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Do you know what a grind box is?

Isabelle is a friend of Marc and Isaac’s at school. She overheard them talking at lunch about the skate park and is excited to help. Isabelle has been skateboarding for a few years and loves the sport.

“She is really good,” Marc says to Isaac.

“Yes, and she is coming over to help us,” Isaac says.

He and Marc have a snack and soon the doorbell rings and it is Isabelle. She has brought her notebook and pencil.

“Hi guys, I have a great idea for the grind box,” Isabelle says, as Isaac’s mom comes into the room.

“What is a grind box?” Isaac’s mom asks.

“It’s a box designed for grinding and sliding tricks,” Marc explains. “We are going to design one today for our park.”

“We can probably build it ourselves,” says Isabelle.

Isaac’s Mom smiles and leaves the room. The three get to work. All is going well until they begin drawing the actual grind box. All three of them have different ideas about the shape the box should be.

Here is Isaac’s box.

Isabelle looks at the drawing and shakes her head.

“Those won’t work guys, the angles are all wrong. You have to have right angles to make this work. The box is only a box if you use rectangles and squares in the design.”

“The angles are alright on this one,” Isaac argues, looking at his design.

“No they aren’t. This is a parallelogram-the angles are not right angles and this box could fall over if you landed incorrectly.”

“We have to design a grind box using squares and rectangles.”

“The angle thing makes sense,” Marc says. “but are you sure that only squares and rectangles will work?”

Before Isabelle can answer, stop right there-this is where you come in. This Concept will teach you all about quadrilaterals. By the end of the Concept, you will be able to answer this question yourself and help the trio with their dilemma.

Guidance

Previously we learned all about triangles. In this Concept, you will learn about quadrilaterals. Let’s begin by learning about identifying a quadrilateral.

What is a quadrilateral?

A quadrilateral is a closed figure with four sides and four vertices . Remember that 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 . We can name the angles by using the angle symbol and the letter of each vertex. Here we have $\angle{A}, \angle{B}, \angle{C}, \angle{D}$ .

A quadrilateral also has four vertices . They are named by letter, $A, B, C$ and $D$ . Naming the quadrilateral uses a small quadrilateral symbol and the four letters of the vertices, $\Box \ ABCD$ .

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

There are several different kinds of quadrilaterals. We can classify and identify quadrilaterals based on angles and side lengths. While each four sided figure is a quadrilateral, sometimes there is a better name for the figure. A name that is more specific can tell us things about the figure.

There are several different types of quadrilaterals. We can figure them out by looking at the relationship between the sides of the quadrilateral and the angles.

Parallelogram – a quadrilateral with opposite sides which are parallel and congruent .

Congruent is a word that you will see a lot in geometry. 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.

What about the angles of 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.

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.

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

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.

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

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

Example A

Solution: A square

Example B

Solution: A trapezoid

Example C

Solution: A parallelogram

Here is the original problem once again. Reread it and then see if you can answer the questions posed about the shapes used in the box.

Isabelle is a friend of Marc and Isaac’s at school. She overheard them talking at lunch about the skatepark and is excited to help. Isabelle has been skateboarding for a few years and loves the sport.

“She is really good,” Marc says to Isaac.

“Yes, and she is coming over to help us,” Isaac says.

He and Marc have a snack and soon the doorbell rings and it is Isabelle. She has brought her notebook and pencil.

“Hi guys, I have a great idea for the grind box,” Isabelle says as Isaac’s mom comes into the room.

“What is a grind box?” Isaac’s mom asks.

“It’s a box designed for grinding and sliding tricks,” Marc explains. “We are going to design one today for our park.”

“We can probably build it ourselves,” says Isabelle.

Isaac’s Mom smiles and leaves the room. The three get to work. All is going well until they begin drawing the actual grind box. All three of them have different ideas about the shape the box should be.

Here is Isaac’s box.

Isabelle looks at the drawing and shakes her head.

“Those won’t work guys, the angles are all wrong. You have to have right angles to make this work. The box is only a box if you use rectangles and squares in the design.”

“The angles are alright on this one,” Isaac argues, looking at his design.

“No they aren’t. This is a parallelogram-the angles are not right angles and this box could fall over if you landed incorrectly.”

“We have to design a grind box using squares and rectangles.”

“The angle thing makes sense,” Marc says. “but are you sure that only squares and rectangles will work?”

First, underline all of the important information.

Next, let’s think about the characteristics of the three figures in the problem.

First, Isaac draws a grind box parallelogram which has opposite sides parallel and congruent. The sum of the angles of a parallelogram equals $360^\circ$ , but the angles are not necessarily right angles.

A rectangle and a square are also parallelograms, but they have angles that are $90^\circ$ .

Isabelle is correct that a parallelogram might tip over because the angles are not right angles.

The grind box pictured uses squares and rectangles because of the stability of the right angle. Isabelle, Marc and Isaac decide to build their grind box using squares and rectangles, that way they can be sure that it is a stable construction.

Vocabulary

closed figure with four sides and four vertices.
Trapezoid
Quadrilateral with one pair of opposite sides parallel.
Rectangle
Parallelogram with four right angles.
Parallelogram
Quadrilateral with opposite sides congruent and parallel.
Square
Four congruent sides and four congruent angles.
Rhombus
Parallelogram with four congruent sides.
Parallel
lines that are equidistant and will never intersect
Congruent
exactly the same, having the same measure

Guided Practice

Here is one for you to try on your own.

What is the name of the figure? Explain how you can be sure this is accurate.

This is a square.

The opposite sides are congruent and parallel. Two right angles have been identified meaning that the other two angles are right angles as well.

Practice

Directions: Look at each image and name the quadrilateral pictured.

1.

2.

3.

4.

5.

Directions: 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.