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Quizzes > High School Quizzes > Mathematics

Unit 8 Parallelograms Practice Quiz

Review Polygons and Quadrilaterals Through Practice Problems

Difficulty: Moderate
Grade: Grade 7
Study OutcomesCheat Sheet
Colorful paper art promoting Parallelogram Power-Up, a dynamic high school trivia quiz.

Which of the following defines a parallelogram?
A quadrilateral with all sides congruent
A quadrilateral with two pairs of parallel sides
A quadrilateral with one pair of parallel sides
A quadrilateral with one pair of parallel angles
A parallelogram is defined as a quadrilateral with both pairs of opposite sides being parallel. This property distinguishes it from other quadrilaterals.
In a parallelogram, which statement is always true about opposite sides?
They are parallel and equal in length
They intersect at a right angle
They are perpendicular
They are congruent but not parallel
Opposite sides in a parallelogram are always parallel and equal in length. This is one of the defining properties of parallelograms.
Which pair of angles is congruent in a parallelogram?
Only one pair of angles
Adjacent angles
All angles
Opposite angles
In a parallelogram, opposite angles are congruent. Adjacent angles are supplementary, but not necessarily equal.
What is the relationship between consecutive angles in a parallelogram?
They are congruent
They are supplementary
They are complementary
They are both right angles
Consecutive angles in a parallelogram add up to 180 degrees, making them supplementary. This is an essential property of parallelograms.
In a diagram of a parallelogram, what do the diagonals do?
They are equal in length
They form right angles
They are parallel
They bisect each other
One important property of parallelograms is that the diagonals bisect each other. This means each diagonal divides the other into two equal parts.
Given a parallelogram with a base of 8 units and a height of 5 units, what is its area?
40 square units
16 square units
13 square units
26 square units
The area of a parallelogram is calculated as base times height. Multiplying 8 by 5 gives 40 square units, which is a fundamental property for area calculations.
If the consecutive angles of a parallelogram are 70° and 110°, which property does this confirm?
They are supplementary
They are complementary
They are equal
They form a linear pair unrelated to parallelograms
Since the consecutive angles add up to 180° (70° + 110° = 180°), they are supplementary. This confirms the key property of parallelograms regarding consecutive angles.
A parallelogram has one angle measuring 65°. What is the measure of its adjacent angle?
65°
115°
90°
120°
In a parallelogram, adjacent angles are supplementary; therefore, if one angle is 65°, the adjacent angle must be 115°. This directly follows from the angle relationships in parallelograms.
Which property of parallelograms ensures that the two diagonals intersect at their midpoint?
Diagonals bisect each other
Opposite sides are equal
The figure is cyclic
Adjacent sides are supplementary
A defining property of parallelograms is that their diagonals bisect each other, which means they intersect at their midpoints. This feature is essential to many proofs and calculations involving parallelograms.
In parallelogram ABCD, if AB = 10 cm and BC = 6 cm, what is the perimeter?
20 cm
32 cm
16 cm
26 cm
The perimeter of a parallelogram is calculated as twice the sum of two adjacent sides. With sides of 10 cm and 6 cm, the perimeter is 2 Ã - (10 + 6) = 32 cm.
When a parallelogram is transformed by a shear transformation, which property remains unchanged?
Perimeter
Side lengths
Area
Angle measures
A shear transformation alters the shape of a parallelogram without changing its area. This invariant property of area under shear is a crucial concept in understanding geometric transformations.
If the coordinates of vertices of a parallelogram are A(1,2), B(4,6), and D(3,3), what is the coordinate of vertex C?
C = (6,7)
C = (5,7)
C = (6,5)
C = (7,7)
In a parallelogram, the fourth vertex can be found using the relation C = B + D - A. Substituting A(1,2), B(4,6), and D(3,3) gives C = (4+3-1, 6+3-2) = (6,7).
A parallelogram has diagonals of lengths 10 cm and 6 cm. Which of the following statements is necessarily true?
They bisect each other
They divide the parallelogram into congruent triangles
They are equal in length
They are perpendicular
One inherent property of parallelograms is that their diagonals bisect each other, meaning they intersect at the midpoint. Although the diagonals have given lengths, only the bisecting property is guaranteed.
Which statement accurately describes a rectangle in the context of parallelograms?
A rectangle does not have parallel sides
A rectangle always has perpendicular diagonals
A rectangle is a quadrilateral with equal sides
A rectangle is a parallelogram with all angles equal to 90°
A rectangle is a special type of parallelogram where every interior angle is 90°. This distinguishes it from other parallelograms, even though not all rectangles have perpendicular diagonals.
Which statement about a rhombus is true?
A rhombus is not a type of parallelogram
A rhombus is a parallelogram with all sides equal and diagonals that intersect at right angles
A rhombus always has a right angle
A rhombus has all angles equal
A rhombus is a specific kind of parallelogram characterized by all sides being equal and its diagonals intersecting at right angles. This differentiates it from a general parallelogram where only opposite sides are equal.
In parallelogram ABCD, if the measure of angle A is 3 times the measure of angle B, what are the measures of angles A and B, respectively?
135° and 45°
100° and 30°
90° and 30°
120° and 60°
Since consecutive angles in a parallelogram are supplementary, if angle A is three times angle B then 3x + x = 180° implies x = 45°. Thus, angle A is 135° and angle B is 45°.
How can the area of a parallelogram be computed if the lengths of the sides and one angle between them are known?
Area = a + b
Area = ab sin(θ)
Area = ab cos(θ)
Area = 2a sin(θ)
When two sides and the included angle of a parallelogram are known, the area can be calculated using the formula area = ab sin(θ). This method employs the sine function to effectively determine the height component.
Which property distinguishes a square from a generic parallelogram?
Its diagonals always intersect at non-right angles
All sides are equal and all angles are 90°
Only opposite sides are equal
It has at least one pair of perpendicular sides
A square is a special type of parallelogram where all sides are equal and every angle is 90°. This set of properties distinguishes it clearly from a general parallelogram.
If a parallelogram has a base of length 12 units and an area of 72 square units, what is the height corresponding to that base?
6 units
10 units
8 units
5 units
The area of a parallelogram is given by the product of its base and height. Dividing the area (72 square units) by the base (12 units) yields a height of 6 units.
In a parallelogram, if one diagonal separates it into two congruent triangles, what can we infer about the properties of these triangles?
They have equal perimeters
They are congruent by the side-angle-side postulate
They are similar but not congruent
They always have a right angle
When a diagonal divides a parallelogram, it creates two triangles that are congruent, typically demonstrated using the side-angle-side postulate. This ensures that both triangles have the same side lengths and angle measures.
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Study Outcomes

