Properties of Quadrilaterals and Parallelograms
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Questions and Answers

A quadrilateral whose diagonals bisect each other and are perpendicular is a

  • rhombus (correct)
  • parallelogram
  • rectangle
  • trapezoid

    • In a certain quadrilateral, two opposite sides are parallel, and the other two opposite sides are not congruent. This quadrilateral could be a

  • square
  • trapezoid (correct)
  • parallelogram
  • rhombus

    • Which quadrilateral has diagonals that always bisect its angles and also bisect each other?

  • rhombus (correct)
  • isosceles trapezoid
  • parallelogram
  • rectangle

    • The diagonals of a quadrilateral are congruent but do not bisect each other. This quadrilateral is

    <p>rectangle (A)</p> Signup and view all the answers

    Given three distinct quadrilaterals, a square, a rectangle, and a rhombus, which quadrilaterals must have perpendicular diagonals?

    <p>the rhombus and the square (B)</p> Signup and view all the answers

    Which reason could be used to prove that a parallelogram is a rhombus?

    <p>Diagonals are perpendicular (C)</p> Signup and view all the answers

    If ABCD is a parallelogram, which statement would prove that ABCD is a rhombus?

    <p>AC ⊥ BD (B)</p> Signup and view all the answers

    A parallelogram is always a rectangle if

    <p>the diagonals intersect at right angles (C)</p> Signup and view all the answers

    If the midpoints of the sides of a quadrilateral are joined consecutively, the resulting figure will always be a

    <p>parallelogram (B)</p> Signup and view all the answers

    In rhombus ABCD, the diagonals AC and BD intersect at E. If AE = 5 and BE = 12, what is the length of AB?

    <p>13 (C)</p> Signup and view all the answers

    Flashcards

    Quadrilateral with perpendicular diagonals

    A quadrilateral whose diagonals bisect each other and are perpendicular is a rhombus.

    Quadrilateral with non-congruent sides

    A quadrilateral having two opposite sides parallel and not congruent could be a trapezoid.

    Diagonals bisect angles

    The quadrilateral that has diagonals that always bisect its angles and each other is a rhombus.

    Quadrilateral with congruent diagonals

    If the diagonals of a quadrilateral are congruent but do not bisect each other, it is an isosceles trapezoid.

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    Perpendicular diagonals in quadrilaterals

    Among a square, rectangle, and rhombus, the square and rhombus must have perpendicular diagonals.

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    Proving a parallelogram is a rhombus

    To prove a parallelogram is a rhombus, you can show the diagonals are perpendicular.

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    Line segments in quadrilaterals

    If ABCD is a parallelogram, the statement $ ar{AC} ot ar{BD} $ proves it is a rhombus.

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    Parallelogram vs rectangle

    A parallelogram is always a rectangle if the diagonals are congruent.

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    Midpoints of a quadrilateral

    Connecting the midpoints of the sides of a quadrilateral results in a parallelogram.

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    Length calculation in rhombus

    In rhombus ABCD, if $ AE = 5 $ and $ BE = 12 $, then $ ar{AB} = 13 $ using the Pythagorean theorem.

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    Properties of rhombus

    A rhombus has diagonals that bisect each other and are perpendicular, and all sides are equal.

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    Trapezoid characteristics

    A trapezoid has at least one pair of parallel sides, but the other sides are not necessarily equal.

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    Rectangle properties

    A rectangle has opposite sides equal and all angles equal to 90 degrees.

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    Square features

    A square is a special rectangle with all sides equal and all angles being right angles.

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    Parallelogram angle characteristics

    In a parallelogram, opposite angles are equal, which helps define its properties.

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    Isosceles trapezoid traits

    An isosceles trapezoid has congruent legs and bases that are parallel.

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    Congruent diagonals

    Congruent diagonals are found in rectangles and isosceles trapezoids, affecting their shape.

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    Diagonal intersections

    The diagonals of a quadrilateral can tell us important properties about the shape, like whether it's a rhombus or rectangle.

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    Quadrilaterals with congruent angles

    If all angles in a quadrilateral are equal, it's a rectangle or square.

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    Study Notes

    Quadrilateral Properties

    • A quadrilateral with diagonals that bisect each other and are perpendicular is a rhombus.
    • A quadrilateral with two pairs of parallel sides and the opposite sides not congruent is a trapezoid.
    • A quadrilateral with diagonals that bisect its angles and bisect each other is a rhombus.
    • A parallelogram with congruent diagonals is a rectangle.
    • Diagonals of a rhombus are perpendicular.
    • Diagonals of a rectangle are congruent.
    • Joining the midpoints of a quadrilateral consecutively results in a parallelogram.

    Parallelograms and Rhombuses

    • Opposite sides of a parallelogram are parallel.
    • Opposite angles of a parallelogram are congruent.
    • Diagonals of a parallelogram bisect each other.
    • A parallelogram is a rhombus if its diagonals are perpendicular.
    • In a parallelogram ABCD, if AC is perpendicular to BD, then the parallelogram is a rhombus.

    Rhombus Diagonals

    • In rhombus ABCD, diagonals AC and BD intersect at E. If AE = 5 and BE = 12, then AB = 13. (Pythagorean theorem)

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    Description

    Test your understanding of quadrilateral properties, focusing on rhombuses and parallelograms. This quiz will challenge your knowledge on the relationships and characteristics of these shapes, including their diagonals and sides. Ideal for students studying geometry, this quiz covers essential concepts for mastering the topic.

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