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## Categories of quadrilaterals based on sides and angles.

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David is cutting out shapes in art class. He cuts a variety of shapes. One of the shapes that he cuts out has opposite sides that are congruent and four 90 degree angles. He notices that one pair of opposite sides is longer than the other pair of opposite sides. What type of shape did David cut out?

### Guidance

A quadrilateral is a closed figure with four sides and four vertices. The vertex is a point where line segments meet. The points where the sides of a quadrilateral meet are called the vertices (which is the plural form of "vertex"). The prefix “quad” means four. You can always remember that a quadrilateral is a four sided figure because of this prefix.

Each quadrilateral has four sides and four angles.

A quadrilateral has four angles. You can name the angles by using the angle symbol and the letter of each vertex. This figure is \begin{align*}\angle{A}, \angle{B}, \angle{C}, \angle{D}\end{align*}.

A quadrilateral has four vertices. They are named by the letters, \begin{align*}A, B, C\end{align*} and \begin{align*}D\end{align*}. Naming the quadrilateral uses a small quadrilateral symbol and the four letters of the vertices, \begin{align*}\Box \ ABCD\end{align*}.

A quadrilateral has four sides. The sides are named by the endpoints of each line segment. The sides are \begin{align*}\overline{AB}, \overline{BD}, \overline{CD}, \overline{AC}\end{align*}.

There are several different kinds of quadrilaterals. Quadrilaterals are classified and identified based on angles and side lengths. While each four sided figure is a quadrilateral, sometimes there is a name that is more specific.

A parallelogram is a quadrilateral with opposite sides that are parallel and congruent. Congruent means exactly the same. In this case, a parallelogram has opposite sides that are parallel and congruent meaning that they have the same length.

Here is a parallelogram.

Notice that the opposite angles in this parallelogram are the same. In this figure, one pair of angles is acute and one pair of angles is obtuse. The arrows indicate the obtuse angles. The other two angles are acute. This is sometimes true but not always. Let’s look at a rectangle.

A rectangle is a parallelogram with four right angles. This means that a rectangle has opposite sides parallel and congruent and it also has four right angles.

A square is a rectangle with four congruent sides and four right angles. Notice that the properties of a parallelogram applies to squares, too.

A rhombus is a parallelogram with four congruent sides. Notice that a rhombus does not ALWAYS have right angles like a square, but it does always have congruent sides.

A trapezoid is a parallelogram with one pair of opposite sides parallel. Here is an example of a trapezoid.

Notice that this trapezoid has two acute angles and two obtuse angles. Be sure to check out the angles when you look at a trapezoid.

### Guided Practice

What is the name of the figure?

First, look to see if any of the sides are the same length.

Yes, all four sides

Next, look to see if there are any right angles.

Yes, all four angles

Then, classify the shape.

Square

### Examples

Identify each of the following quadrilaterals. Be as specific as you can.

#### Example 1

First, look to see if any of the sides are the same length.

Yes, all four sides

Next, look to see if there are any right angles.

Yes, all four angles

Then, classify the shape.

Square

#### Example 2

First, look to see if any of the sides are the same length.

No

Next, count the pairs of parallel sides.

One

Then, classify the shape.

Trapezoid

#### Example 3

First, look to see if any of the sides are the same length.

Yes, the opposite sides

Next, look to see if there are any right angles.

No

Then, classify the shape.

Parallelogram

Remember David and his shapes? One of his shapes has two pairs of parallel sides that are congruent, four 90 degree angles and one pair of opposite sides is longer than the other pair. What type of shape did David cut out?

First, note if any of the sides are the same length.

Yes, opposite sides are the same length.

Next, determine if there are any right angles.

Yes, all four angles

Then, classify the shape.

Rectangle

### Explore More

Look at each image and name the quadrilateral pictured.

1.

2.

3.

4.

5.

Name the geometric figure described below or answer each question.

6. Has four sides and four angles

7. Has one pair of opposite sides that are parallel

8. Has four right angles and four congruent sides

9. A parallelogram with four right angles.

10. A parallelogram with four congruent sides

11. Has four right angles

12. Has four sides

13. The word "parallel is part of it's name

14. True or false. A parallelogram is also a square.

15. True or false. A square is also a rectangle.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 9.10.

### Vocabulary Language: English

Congruent

Congruent

Congruent figures are identical in size, shape and measure.
Kite

Kite

Parallel

Parallel

Two or more lines are parallel when they lie in the same plane and never intersect. These lines will always have the same slope.
Parallelogram

Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides.

A quadrilateral is a closed figure with four sides and four vertices.
Rectangle

Rectangle

A rectangle is a quadrilateral with four right angles.
Rhombus

Rhombus

A rhombus is a quadrilateral with four congruent sides.
Square

Square

A square is a polygon with four congruent sides and four right angles.
Trapezoid

Trapezoid

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