  1. Describe the key properties of parallelograms, including parallel sides and congruent opposite angles.
  2. Identify and classify various quadrilaterals based on their geometric properties.
  3. Apply formulas to calculate the area and perimeter of parallelograms.
  4. Synthesize geometric concepts to solve problem-based scenarios involving parallelograms.
  5. Evaluate the effects of transformations on the shape and properties of parallelograms.

Unit 8: Polygons, Quadrilaterals & Parallelograms Cheat Sheet

  1. Opposite sides are parallel and equal - This fundamental property ensures the shape's balance and symmetry: each pair of opposite sides run in the same direction and have the same length. You can lean on this fact to verify parallelograms in both proofs and coordinate geometry. So next time you sketch one, trust your parallel sides to keep things even! MathPlanet: Parallelogram Properties
  2. Opposite angles are congruent - In any parallelogram, two angles facing each other always have the same measure, making it easy to solve for unknown angles by setting up simple equality relationships. This trick is a lifesaver when working through complex angle-chasing problems. Plus, once you spot one congruent pair, the rest falls into place faster than you'd expect! MathPlanet: Parallelogram Properties
  3. Consecutive angles are supplementary - Any two angles that share a side in a parallelogram add up to 180°, letting you quickly find missing angle values by simple subtraction. This property is perfect for angle puzzles and geometry proofs alike. Think of it as a friendly team: when one angle grows, its neighbor shrinks to keep the sum in check! MathPlanet: Parallelogram Properties
  4. Diagonals bisect each other - The diagonals of a parallelogram cut each other exactly in half, meaning the intersection point divides each diagonal into equal halves. You can use this to find midpoint coordinates or prove congruent triangles inside. Imagine it as a perfectly fair split - no favoritism here! MathPlanet: Parallelogram Properties
  5. Diagonals form congruent triangles - When you draw a diagonal, it splits the parallelogram into two identical triangles, so you can apply triangle congruence theorems to unlock more properties. This method often simplifies area and side-length calculations. It's like folding a paper shape and seeing two mirror-image triangles at once! MathPlanet: Parallelogram Properties
  6. Area equals base times height - To find the space inside, multiply the length of any base by the corresponding perpendicular height. Remember, that height is the straight-line distance from the base to the opposite side, not a slanted edge. Once you nail this, area problems become a piece of cake! AMU Guide: Parallelogram Formulas
  7. Perimeter is 2×(length + width) - Just add together one pair of adjacent sides and double the sum to trace the entire boundary. This handy formula helps in fencing problems and when dealing with real-life parallelogram shapes. No complicated calculus needed - just simple addition and multiplication! AMU Guide: Parallelogram Formulas
  8. One right angle makes a rectangle - The moment a parallelogram has a 90° angle, all its angles turn right, and voilà - you've got a rectangle with parallel opposite sides. This special case is useful for spotting rectangles hidden in complex figures. It's like discovering a secret identity inside the shape! MathPlanet: Parallelogram Properties
  9. Parallelogram law (sums of squares) - The sum of the squares of all four sides equals the sum of the squares of its diagonals, forming a neat algebraic identity. This law pops up in physics and vector geometry, linking geometry with algebra. Think of it as a geometry-themed math magic trick! Wikipedia: Parallelogram Law
  10. Rotational symmetry of order 2 - Rotate it 180° around its center, and the parallelogram looks exactly the same, showcasing its neat halfway-turn symmetry. This property deepens your understanding of geometric transformations. It's like the shape's own dress-rehearsal for a perfect half-turn spin! Wikipedia: Parallelogram
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