinder. The height of the can will be $3$ times the radius of the can. The volume of the can will be $350$ cubic centimeters.

Which measurement, in centimeters, is closest to the radius of the soup can?

A.

$2.3$
B.

$3.3$
C.

$6.9$
D.

$9.9$

Correcti1

Assumed the radius is $3$ times the height and calculated the heighti3

Assumed the radius is $3$ times the heighti4

Calculated the height instead of the radius

In the figure shown, point $M$ is the midpoint of segment R S.

Which value best represents the length of segment R M?

A.

$41.6$
B.

$24.0$
C.

$20.8$
D.

$12.0$

Correcti1

Finds the length of the longer leg of a $30 - 60 - 90$ right triangle.i2

Finds the length of segment $R S$ i3

Finds half the length of the longer leg of a $30 - 60 - 90$ right triangle.

Using a three-dimensional printer, an artist will produce several models of hemispheres. The material used to make each model is 1 centimeter thick, as shown in the diagram.

The artist will fill each model with colored paint to its maximum capacity. The colored paint costs $0.01 per cubic centimeter.

Write an expression that represents the cost, in dollars, of the colored paint needed to fill any model based on x, the outer radius, in centimeters, of the model.

Enter your expression in the space provided. Enter only your expression.

Right triangle $X Y Z$ is shown in the xy-plane. Vertex $X$ has coordinates $2 comma 3.$ The length of segment X Y is 2 units, and the length of segment X Z is 6 units.

A student’s work for finding the slope of the perpendicular bisector of segment Y Z is shown.

The slope of is or The opposite of the reciprocal of is So, the perpendicular bisector of has a slope of

Describe the student’s mistake.

Find the equation of the line that represents the perpendicular bisector of segment Y Z. Show your work or explain how you found the equation.

Enter your answer and your work or explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

Holistic Rubric for 4-Point Reasoning Constructed Response Items
Sample Top Score Response The student’s mistake was using a slope of $\frac{1}{3}$ instead of $- \frac{1}{3}$ since side $Y Z$ points down and to the right. The opposite of the reciprocal of $- \frac{1}{3}$ is $3 .$ The perpendicular bisector of side $Y Z$ passes through the midpoint of side $Y Z .$ The coordinates of the midpoint are $\begin{matrix} x = 2 + \frac{6}{2} = 5 \\ y = 3 + \frac{2}{2} = 4 \end{matrix} .$ The equation of the perpendicular bisector is $\begin{matrix} x â 4 = 3 (x â 5) \\ y = 3 x â 15 + 4 \\ y = 3 x â 11 \end{matrix} .$ Scoring
Points	Sample Characteristics
4 Points	A four-point response for reasoning items provides evidence of correct, complete, and appropriate mathematical reasoning. The response may: be clear and well developed with logical reasoning communicated by the use of precise and appropriate representations, symbols, drawings, or mathematical vocabulary. demonstrate a thorough understanding of the mathematics. contain minor flaws that do not detract from the correct reasoning or demonstration of a thorough understanding.
3 Points	A three-point response for reasoning items provides evidence of essentially correct, essentially complete, and essentially appropriate mathematical reasoning. The response may: be clear and developed with logical reasoning communicated by the use of essentially precise and appropriate representations, symbols, drawings, or mathematical vocabulary. contain minor flaws that detract from the correct reasoning or demonstration of a thorough understanding.
2 Points	A two-point response for reasoning items provides evidence of partially correct mathematical reasoning. The response may: display an incomplete reasoning process. contain mathematical flaws.
1 Point	A one-point response for reasoning items provides limited evidence of correct mathematical reasoning. The response may: demonstrate the beginning of a valid chain of reasoning. reflect a lack of essential understanding of the underlying mathematical concepts. contain the correct solution, but work is limited or missing. contain errors in the fundamental mathematical procedures or reasoning. contain omissions or irregularities that lead to an inadequate solution.
0 Point	A zero-point response is completely incorrect, incoherent or irrelevant.

