# 8.1: Angle Classification

**Basic**Created by: CK-12

**Practice**Angle Classification

Have you ever been on a field trip to an art museum?

Mrs. Gilson is taking her math class to the art museum on a field trip. Before leaving for the museum, Mrs. Gilson posed some questions to her class.

“Does anyone know why we are going to an art museum for math class?” She asked.

Matt, who tended to like to make jokes, was the first one to speak up.

“So we can count the paintings?” he joked.

Mrs. Gilson smiled as if she was expecting just such an answer. She looked around and waited for any other responses. Kyle was the next one to speak.

“I think it has to do with geometry. Isn’t most art based on some kind of geometry?” he asked.

“Very nice, and yes you are correct. We can find geometry in many different paintings forms and figures. Some of the first painters, sculptures, actually all kinds of artists used geometry to design their work. Here is a slide that we can look at together,” Mrs. Gilson said putting an image up on the board from the computer.

“This is to help you practice before we go,” Mrs. Gilson explained. “This is a piece of a stained glass window. You can see the flowers in the painting, but can you see the geometry? Take out your notebooks and make a note of any place that you see angles. Let’s start with angles after all it is a building block of geometry.”

The students all took out their notebooks and began to work.

**While the students make their notes, you make some notes too. What do you already know about angles?**

**This Concept is all about identifying angles as right, straight, acute and obtuse. After you learn about these classifications, you will be able to identify each type of angle in the stained glass. Pay attention and at the end of the Concept you will be able to find the geometry in the stained glass.**

### Guidance

The word “** angle**” is one of those words that we hear all the time. You might hear someone say “What is the angle of that corner?” or a photographer could use the term “wide angle lens.” Have you ever tried to angle a sofa through a doorway? The way that you turn the sofa makes a big difference in whether the sofa fits through the doorway or not. Notice that the same word “angle” is being used in each of these examples, but each example uses it differently.

**In geometry, we use the word angle too. Understanding angles in geometry can help you when you use the angles in real life.**

**What is an angle?**

**An** *angle***is a when two lines, line segments or rays connect at a common point.** The angle is created by the space between the two lines. We can say that this space “forms the angle.”

We measure an angle in ** degrees.** What we are actually measuring is the distance between the two lines. The space between them near the point where they connect forms the angle. This may sound confusing, but it will make more sense as we continue.

**Why are degrees important?**

Degrees are important because we classify angles by their size. Knowing the degrees of an angle can help us to be sure that we are using the correct words to classify or identify them. **The** \begin{align*}^\circ\end{align*} **symbol means “degrees.” The number of degrees tells how open or closed the angle is.** The smaller the number of degrees, the smaller or more closed the angle is. Angle sizes can range from 0 to \begin{align*}360^\circ\end{align*}, a complete circle.

Here is a diagram that shows some angle measurements.

As you can see, an angle of \begin{align*}360^\circ\end{align*} makes a complete circle. An angle of \begin{align*}270^\circ\end{align*} is three-quarters of a circle, and an angle of \begin{align*}180^\circ\end{align*} is half a circle. A \begin{align*}180^\circ\end{align*} angle is a straight line. Most angles that we deal with are between 0 and \begin{align*}180^\circ\end{align*}.

**We classify angles by their size, or number of degrees.**

**We classify angles as** *acute, right, obtuse, or straight.*

Let’s find out what these names mean.

**An** *acute angle***measures less than** \begin{align*}90^\circ\end{align*}. **If its measure is** \begin{align*}1^\circ\end{align*} **or** \begin{align*}89^\circ\end{align*} **or anywhere in between, we call it an acute angle.**

** Obtuse angles** measure more than \begin{align*}90^\circ\end{align*}. Angles greater than \begin{align*}90^\circ\end{align*} and less than \begin{align*}180^\circ\end{align*} are obtuse angles.

Most angles are either acute or obtuse. However, there are two special angles with exact measurements. **A** *right angle***measures exactly \begin{align*}90^\circ\end{align*}.** Right angles are one of the most important concepts you need to know about geometry. We find them in squares, rectangles, and triangles. They are everywhere in the real world too.

**There are many places in the real world where you can see acute, obtuse and right angles. Here are a few examples. Can you identify the angles?**

Notice that we use a small box to show when an angle is a right angle.

http://www.flickr.com/photos/ghost_of_kuji/394579484/ (attribution)

http://www.flickr.com/photos/basykes/6558444/ (attribution)

If you look at each of these pictures, you can see the right angles clearly. Also notice that the wires of the bridge stretch to create acute angles on each side of the center beam.

The other special angle is called a *straight angle.***A** *straight angle***measures exactly** \begin{align*}180^\circ\end{align*}. We have already seen that a straight angle forms a line.

Now that we know each kind of angle, let’s try classifying some.

Classify each angle below.

