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This page shows questions in the Geometry Similarity, Right Triangles, and Trigonometry public release module at MSDE. Geometry
"Geometry Similarity, Right Triangles, and Trigonometry"

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Expand All Official Practice Test

This is a multiple choice question that allows you to select several options.

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

The figure shows right triangle J K L and right triangle Q R S. The two triangles appear to be similar such that segment J K corresponds to segment Q R, segment K L corresponds to segment R S, and segment L J corresponds to segment S Q,. Angle J and angle Q are right angles.

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

Select all that apply.

  1. A.
    angle J
  2. B.
    angle K
  3. C.
    angle L
  4. D.
    angle Q
  5. E.
    angle R
  6. F.
    angle S

This is a drag and drop question that allows you to select text and place it in an appropriate answer space.

An incomplete proof is shown.

The figure shows triangle P Q R, and triangle J K L. The two triangles appear to be similar, such that segment J K corresponds to segment P Q, segment K L corresponds to segment Q R, and segment L J corresponds to segment R P,. In triangle P Q R, segment Q R has a length of 7, and segment R P has a length of 5,. In triangle J K L, segment L J has a length of 2.

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 KLQR=JLPR
4 KL7=25 Substitution
5 KL=145=2.8 Multiplication property of equality

This is a multiple choice question that allows you to select only one option.

Two triangles are shown.

The figure shows triangle J K L and triangle Q R S. The two triangles appear to be similar, such that segment J K corresponds to segment Q R, segment J L, corresponds to segment Q S, and segment L K, corresponds to segment S R. In triangle J K L, segment J K, has a length of 18, segment J L, has a length of 8, and segment L K, has a length of 12. In triangle Q R S, side Q S, has a length of 8 and side S R, has a length of 12.

Which statement describes the additional information needed to prove QR=18?

  1. A.
    If angle J is congruent to angle Q, then triangle J K L is congruent to triangle Q R S because of SSS postulate, and therefore, Q R equals 18 because corresponding sides of congruent triangles are congruent.
  2. B.
    If angle J is congruent to angle Q, then triangle J K L is congruent to triangle Q R S because of SAS postulate, and therefore, Q R equals 18 because corresponding sides of congruent triangles are congruent.
  3. C.
    If angle L is congruent to angle S, then triangle J K L is congruent to triangle Q R S because of SSS postulate, and therefore, Q R equals 18 because corresponding sides of congruent triangles are congruent.
  4. D.
    If angle L is congruent to angle S, then triangle J K L is congruent to triangle Q R S because of SAS postulate, and therefore, Q R equals 18 because corresponding sides of congruent triangles are congruent.
i4

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

This is a question with 2 parts, including a fill in the blank question that allows you to enter only numbers.

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

The figure pre sents triangles A B C, and D E F, with triangle A B C, larger, and to the left, of triangle D E F. Sides A C, and D F, are horizontal. There is a point labeled N, to the right of both triangles. There are 3 dashed lines originating from point N, to vertices A, B, and see. Of the 3 dashed lines, one runs from N, to vertex A, passing through vertex D, one runs from N, to vertex B, passing through vertex E, and the third dashed line runs from N, to vertex see, passing through vertex F.

If EN=16 and BN=20 , what is the scale factor of the dilation?

Enter your answer as a fraction in the spaces provided.

fraction bar

This is a question with 2 parts, including a fill in the blank question that allows you to enter only numbers.

In right ΔXYZ, the length of the hypotenuse YZ¯ is 85 inches and tanZ=34.

What are the lengths, in inches, of the legs XY¯ and XZ¯?

Enter your answers in the spaces provided.

XY= inches

XZ= inches

This is a multiple choice question that allows you to select only one option.

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

The figure shows right triangle Q R S, and point M,.  Angle S is a right angle, and angle R is labeled, 60 degrees,. The length of segment Q, R, is labeled, 48,.  Point M lies at the midpoint of segment R S.

Which value best represents the length of segment R M?

  1. A.
    41.6
  2. B.
    24.0
  3. C.
    20.8
  4. D.
    12.0
i4

Correcti1

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

Finds the length of segment RS i3

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

This is a question with 3 parts, including a question with drop-down menus from which you must select an option to fill in the blank.

The following proof is incomplete.

The figure shows triangle J K L, where segment J L is horizontal, and vertex K is above segment J L,.  A point P, lies on segment J K, and a point Q, lies on segment K L,. Horizontal line segment P Q is drawn, forming a triangle P K Q.

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 JP+PKPK=LQ+QKQK Segment addition postulate
6 JPPK+PKPK=LQQK+QKQK Distributive property
7 JPPK=LQQK 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  
  .

This is a matching question that allows you to match elements from one list with those on another list.

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

The figure shows right triangle J K L, and right triangle S R Q. In triangle J K L, angle J is a right angle. Segment J K has a length of 20, and segment J L has a length of 21. In triangle S R Q, angle S is a right angle.

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)