# 6.3: Quadrilaterals and Angles

**At Grade**Created by: CK-12

## Introduction

*Figuring out a Yurt*

“This is very cool!” Marcus exclaimed when looking through a book on different types of houses.

“What do you see?” Lynne asked leaning over her desk to look at the book that Marcus was holding.

Lynne and Marcus are both students in Mrs. Patterson’s World Cultures class. Like Jaime, they are also working on projects. Marcus has discovered a yurt. A yurt is a type of home common in Mongolia. There is a lattice structure that is build and then a canvas is used to cover the frame.

“That is cool, what is it?” Lynne asked.

“It’s called a yurt. I think that this is what I am going to do my project on,” Marcus said studying the picture.

Marcus took out a piece of paper and a pencil and he began to draw the lattice of the yurt.

**As Marcus draws his design, take a look at the lattice and hunt for the quadrilaterals that are used in the design. In this lesson you will learn all about different types of quadrilaterals so that by the end of the lesson, you will be able to identify the ones that Marcus will need to use.**

*What You Will Learn*

In this lesson, you will learn how to do the following skills.

- Classify quadrilaterals (trapezoid, parallelogram, rhombus, rectangle and square)
- Confirm that the sum of the angle measures of any quadrilateral is \begin{align*}360^\circ\end{align*}, using two triangles formed by the diagonal of the quadrilateral.
- Describe and analyze quadrilaterals and associated angle measures using known classifications and sufficient given info, using variable expressions.
- Describe and classify quadrilaterals found in real-world objects and architecture.

*Teaching Time*

I. **Classify Quadrilaterals (Trapezoid, Parallelogram, Rhombus, Rectangle and Square)**

In the last lesson, we learned all about triangles. You learned how to identify different types of triangles and how to problem solve angle measures. In this lesson, we are going to work with ** quadrilaterals**.

**What is a quadrilateral?**

**A** *quadrilateral***is any four-sided figure.** In the word “quadrilateral”, we find the word “quad” which means four. This means that any four-sided figure is considered a quadrilateral. Now, there are different types of quadrilaterals that we are going to learn about in this lesson.

**We can say that a quadrilateral is any four-sided figure. We could consider this an umbrella category meaning that there are different types of quadrilaterals that we can identify in a specific way even though they are still quadrilaterals too.** Let’s look at identify the types of quadrilaterals.

**The first type of quadrilateral to learn about is called a parallelogram. A** *parallelogram***is a quadrilateral with opposite sides parallel and congruent.** Here is a picture of a parallelogram.

When you look at this picture, you can see that the opposite sides of the figure are parallel. They are also the **same length-meaning** *congruent***.**

**There are three main kinds of parallelograms.** Parallelograms can be plain old parallelograms like the one in the picture. They can also be rectangles, squares and rhombi.

**A** *rectangle***is a parallelogram with four right angles, where opposite sides are congruent and parallel.** You have been looking at rectangles for a long time, but now you need to notice that there are specific properties that make a rectangle a rectangle.

**A** *rhombus***is a parallelogram with four congruent sides, but not necessarily four right angles. A rhombus can look like a square, but while a square is always a rhombus, a rhombus is not necessarily a square. A rhombus can only be a square if it has four right angles.**

**A** *square***is a parallelogram too. The big difference between a square and a rectangle is that a square has four congruent sides. It also has four right angles though just like a rectangle.**

**There is one other type of quadrilateral. This quadrilateral is NOT a parallelogram. It is a special kind of quadrilateral. It is called a** *trapezoid***. A** *trapezoid***is a quadrilateral with one pair of opposite sides parallel.**

*Write these definitions and draw a picture of each figure in your notebook.*

The best way to remember the different types of quadrilaterals is to spend a little time studying the definitions. Then you will be able to identify them and answer questions about the different types with ease.

II. **Confirm that the Sum of the Angle Measures of any Quadrilateral is \begin{align*}\underline{360^\circ,}\end{align*} using Two Triangles Formed by the Diagonal of the Quadrilateral**

In the last lesson on triangles, you discovered that the sum of the angles of any triangles will always add up to be \begin{align*}180^\circ\end{align*}. We can find a similar rule with quadrilaterals. Take a look.

