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

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

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

Prove: Parallelogram PQRS is a rectangle.

The figure shows parallelogram P Q R S.

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

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.

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° about the origin to obtain the image ΔR'S'T'.

Complete each sentence with the best description of the relationship between ΔRST and ΔR'S'T'.

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

The length of each side of ΔR'S'T' is  
  the length of the corresponding side of ΔRST.
The area of ΔR'S'T' is  
  the area of ΔRST.
The location of each vertex of ΔR'S'T'  
  the same as the location of the corresponding vertex of ΔRST.

This is a fill in the blank question that allows you to enter only numbers.

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

The figure shows a line segment and two rays. The endpoints of the line segment are labeled, Q and P, and a ray extends from each point. The rays intersect at a point R so that triangle P Q R is formed. Angle P Q R is labeled, x degrees. The angle opposite P R Q, formed by the intersection of the two rays outside the triangle, is labeled, 70 degrees.

What is the value of x?

Enter your answer in the space provided.

x=

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

Starting with PQ¯ (not shown) a square will be constructed using a compass and straightedge so that PQ¯ is one of its sides.

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

Select all that apply.

  1. A.
    A bisector of a right angle
  2. B.
    The perpendicular bisector of PQ¯
  3. C.
    A right angle with vertex P or vertex Q
  4. D.
    A segment with the same length as PQ¯
  5. E.
    An arc of a circle with center P or center Q

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

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

The figure shows triangle X Y Z, and line k. Line k passes through point Y, and is parallel to segment X Z,. Angles 1, 2, and 3 are labeled where segments X Y, and Y Z, intersect with line k, such that angle X Y Z is labeled, 2, and angles 1 and 3 are outside the triangle, to the left and right, respectively.

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?

  1. A.
    the measure of angle 1 equals the measure of angle X
  2. B.
    the measure of angle 1 equals the measure of angle 3
  3. C.
    the measure of angle 2 equals 90 degrees
  4. D.
    the measure of angle 2 plus the measure of angle Z, equals 180 degrees
i1

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

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

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

The figure shows intersecting lines k, and n. Line k extends upward and to the right, and line n is horizontal. The angles formed at the intersection of the two lines are labeled, clockwise, from upper left, 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°.
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?

  1. A.
    Substitution
  2. B.
    Reflexive Property
  3. C.
    Subtraction Property of Equality
  4. D.
    Definition of Supplementary Angles
i3

Correct i1

Misunderstood the reason i2

Misunderstood the reason i4

Misunderstood the reason

This is test content.

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.