A student claims that any quadrilateral with two right angles must be a rectangle.

Which figures can be used to show the student is incorrect?

Select all that apply.

A student is standing next to a vertical flagpole. The top of the student’s shadow coincides with the top of the flagpole’s shadow as shown.

The student is $62$ inches tall. The student estimates that the distance from the flagpole to the point where the student is standing is between $21$ and $22$ feet. The student also estimates that the length of the student’s shadow is between $7$ and $7 and 1 half$ feet.

Based on the given information, what are the least and greatest possible heights, in feet, of the flagpole? Explain how you arrived at your answers.

Enter your answers and explanations in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

Holistic Rubric for 4-Point Modeling Constructed Response Items
Sample Top Score Response The least height for the flagpole will be when the shadow is longest ( $7.5$ feet or $90$ inches) and the distance between Isabella and the flag is shortest ( $21$ feet or $252$ inches). The following proportion can be solved to arrive at the least height of the flagpole, in inches. $\frac{62}{90} = \frac{x}{90 + 252} 62 (342) = 90 x$ $x = 235.6$ inches or $19.63$ feet. The greatest height for the flagpole will be when the shadow is shortest ( $7$ feet of $84$ inches) and the distance between Isabella and the flag is longest ( $22$ feet or $264$ inches). The following proportion can be solved to arrive at the greatest height of the flagpole, in inches. $\frac{62}{84} = \frac{x}{84 + 264} 62 (348) = 84 x$ $x = 256.9$ inches or $21.4$ feet.
Points	Sample Characteristics
4 Points	A four-point response provides full and complete evidence of the modeling process used to solve a real-world problem. The response may: identify the problem that needs to be solved. determine information that is needed to solve the problem. communicate an accurate, organized solution path that is aligned to the problem using appropriate, effective, and essentially precise representations. contain minor flaws that do not detract from correct modeling or demonstration of a thorough understanding. evaluate or validate a partial or complete solution and show how to improve or refine the solution.
3 Points	A three-point response provides evidence of the modeling process used to solve a real-world problem. The response may: identify most of the problem that needs to be solved. determine most of the information that is needed to solve the problem. communicate an accurate, organized solution path that is aligned to the problem using appropriate, effective, and precise representations with minor flaws. evaluate or validate a partial or complete solution and show how to improve or refine the solution, but the improvement or refinement may include minor flaws.
2 Points	A two-point response provides partial evidence of the modeling process used to solve a real-world problem. The response may: partially identify the problem that needs to be solved. determine some of the information that is needed to solve the problem. include a partial solution path that may be incomplete. contain some errors in identifying the mathematics that is needed to solve the problem. evaluate or validate a partial or complete solution and attempt to improve or refine the solution.
1 Point	A one-point response provides limited evidence of the modeling process used to solve a real-world problem. The response may: partially or incorrectly identify the problem that needs to be solved. determine a minimal amount of the information that is needed to solve the problem. include an incomplete or unorganized solution path. contain errors in identifying the mathematics that is needed to solve the problem. contain the correct solution, but work is limited or missing. evaluate or validate a partial or complete solution but does not show how to improve or refine the solution.
0 Point	A zero-point response is completely incorrect, incoherent or irrelevant.

A 30-inch chord in a circle is 8 inches from the center of the circle, as shown in the following figure.

What is the length, in inches, of the radius of the circle?

Enter your answer in the space provided.

inches

The following figure shows a circle that has center $O,$ diameters line segment P S and line segment R T, and chords line segment P Q and line segment Q R.

If the measure of arc R S equals 74 degrees and the measure of arc S T equals 106 degrees, what is the measure of angle P Q R?

Select one answer.