**One important thing to remember about quadrilaterals is that their four angles always have a sum of \begin{align*}360^\circ\end{align*}. This is true no matter what shape or size the quadrilateral is.**

Notice how different the angles and the sides of the quadrilaterals are. Look closely, though. If you add up the measures of the four angles, they always equal \begin{align*}360^\circ\end{align*}. **This is because every quadrilateral is actually two triangles put together. As we know, the three angles in all triangles always add up to \begin{align*}180^\circ\end{align*}.**

This quadrilateral has been divided into two congruent triangles, each with angles of \begin{align*}120^\circ, 25^\circ\end{align*}, and \begin{align*}35^\circ\end{align*}. If we add these angles together, we get a sum of \begin{align*}180^\circ\end{align*}. If we step back and look at the whole quadrilateral, we see that it has two \begin{align*}120^\circ\end{align*} angles and two \begin{align*}60^\circ\end{align*} angles \begin{align*}(25^\circ + 35^\circ = 60^\circ)\end{align*}. When we add these together, we get a sum of \begin{align*}360^\circ\end{align*}: \begin{align*}60^\circ + 120^\circ + 60 + 120^\circ = 360^\circ\end{align*}. This will be true no matter what size each angle in the quadrilateral measures.

III. **Describe and Analyze Quadrilaterals and Associated Angle Measures Using Known Classifications and Sufficient Given Information, using Variable Expressions**

We can use what we know about quadrilaterals to analyze them. When we analyze quadrilaterals, we can find the measure of an unknown angle or side. Remember, one of the most important things to know about quadrilaterals is that their angles always add up to \begin{align*}360^\circ\end{align*}. That means that if we know the measure of any three angles, we can set up an equation to solve for the measure of the fourth. Let’s see how this works.

Example

*Find the measure of the unknown angle in the quadrilateral below.*

**We know that the four angles must have a sum of \begin{align*}360^\circ\end{align*}, so we can add the four angles, using \begin{align*}m\end{align*} to represent the unknown angle.**

\begin{align*}55+90+105+m &= 360\\ 250+m &= 360\\ m &= 360-250\\ m &= 110^\circ\end{align*}

**By solving for \begin{align*}m\end{align*}, we have found that the fourth angle has a measure of \begin{align*}110^\circ\end{align*}.**

**We can check our work by adding the four angles to see if they total \begin{align*}360^\circ\end{align*}.**

\begin{align*}55^\circ + 90^\circ + 105^\circ + 110^\circ = 360^\circ\end{align*}

**Our calculation was correct. We can always use this method when given three out of the four angles in a quadrilateral.**

Example

*Find the measures of the unknown angles in the quadrilateral below.*

**This time we have only been given the measures of two angles and we need to solve for the other two.** First let’s determine what we know about the figure. What kind of quadrilateral is it? It has two pairs of parallel sides, so it must be a parallelogram. It doesn’t have \begin{align*}90^\circ\end{align*} angles, so it’s not a rectangle or square. Finally, the side lengths are not all congruent so it cannot be a rhombus. It is a regular parallelogram.

**Now, what do we know about the angles of parallelograms? Not only do they add up to \begin{align*}360^\circ\end{align*}, they fall into two congruent pairs.** The congruent angles are opposite each other. Take a look back at the figure.

**Angle \begin{align*}x\end{align*} is opposite the \begin{align*}56^\circ\end{align*} angle. Therefore it must also be \begin{align*}56^\circ\end{align*}. Angle \begin{align*}y\end{align*} is opposite the \begin{align*}124^\circ\end{align*} angle, so it must also be \begin{align*}124^\circ\end{align*}. This gives us two pairs of congruent angles.**

**Let’s check to make sure these are the correct measurements by adding them to see if they total \begin{align*}360^\circ\end{align*}.**

\begin{align*}124^\circ + 124^\circ + 56^\circ + 56^\circ = 360^\circ\end{align*}

**They do, so our answers are correct.**

Often we can use what we know about the properties of quadrilaterals to find unknown measures without having to set up an equation. We can simply use reasoning to put the pieces together.

