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Quadrilaterals

Objective

In this lesson, you will study the relationship of the angles, sides, and diagonals in special quadrilaterals.

Previously Covered:

  • A line segment whose endpoints lie on two nonconsecutive vertices of the polygon is called a diagonal.
  • The parallel postulate states: Given a point not on a line, there is exactly one line parallel to the given line containing that point.
  • If two parallel lines are cut by a transversal, then any pair of alternate interior angles is congruent.

What is a quadrilateral?

Quadrilaterals are the simplest polygons after triangles; they are polygons with four sides. There is a special class of quadrilaterals for which we have a plethora of theorems. In fact, there are so many we will mention just a handful of them here. But first, let’s get comfortable with quadrilateral terminology.

Which quadrilaterals are special?

In a quadrilateral, two sides are opposite if they do not share a common endpoint. Two sides are consecutive if they share a common endpoint.

Special quadrilaterals have at least one pair of opposite sides parallel.

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Other properties of parallelograms include:

  • Opposite sides are congruent
  • Opposite angles are congruent.
  • Consecutive angles are supplementary.
  • The diagonals bisect each other.

As you probably know, the most famous of quadrilaterals, a rectangle, is a quadrilateral with four right angles.

The less famous rhombus is a parallelogram with four congruent sides. In any rhombus, the diagonals are perpendicular to each other.

A square is a rectangle with congruent sides.

Question

Which of the following statements is false?

  1. Every rhombus is a parallelogram.
  2. Every square is a parallelogram.
  3. Every rectangle is a square.
  4. Every square is a rhombus.

Reveal Answer

The correct answer is C. The truth of A and B follow directly from the definitions. The truth of D requires more thought; but according to the definitions, a square can also be thought of as a rhombus with four congruent angles.

What are some properties of
trapezoids?

A trapezoid is a quadrilateral with exactly parallel segments called the bases; the nonparallel sides are called legs. Each base forms two angles called, appropriately, base angles. Thus, each trapezoid contains two pairs of base angles.

An isosceles trapezoid is a trapezoid with congruent legs.

  • Each pair of base angles in an isosceles trapezoid is congruent.
  • The diagonals of an isosceles trapezoid are congruent.

Important Tidbit

You can prove these theorems about isosceles trapezoids using the following corollary of the theorem on transversals cutting parallel lines: If two lines are parallel, the distance between the two lines is constant.

Remember, the distance between two lines is the distance from any one point of the line to the other.

Question

Which of the following quadrilaterals does not necessarily have congruent diagonals?

  1. Rectangle
  2. Square
  3. Rhombus
  4. Isosceles trapezoid

Reveal Answer

The correct answer is C. A rhombus that is not a square will have non-congruent diagonals.

Review of New Vocabulary and Concepts

  • When the endpoints of a line segment lie on two nonconsecutive vertices of a polygon, that line segment is called a diagonal.
  • The sum of the measures of the interior angles of a convex polygon with n sides is 180(n – 2).
  • The sum of the measures of the exterior angles of a convex polygon with n sides is 360°.
  • A trapezoid is a quadrilateral with exactly one pair of parallel sides. Isosceles trapezoids have the following properties:
    • Each pair of base angles in an isosceles trapezoid is congruent.
    • The diagonals of an isosceles trapezoid are congruent.
  • A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
  • A rectangle is a quadrilateral with four right angles.
  • A rhombus is a parallelogram with four congruent sides.
  • A square is a rectangle with congruent sides.

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