The vertices of rectangle PQRS are graphed in the xy-plane with coordinates of vertex P at negative 3 comma negative 2, vertex Q at negative 3 comma 3, vertex R at 4 comma 3, and vertex S at 4 comma negative 2

The rectangle PQRS is reflected across the y-axis and rotated 90° counterclockwise about the origin to form rectangle P prime Q prime R prime S prime.

Plot the location of point P prime.

Five points are shown on a number line.

Information about some of the lengths of different segments is given.

The length of line segment P Q is 5 units.

The length of line segment R S is equal to the length of line segment P R.

The length of line segment S T is 3 times the length of line segment Q R.

The length of line segment P T is 25 units.

Which statements are correct?

Select all that apply.

Correct i2

Correct i5

Correct i3

Incorrectly calculated $Q R$ to be $7$ i4

Incorrectly calculated $Q R$ to be $2$

Mahari will dig a hole in which to plant a new tree. The root ball of the tree has a diameter of $18$ inches and will be placed in the hole, as shown in the following figure.

Part A

The hole needs to be twice as wide as the diameter of the root ball and deep enough for the entire root ball to fit inside. After placing the tree in the hole, Mahari will fill the rest of the hole with soil.

Approximate the amount of soil, in cubic feet, that Mahari needs to fill the hole. Show how you arrived at your answer.

Enter your answer and work in the space provided. You may also use the drawing tool to help explain or support your answer.

Part B

After filling the hole with soil, Mahari will place mulch within a circle around the tree. The diameter of the circle is the diameter of the hole. The diameter of the base of the tree trunk is $6$ inches.

What is the area, in square feet, that the mulch will cover? Show how you arrived at your answer.

Enter your answer and your work in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

NOTE for ETS Editorial:

There is an error in the rubric that was corrected in the paper and online Practice Test keys. The measurement at the end of the third sentence should be "cubic inches" instead of "inches". This rubric will be updated after the Spring 2022 Practice Test is no longer available.

Holistic Rubric for 4-Point Modeling Constructed Response Items
Sample Top Score Response Part A The hole must have a diameter that measures approximately $36$ inches and a height that measures approximately $18$ inches. Thus the volume of the hole is approximately $V = π {(\frac{36}{2})}^{2} (18) = 5,832 π$ cubic inches. The root ball has an approximate volume of $V = \frac{4}{3} π {(\frac{18}{2})}^{3} = 972 π$ cubic inches. So approximately $(5,832 â 972) π â 15,268$ cubic inches of soil will be needed. Since $12$ inches $= 1$ foot, $1728$ cubic inches $= 1$ cubic foot, and $15, 268$ cubic inches $= \frac{15, 268}{1728}$ cubic feet $â 8.8$ cubic feet. Part B The part that will be covered by mulch has the form of a circle of radius $18$ inches. And the area occupied by the tree trunk has radius of $3$ inches. So the total area to be covered by mulch is $(18^{2} â 3^{2}) π â 989.6$ square inches. Since $12$ inches $= 1$ foot, $144$ square inches $= 1$ square foot, and $989.6$ square inches $= \frac{989.6}{144}$ square feet $â 6.9$ square feet. Scoring
Points	Sample Characteristics
4 Points	A four-point response provides full and complete evidence of the modeling process used to solve a real-world problem. The response may: identify the problem that needs to be solved. determine information that is needed to solve the problem. communicate an accurate, organized solution path that is aligned to the problem using appropriate, effective, and essentially precise representations. contain minor flaws that do not detract from correct modeling or demonstration of a thorough understanding. evaluate or validate a partial or complete solution and show how to improve or refine the solution.
3 Points	A three-point response provides evidence of the modeling process used to solve a real-world problem. The response may: identify most of the problem that needs to be solved. determine most of the information that is needed to solve the problem. communicate an accurate, organized solution path that is aligned to the problem using appropriate, effective, and precise representations with minor flaws. evaluate or validate a partial or complete solution and show how to improve or refine the solution, but the improvement or refinement may include minor flaws.
2 Points	A two-point response provides partial evidence of the modeling process used to solve a real-world problem. The response may: partially identify the problem that needs to be solved. determine some of the information that is needed to solve the problem. include a partial solution path that may be incomplete. contain some errors in identifying the mathematics that is needed to solve the problem. evaluate or validate a partial or complete solution and attempt to improve or refine the solution.
1 Point	A one-point response provides limited evidence of the modeling process used to solve a real-world problem. The response may: partially or incorrectly identify the problem that needs to be solved. determine a minimal amount of the information that is needed to solve the problem. include an incomplete or unorganized solution path. contain errors in identifying the mathematics that is needed to solve the problem. contain the correct solution, but work is limited or missing. evaluate or validate a partial or complete solution but does not show how to improve or refine the solution.
0 Point	A zero-point response is completely incorrect, incoherent or irrelevant.

