# Parallelogram Classification

## Rectangles, rhombuses, and squares are specific parallelograms.

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Parallelogram Classification

What if you were designing a patio for you backyard? You decide to mark it off using your tape measure. Two sides are 21 feet long and two sides are 28 feet long. Explain how you would only use the tape measure to make your patio a rectangle.

### Parallelogram Classification

Rectangles, rhombuses (the plural is also rhombi) and squares are all more specific versions of parallelograms.

Rectangle Theorem: A quadrilateral is a rectangle if and only if it has four right (congruent) angles.

Rhombus Theorem: A quadrilateral is a rhombus if and only if it has four congruent sides.

Square Theorem: A quadrilateral is a square if and only if it has four right angles and four congruent sides.

From the Square Theorem, we can also conclude that a square is a rectangle and a rhombus.

Recall that diagonals in a parallelogram bisect each other. Therefore, the diagonals of a rectangle, square and rhombus also bisect each other. The diagonals of these parallelograms also have additional properties.

#### Investigation: Drawing a Rectangle

Tools Needed: pencil, paper, protractor, ruler

1. Draw two lines on either side of your ruler, to ensure they are parallel. Make these lines 3 inches long.
2. Remove the ruler and mark two angles, 2.5 inches apart on the bottom line drawn in Step 1. Then, draw the angles to intersect the top line. This will ensure that all four angles are . Depending on your ruler, the sides should be 2.5 inches and 1 inch.
3. Draw in the diagonals and measure them. What do you discover?

Theorem: A parallelogram is a rectangle if and only if the diagonals are congruent.

#### Investigation: Drawing a Rhombus

Tools Needed: pencil, paper, protractor, ruler

1. Draw two lines on either side of your ruler, to ensure they are parallel. Make these lines 3 inches long.
2. Remove the ruler and mark a angle, at the left end of the bottom line drawn in Step 1. Draw the other side of the angle and make sure it intersects the top line. Measure the length of this side.
3. The measure of the diagonal (red) side should be about 1.3 inches (if your ruler is 1 inch wide). Mark this length on the bottom line and the top line from the point of intersection with the angle. Draw in the fourth side. It will connect the two endpoints of these lengths.
4. By the way we drew this parallelogram; it is a rhombus because all four sides are 1.3 inches long. Draw in the diagonals.

Measure the angles created by the diagonals: the angles at their point of intersection and the angles created by the sides and each diagonal. You should find the measure of 12 angles total. What do you discover?

Theorem: A parallelogram is a rhombus if and only if the diagonals are perpendicular.

Theorem: A parallelogram is a rhombus if and only if the diagonals bisect each angle.

We know that a square is a rhombus and a rectangle. So, the diagonals of a square have the properties of a rhombus and a rectangle.

#### Classifying Parallelograms

What type of parallelogram are the ones below?

a)

All sides are congruent and one angle is , meaning that the angles are not congruent. By the Rhombus Theorem, this is a rhombus.

b)

This quadrilateral has four congruent angles and all the sides are not congruent. By the Rectangle Theorem, this is a rectangle.

#### Understanding the Definition of a Rhombus

Is a rhombus SOMETIMES, ALWAYS, or NEVER a square? Explain your reasoning.

A rhombus has four congruent sides, while a square has four congruent sides and angles. Therefore, a rhombus is only a square when it also has congruent angles. So, a rhombus is SOMETIMES a square.

#### Listing the Properties of a Square

List everything you know about the square .

A square has all the properties of a parallelogram, rectangle and rhombus.

Properties of Parallelograms Properties of Rhombuses Properties of Rectangles

#### Patio Problem Revisited

In order for the patio to be a rectangle, first the opposite sides must be congruent. So, two sides are 21ft and two are 28 ft. To ensure that the parallelogram is a rectangle without measuring the angles, the diagonals must be equal. You can find the length of the diagonals by using the Pythagorean Theorem.

### Examples

#### Example 1

Is a rectangle SOMETIMES, ALWAYS, or NEVER a parallelogram? Explain why.

A rectangle has two sets of parallel sides, so it is ALWAYS a parallelogram.

#### Example 2

Is a rhombus SOMETIMES, ALWAYS, or NEVER equiangular? Explain why.

Any quadrilateral, including a rhombus, is only equiangular if all its angles are . This means are rhombus is SOMETIMES equiangular, only when it is a square.

#### Example 3

Is a quadrilateral SOMETIMES, ALWAYS, or NEVER a pentagon? Explain why.

A quadrilateral has four sides, so it will NEVER be a pentagon with five sides.

### Review

1. is a rectangle. Find:
2. is a rhombus. Find:
3. Draw a square and label it . Mark the point of intersection of the diagonals . Find:

For questions 4-12, determine if the quadrilateral is a parallelogram, rectangle, rhombus, square or none. Explain your reasoning.

For problems 13-15, find the value of each variable in the figures.

For questions 16-19 determine if the following are ALWAYS, SOMETIMES, or NEVER true. Explain your reasoning.

1. A rectangle is a rhombus.
2. A square is a parallelogram.
3. A parallelogram is regular.
4. A square is a rectangle.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
converse If a conditional statement is $p \rightarrow q$ (if $p$, then $q$), then the converse is $q \rightarrow p$ (if $q$, then $p$. Note that the converse of a statement is not true just because the original statement is true.
Parallelogram A parallelogram is a quadrilateral with two pairs of parallel sides.
Reflexive Property of Congruence $\overline{AB} \cong \overline{AB}$ or $\angle B \cong \angle B$