**For each angle, it may help to ask yourself: “Is it bigger or smaller than a right angle?”**

Remember, right angles always measure \begin{align*}90^\circ\end{align*}, and we use \begin{align*}90^\circ\end{align*} to tell whether an angle is acute or obtuse.

**Is Figure 1 larger or smaller than a right angle? A right angle looks like a perfect corner, often with one arm pointing straight up. This angle is wider than that, so it is an obtuse angle.**

**The angle in Figure 2 looks like a straight line... you know what that means! It must be a straight angle.**

**Is Figure 3 larger or smaller than a right angle? It is smaller than** \begin{align*}90^\circ\end{align*}, **so it is an acute angle.**

**The angle in Figure 4 does resemble a perfect corner, so it could be a right angle. Now take a closer look. The small box tells you that it definitely is a right angle.**

**We can also identify an angle by using a symbol. Here is the symbol for angle.**

\begin{align*}\angle A\end{align*}

**This means “Angle** \begin{align*}A\end{align*}”. **You will see this symbol used when we work with angles.**

Identify each type of angle described.

#### Example A

An angle greater than \begin{align*}90^\circ\end{align*}

**Solution: Obtuse**

#### Example B

An angle that measures \begin{align*}15^\circ\end{align*}

**Solution: Acute**

#### Example C

An angle that measures exactly \begin{align*}90^\circ\end{align*}

**Solution: Right**

Now you have learned all about angles. Here is the original problem once again. Reread it and then pay attention to Mrs. Gilson’s instructions.

Mrs. Gilson is taking her math class to the art museum on a field trip. Before leaving for the museum, Mrs. Gilson posed some questions to her class.

“Does anyone know why we are going to an art museum for math class?” She asked.

Matt, who tended to like to make jokes, was the first one to speak up.

“So we can count the paintings?” he joked.

Mrs. Gilson smiled as if she was expecting just such an answer. She looked around and waited for any other responses. Kyle was the next one to speak.

“I think it has to do with geometry. Isn’t most art based on some kind of geometry?” he asked.

“Very nice, and yes you are correct. We can find geometry in many different paintings forms and figures. Some of the first painters, sculptures, actually all kinds of artists used geometry to design their work. Here is a slide that we can look at together,” Mrs. Gilson said putting an image up on the board from the computer.

“This is to help you practice before we go,” Mrs. Gilson explained. “This is a piece of a stained glass window. You can see the flowers in the painting, but can you see the geometry? Take out your notebooks and make a note of any place that you see angles. Let’s start with angles after all it is a building block of geometry.”

The students all took out their notebooks and began to work.

**Can you find an example of each of the different types of angles in this stained glass? Make a few notes in your notebook. It may be helpful to draw them too.**

While the students worked, Mrs. Gilson walked around the room. When most seemed finished, Mrs. Gilson gave the students this instruction.

“Now find a partner and share the angles that you found in the painting.”

**You do this too. Find a partner and share the angles that you found. This is the best way to check your work for accuracy. If you and your partner both selected the same angle, then choose a new one together.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Acute Angle
- an angle whose measure is less than \begin{align*}90^\circ\end{align*}

- Obtuse Angle
- an angle whose measure is greater than \begin{align*}90^\circ\end{align*}

- Right Angle
- an angle whose measure is equal to \begin{align*}90^\circ\end{align*}

- Straight Angle
- an angle whose measure is equal to \begin{align*}180^\circ\end{align*}

- Degrees
- how an angle is measured

### Guided Practice

Here is one for you to try on your own.

True or false. An acute angle can also be a right angle.

**Answer**

False. An acute angle measures 90 degrees while a right angle measures exactly 90 degrees.

### Video Review

Here is a video for review.

- This is a James Sousa video on the types of angles.

### Practice

Directions: Label each angle as acute, obtuse, right, or straight.

9. \begin{align*}55^\circ\end{align*}

10. \begin{align*}102^\circ\end{align*}

11. \begin{align*}90^\circ\end{align*}

12. \begin{align*}180^\circ\end{align*}

13. \begin{align*}10^\circ\end{align*}

14. \begin{align*}87^\circ\end{align*}

15. \begin{align*}134^\circ\end{align*}

Acute Angle

An acute angle is an angle with a measure of less than 90 degrees.Obtuse angle

An obtuse angle is an angle greater than 90 degrees but less than 180 degrees.Perpendicular

Perpendicular lines are lines that intersect at a angle. The product of the slopes of two perpendicular lines is -1.Right Angle

A right angle is an angle equal to 90 degrees.Straight angle

A straight angle is a straight line equal to .vertical angles

Two non-adjacent angles formed by intersecting lines.Riemann sum

A Riemann sum is an approximation of the area under a curve, calculated by dividing the region up into shapes that approximate the space.### Image Attributions

Here you'll learn to classify angles as acute, obtuse, right or straight.