Properties of Special Parallelograms

Open Ended

Why is it important to understand the properties of rectangles when solving geometric problems?

Multiple Choice

Which of the following is NOT a key property of rectangles?

• Opposite sides are parallel and congruent

• All four angles are right angles

• Diagonals are congruent and bisect each other

• All angles are acute

Open Ended

Explain how you would find the value of x if US = 6x + 3 and RT = 7x - 2 in a rectangle.

Multiple Choice

Which property of rectangles allows us to set US = RT in the problem involving quadrilateral RUTS?

• Rectangles have equal diagonals

• Rectangles have equal sides

• Rectangles have right angles

• Rectangles have parallel sides

Multiple Choice

If m∠STR = 8x + 3 and m∠UTR = 16x - 9 in a rectangle, and their sum is 90°, what is the value of x?

• 2

• 3

• 4

• 5

Multiple Choice

Which of the following statements about the angles in a rectangle is correct?

• Each angle in a rectangle is 90 degrees

• Opposite angles in a rectangle are supplementary

• Adjacent angles in a rectangle are complementary

• All angles in a rectangle are acute

Multiple Select

In any rectangle PQRS, which pairs of sides are parallel?

• PQ and SR

• PS and SR

• QR and PS

• PQ and QR

Multiple Choice

What is the measure of each interior angle in a rectangle?

• 60 degrees

• 90 degrees

• 120 degrees

• 180 degrees

Multiple Select

Select all true properties of diagonals in a rectangle.

• Diagonals are congruent

• Diagonals bisect each other

• Diagonals are perpendicular

• Diagonals bisect opposite angles equally

Drag and Drop

Consecutive interior angles along one side of a rectangle are

Placeholder

.

Drag these tiles and drop them in the correct blank above

supplementary

complementary

equal

right

Fill in the Blank

Quadrilateral RUTS is a rectangle. If diagonal US = 6x + 3 and diagonal RT = 7x − 2, find the value of x.

Multiple Choice

Quadrilateral RUTS is a rectangle. If m∠STR = 8x + 3 and m∠UTR = 16x − 9, find m∠STR.

• 35°

• 37°

• 33°

• 40°

Multiple Choice

In rectangle RUTS, angle UTS is a right angle. Which equation correctly uses this fact to relate m∠STR and m∠UTR?

• m∠STR + m∠UTR = 180

• m∠STR − m∠UTR = 90

• m∠STR = m∠UTR

• m∠STR + m∠UTR = 90

Fill in the Blank

A rhombus is a quadrilateral with ___ congruent sides.

Open Ended

List two properties that squares share with both rectangles and rhombi.

Multiple Choice

In rhombus ABCD, m∠BAC = 32°. Which of the following statements about the numbered angles in the diagram is correct?

• m∠4 = 90°, m∠1 = 58°, m∠2 = 58°, m∠3 = 32°

• m∠4 = 90°, m∠1 = 32°, m∠2 = 58°, m∠3 = 58°

• m∠4 = 58°, m∠1 = 90°, m∠2 = 32°, m∠3 = 58°

• m∠4 = 32°, m∠1 = 58°, m∠2 = 90°, m∠3 = 58°

Open Ended

Explain how the properties of diagonals in a rhombus help in solving for unknown angles or side lengths in rhombus-related problems. Use examples from both diagrams to support your answer.

Multiple Choice

A quadrilateral has four equal sides and its diagonals are perpendicular but not congruent. Which classification fits best?

• Square in all cases

• Rhombus but not a square

• Rectangle but not a square

• Parallelogram only

Multiple Choice

In rhombus ABCD, diagonals AC and BD intersect at E and are perpendicular. If m∠BAC = 32°, what is m∠4 at E (the angle between the diagonals)?

• 32°

• 122°

• 58°

• 90°

Multiple Choice

Using the right triangle ABE formed by the perpendicular diagonals, find m∠1 at ABE when m∠BAC = 32°.

• 90°

• 148°

• 58°

• 32°

Multiple Choice

Diagonals of a rhombus bisect vertex angles. If m∠1 in triangle ABE is 58°, what is m∠2 at vertex B adjacent to ∠1?

• 32°

• 58°

• 116°

• 64°

Multiple Select

Along diagonal AC, angles 1 and 3 are formed with sides AB and BC. Given m∠BAC = 32° and the diagonals are perpendicular, which statements are correct? Select all that apply.

• m∠1 equals 58° using right triangle sum

• m∠3 equals 32° by alternate interior angles

• m∠2 equals 32° by angle bisection

• m∠4 equals 90° by perpendicular diagonals

Multiple Choice

Quadrilateral ABCD is a rhombus. If m∠ABD = 60°, what is m∠DBC?

• 120°

• 30°

• 60°

• 90°

Fill in the Blank

In rhombus ABCD, diagonal BD bisects angle B. If m∠BAD = 8x and m∠ABD = 8°, find x so that BD bisects ∠B.

.

Multiple Choice

Rhombus ABCD has side AB = 26 and diagonal BD = 20. Let E be the intersection of the diagonals. Find AE.

• 10

• 12

• 16

• 13

Multiple Choice

PRYZ is a rhombus. Given RK = 5, RY = 13, and m∠YRZ = 67°, find KY.

• 13

• 9

• 8

• 10

Multiple Choice

In rhombus PRYZ with RK = 5, RY = 13, and m∠YRZ = 67°, find PK.

• 5

• 8

• 10

• 13

Properties of squares

• Four equal sides

• Four right angles

• Two sets of parallel lines

Diagonals

Squares have two congruent diagonals that bisect each other at a right angle.

The diagonals also bisect the four corner angles

Multiple Select

Which properties hold for the diagonals of a square? Select all that apply.

• They are congruent segments

• They are perpendicular lines

• They bisect each other

• They are parallel lines

Multiple Choice

In a square ABCD, what is true about the angles at each vertex?

• Each is an acute angle

• Opposite angles are obtuse

• Adjacent angles are supplementary

• Each is a right angle

Fill in the Blank

Fill in the blank: The diagonals of a square ______ the vertex angles.

Open Ended

Reflecting on what you learned today, how do the properties of rectangles help in solving geometric problems?