In triangle K L M and triangle X Y Z shown, segment K L is congruent to segment X Y and angle L is congruent to angle Y.

What additional information is needed to prove that triangle K L M is congruent to triangle X Y Z?

Select from the drop-down menus to correctly complete each statement.

If then by the postulate.

A farmer wants to build a garden using fence. A total of $60$ feet of fence will be used to enclose the garden. The farmer calculates that the area of the garden will be $225$ square feet if a square-shaped garden is built using the fence. The farmer also considers building the garden in different shapes to increase the area enclosed by the fence.

Determine how building the garden in different shapes affects the area of the garden.

Select the box to compare the area of the gardens. Select only one box per row.

Shape of the garden	Less than $225$ square feet	Equal to $225$ square feet	Greater than $225$ square feet
Equilateral triangle
Rectangle with dimensions $18$ feet by $12$ feet

The following proof is incomplete.

Given: segment P Q is parallel to segment J L

Prove: the fraction with a numerator of J P and a denominator P K equals the fraction with a numerator of L Q and a denominator of Q K

Step	Statement	Reason
1	segment P Q is parallel to segment J L	Given
2	angle K P Q is congruent to angle K J L, angle K Q P is congruent to angle K L J	?
3	triangle P K Q is similar to triangle J K L	AA criterion
4	the fraction with a numerator of J K and a denominator of P K equals the fraction with a numerator of L K and a denominator of Q K	?
5	$\frac{J P + P K}{P K} = \frac{L Q + Q K}{Q K}$	Segment addition postulate
6	$\frac{J P}{P K} + \frac{P K}{P K} = \frac{L Q}{Q K} + \frac{Q K}{Q K}$	Distributive property
7	$\frac{J P}{P K} = \frac{L Q}{Q K}$	Addition property of equality

Which reasons for Step 2 and Step 4 complete the proof?

Select from the drop-down menus to complete the proof.

Step 2: When two parallel lines are cut by a transversal, angles are congruent.

Step 4: Corresponding sides of triangles are .

In the figure, the measure of angle K plus the measure of angle Q equals 90 degrees.

Describe each of the trigonometric ratios in the table.

Select the boxes to identify the equivalent trigonometric ratios. Select only one box per row.

Trigonometric ratio	Equivalent to $sin (L)$	Equivalent to $cos (L)$
$\sin (Q)$
$\cos (Q)$
$\sin (R)$

This page shows questions in the Math Geometry 2024 public release module at MSDE. Geometry "Math Geometry 2024"

Icon representing a test question. 1. Geometry 2024 Release, Question 1 (G.GPE.A.1) - No Calculator.

Icon representing a test question. 2. Geometry 2024 Release, Question 2 (G.CO.C.11) - No Calculator.

Icon representing a test question. 3. Geometry 2024 Release, Question 3 (G.SRT.C.7) - No Calculator.

Icon representing a test question. 4. Geometry 2024 Release, Question 4 (G.CO.B.6) - No Calculator.

Icon representing a test question. 5. Geometry 2024 Release, Question 5 (G.SRT.B.5) - No Calculator.

