Special Parallelograms



Special Parallelograms

Lesson 6-4A: The Rectangle

Warm-Up

List 6 properties of the parallelogram.

The Quadrilateral Families

The set of quadrilaterals (four-sided figures) may be divided into three main families.

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We began our study of quadrilaterals with a look into the set of parallelograms.

Let’s look more closely into the parallelogram family.

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Because the rhombus, rectangle, and square are all parallelograms, these figures all inherit the properties of a parallelogram.

|Rectangle |Rhombus |Square |

|Parallelogram with one right angle |Parallelogram with two consecutive |Parallelogram with two consecutive |

| |congruent sides |congruent sides and one right angle |

The rhombus, rectangle, and square contain all properties of the parallelogram.

The rhombus, rectangle, and square have:

▪ Both pairs of opposite sides congruent and parallel.

▪ Diagonals that bisect each other

▪ Both pairs of opposite angles congruent and consecutive angles supplementary.

However, the inheritance does NOT go the other way.

For example, some properties of the rhombus are not shared by the parallelogram.

We will call these the special properties of the rhombus. In this lesson, we will study the special properties of the rhombus.

Recall the parallelogram family. This lesson will focus on rectangles and squares.

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Definition of a Rectangle

A rectangle is a parallelogram with one right angle.

Special Properties of a Rectangle

Recall too that because a rectangle is a parallelogram, it has all the characteristics of a parallelogram. But it also has others:

1) All angles of a rectangle are right angles.

Since a rectangle is a parallelogram, opposite angles are congruent and consecutive angles are supplementary. This makes all four angles congruent and 90 degrees.

We can also say a rectangle is equiangular.

2) The diagonals of a rectangle are congruent.

Ex. In rectangle ABCD, we have AC = BD.

Model Problem

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2)

3) Find the measure of each numbered angle in the rectangle.

Exercise

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5) Find the measure of each numbered angle in the rectangle.

Proving that a Quadrilateral is a Rectangle

1) If the diagonals of a parallelogram are congruent, then it is a rectangle.

2) If a parallelogram contains one right angle, then it is a rectangle.

Model Problem

1)

2)

Exercise

1) Is the figure at right a rectangle? Explain why or why not.

2) Determine whether the figure at right is a rectangle. Justify your answer using

coordinate geometry.

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Lesson Summary

A rectangle is a parallelogram with one right angle. A rectangle has all the characteristics of a parallelogram, plus some special characteristics:

|From the Parallelogram: |Special Characteristics: |

| | |

|Opposite sides are parallel. |Diagonals are congruent. |

|Opposite sides are congruent. |Contains four right angles. |

|Opposite angles are congruent. |Diagonals make four isosceles triangles. |

|Consecutive angles are supplementary. | |

|Diagonals bisect each other. | |

|Diagonal divide the parallelogram into 2 congruent triangles. | |

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Kites

Trapezoids

Parallelograms

Quadrilaterals

Parallelograms

Rhombuses

Rectangles

Squares

Parallelograms

Rhombuses

Rectangles

Squares

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