## Categories of quadrilaterals based on sides and angles.

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A yurt is the traditional dwelling of the nomads of Mongolia. This tent-like circular structure is a domed home made from a wooden frame and has lattice walls made of flexible wood. These lattice walls are made in sections of crisscrossed wooden poles attached together with leather ties.

If you were to draw a picture to represent one section of the lattice wall, what type of quadrilateral would you have to draw to represent a section of crisscrossed poles?

In this concept, you will learn to classify quadrilaterals.

A quadrilateral is a two-dimensional (a flat shape) closed shape that has four straight sides. The word quadrilateral defines itself since ‘quad’ means four and ‘lateral’ means sides. Included in this broad category of four-sided figures are special types of quadrilaterals.

The first special quadrilateral is a parallelogram. A parallelogram is a quadrilateral with opposite sides parallel and congruent. The following diagram is a parallelogram.

Notice that sides \begin{align*}\begin{array}{rcl} \overline{AB} & \cong & \overline{CD}\\ \overline{AB} & || & \overline{CD} \end{array}\end{align*} and sides \begin{align*}\begin{array}{rcl} \overline{AC} & \cong & \overline{BD}\\ \overline{AC} & || & \overline{BD} \end{array}\end{align*}. Remember parallel means the parallel sides are always the same distance apart and they will never intersect. The congruent sides are equal in length.

There are three types of parallelograms.

One type of parallelogram is called a rectangle. A rectangle is a parallelogram where every angle is a right angle. Also the opposite sides of a rectangle are congruent and parallel. The following diagram is a rectangle.

Notice that sides \begin{align*}\begin{array}{rcl} \overline{AB} & \cong & \overline{CD}\\ \overline{AB} & || & \overline{CD} \end{array}\end{align*} and sides \begin{align*}\begin{array}{rcl} \overline{AC} & \cong & \overline{BD}\\ \overline{AC} & || & \overline{BD} \end{array}\end{align*}. In addition to the relationships between the opposite sides of the rectangle, all of the angles of the rectangle are right angles.

A second type of parallelogram is a square. A square is a parallelogram where every angle is a right angle. Opposite sides of the square are parallel and all sides are the same length.

Notice that sides \begin{align*}\begin{array}{rcl} \overline{AB} & || & \overline{CD}\\ \overline{AC} & || & \overline{BD} \end{array}\end{align*} and sides \begin{align*}\overline{AC} \cong \overline{BD} \cong \overline{AB} \cong \overline{CD}\end{align*}. In addition to the relationships between the opposite sides and all sides of the rectangle, all of the angles of the rectangle are right angles.

A third type of parallelogram is a rhombus. A rhombus is a parallelogram with every side equal in length. Opposite sides of the rhombus are parallel but the angles are not right angles. The following diagram is a rhombus.

Notice that sides \begin{align*}\begin{array}{rcl} \overline{AB} & || & \overline{CD}\\ \overline{AC} & || & \overline{BD} \end{array}\end{align*} and sides \begin{align*}\overline{AC} \cong \overline{BD} \cong \overline{AB} \cong \overline{CD}\end{align*}.

Another type of quadrilateral that is not a parallelogram is a trapezoid. A trapezoid is a quadrilateral that has one pair of opposite sides parallel. The non-parallel sides are called the legs of the trapezoid. The angles on the same side of a leg are adjacent angles such that \begin{align*}\angle A\end{align*} and \begin{align*}\angle C\end{align*} are supplementary. This means \begin{align*}\angle A+\angle C=180^\circ\end{align*}. The following diagram is a trapezoid.

Notice that side \begin{align*}\overline{AB} || \overline{CD}\end{align*}.

What type of quadrilateral has opposite sides congruent and parallel? Every angle is not a right angle.

A parallelogram has opposite sides congruent and parallel and every angle is not a right angle.

Another type of quadrilateral is called a kite. A kite is a flat shape such that it has two pairs of congruent adjacent sides. The angles where the two pairs meet are also congruent in measure. The following diagram is a kite.

### Examples

#### Example 1

Remember the yurt? You need to figure out the type of quadrilateral used to form the lattice walls.

The sides of each figure created by the lattice are all equal in length. This may make you think that this is definitely a square. However, if you look at the angles, the angles are not right angles. Therefore, it can’t be a square. In fact, it is actually a rhombus. Remember a rhombus has four sides of equal length, opposite sides are equal and parallel, but every angle is not a right angle.

#### Example 2

What is the name of the four-sided figure that has all sides equal in length, opposite sides parallel and four right angles?

The name of the figure is a square.

#### Example 3

How are a parallelogram and a rectangle different?

Remember there are several types of parallelograms. A regular parallelogram has no right angles but every angle of a rectangle is a right angle.

#### Example 4

What is the name of the quadrilateral that has one pair of opposite sides parallel?

A trapezoid has one pair of opposite sides parallel.

#### Example 5

What name is given to the quadrilateral that has two pairs of equal adjacent sides?

This quadrilateral is called a kite.

### Review

Identify each quadrilateral based on the description provided.

1. A figure with four equal sides and four right angles.

2. A figure with opposite sides congruent and parallel.

3. A figure with opposite sides congruent and parallel and four right angles.

4. A figure with four sides.

5. A figure with four equal sides which may or may not have four right angles.

Use what you have learned about quadrilaterals to answer each of the following questions true or false.

6. A quadrilateral can be any four sided figure.

7. A rectangle is also a parallelogram, but a parallelogram is not necessarily a rectangle.

8. A square is never a parallelogram.

9. A rhombus can be a square.

10. A square is always a rhombus.

11. A rhombus is a parallelogram.

12. A quadrilateral is a type of parallelogram.

13. A trapezoid has opposite sides parallel and congruent.

14. What does the sum of the angles of a quadrilateral add up to be?

15. What are all four angle measures of a rectangle?

16. What are all four angle measures of a square?

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### Vocabulary Language: English

TermDefinition
Congruent Congruent figures are identical in size, shape and measure.
Parallelogram A parallelogram is a quadrilateral with two pairs of parallel sides.
Quadrilateral A quadrilateral is a closed figure with four sides and four vertices.
Rectangle A rectangle is a quadrilateral with four right angles.
Rhombus A rhombus is a quadrilateral with four congruent sides.
Square A square is a polygon with four congruent sides and four right angles.
Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel opposite sides.

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