Icon representing a test question. 6. Geometry 2024 Release, Question 6 (G.CO.C.10) - No Calculator.

Icon representing a test question. 7. Geometry 2024 Release, Question 7 (G.SRT.B.5) - No Calculator.

Icon representing a test question. 8. Geometry 2024 Release, Question 8 (G.CO.D.13) - No Calculator.

Icon representing a test question. 9. Geometry 2024 Release, Question 9 (G.SRT.A.1.b) - No Calculator.

Icon representing a test question. 10. Geometry 2024 Release, Question 10 (G.GMD.B.4-1) - No Calculator.

Icon representing a test question. 11. Geometry 2024 Release, Question 11 (G.CO.C.10) - No Calculator.

Icon representing a test question. 12. Geometry 2024 Release, Question 12 (G.GPE.B.7) - No Calculator.

Icon representing a test question. 13. Geometry 2024 Release, Question 13 (G.CO.C.9) - No Calculator.

Icon representing a test question. 14. Geometry 2024 Release, Question 14 (G.GPE.B.5) - Calculator.

Icon representing a test question. 15. Geometry 2024 Release, Question 15 (G.SRT.C.8) - Calculator.

Icon representing a test question. 16. Geometry 2024 Release, Question 16 (G.M.6-1, G.GMD.A.3) - Calculator.

Icon representing a test question. 17. Geometry 2024 Release, Question 17 (G.R.6, G.GPE.B.7) - Calculator.

Icon representing a test question. 18. Geometry 2024 Release, Question 18 (G.M.1, G.SRT.C.8) - Calculator.

Icon representing a test question. 19. Geometry 2024 Release, Question 19 (G.R.3, G.GMD.A.3) - Calculator.

Icon representing a test question. 20. Geometry 2024 Release, Question 20 (G.C.B.5) - Calculator.

Icon representing a test question. 21. Geometry 2024 Release, Question 21 (G.MG.A.3) - Calculator.

Icon representing a test question. 22. Geometry 2024 Release, Question 22 (G.SRT.C.6) - Calculator.

Icon representing a test question. 23. Geometry 2024 Release, Question 23 (G.M.6, G.GMD.A.3, G.MG.A.2) - Calculator.

Icon representing a test question. 24. Geometry 2024 Release, Question 24 (G.R.7, G.CO.C.9, G.GPE.B.5) - Calculator.

Icon representing a test question. 25. Geometry 2024 Release, Question 25 (G.R.1, G.CO.C.11) - Calculator.

Icon representing a test question. 26. Geometry 2024 Release, Question 26 (G.M.6-2, G.SRT.B.5) - Calculator.

Icon representing a test question. 27. Geometry 2024 Release, Question 27 (G.C.A.2) - Calculator.

Icon representing a test question. 28. Geometry 2024 Release, Question 28 (G.C.A.2) - Calculator.

Icon representing a test question. 29. Geometry 2024 Release, Question 29 (G.CO.A.5) - Calculator.

Icon representing a test question. 30. Geometry 2024 Release, Question 30 (G.R.8, G.GPE.B.6) - Calculator.

Icon representing a test question. 31. Geometry 2024 Release, Question 31 (G.M.6-1, G.GMD.A.3, G.MG.A.1) - Calculator.

Icon representing a test question. 32. Geometry 2024 Release, Question 32 (G.R.5, G.CO.B.8) - Calculator.

Icon representing a test question. 33. Geometry 2024 Release, Question 33 (G.M.3, G.MG.A.3) - Calculator.

Icon representing a test question. 34. Geometry 2024 Release, Question 34 (G.SRT.B.4) - Calculator.

Icon representing a test question. 35. Geometry 2024 Release, Question 35 (G.SRT.C.7) - Calculator.

This page shows questions in the Math Geometry 2024 public release module at MSDE. Geometry
"Math Geometry 2024"