IV. **Describe and Classify Quadrilaterals Found in Real – World Objects and Architecture**

We can use these methods to analyze and classify any quadrilateral, including quadrilaterals we see around us every day. Let’s take a look at a few examples.

Example

*Name the quadrilateral pictured below.*

**Looking at this pool, we can begin to think about the different characteristics of the pool. First, it has opposite sides that are congruent and parallel. It also has four right angles, this makes this figure a rectangle.**

Example

*Name the quadrilateral pictured below.*

**Now let’s examine this picture. We can look for the qualities that identify this quadrilateral. Notice that it has two parallel sides. The other two sides aren’t parallel or congruent. With one pair of parallel sides, this figure must be a trapezoid.**

**Now let’s go back to the problem from the introduction.**

## Real-Life Example Completed

*Figuring out a Yurt*

**Here is the original problem once again. Reread it and then identify the types of quadrilaterals used in the lattice design. Explain how you have identified each quadrilateral. There are two parts to your answer.**

“This is very cool!” Marcus exclaimed when looking through a book on different types of houses.

“What do you see?” Lynne asked leaning over her desk to look at the book that Marcus was holding.

Lynne and Marcus are both students in Mrs. Patterson’s World Cultures class. Like Jaime, they are also working on projects. Marcus has discovered a yurt. A yurt is a type of home common in Mongolia. There is a lattice structure that is build and then a canvas is used to cover the frame.

“That is cool, what is it?” Lynne asked.

“It’s called a yurt. I think that this is what I am going to do my project on,” Marcus said studying the picture.

Marcus took out a piece of paper and a pencil and he began to draw the lattice of the yurt.

*Remember there are two parts to your answer.*

*Solution to Real – Life Example*

**Now look at the picture of the yurt once again.**

**In looking at this diagram, it looks like there is a square being used as the design of the lattice. Examine this more closely and you will see that the sides of each figure created by the lattice are all equal. 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 that a rhombus has four sides of equal length, but it does not have to have right angles.**

**This is the answer.**

## Vocabulary

Here are the vocabulary words that are found in this lesson.

- Quadrilateral
- any four-sided figure.

- Trapezoid
- a quadrilateral with one pair of parallel sides.

- Parallelogram
- a quadrilateral with two pairs of opposite sides that are congruent and parallel.

- Rhombus
- a parallelogram with four congruent sides.

- Rectangle
- a parallelogram with opposites congruent and four right angles.

- Square
- a parallelogram with four congruent sides and four right angles.

- Congruent
- means exactly the same.

## Time to Practice

Directions: Identify each quadrilateral based on the description provided.

- A figure with four equal sides and four right angles.
- A figure with opposite sides congruent and parallel.
- A figure with opposite sides congruent and parallel and four right angles.
- A figure with four sides.
- A figure with four equal sides which may or may not have four right angles.

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

- A quadrilateral can be any four sided figure.
- A rectangle is also a parallelogram, but a parallelogram is not necessarily a rectangle.
- A square is never a parallelogram.
- A rhombus can be a square.
- A square is always a rhombus.
- A rhombus is a parallelogram.
- A quadrilateral is a type of parallelogram.
- A trapezoid has opposite sides parallel and congruent.
- What does the sum of the angles of a quadrilateral add up to be?
- What are all four angle measures of a rectangle?
- What are all four angle measures of a square?

Directions: Use what you have learned about quadrilaterals to figure out the missing angle measure of each quadrilateral based on three given angles.

- \begin{align*}120^\circ, 120^\circ, 60^\circ,?\end{align*}
- \begin{align*}50^\circ, 70^\circ, 130^\circ,?\end{align*}
- \begin{align*}52^\circ, 128^\circ, 52^\circ,?\end{align*}
- \begin{align*}47^\circ, 55^\circ, 120^\circ,?\end{align*}

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