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Angles

Classification of angles, such as right or vertical, and their relation to lines and rays.

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Angles and Lines

Angles are formed by intersecting lines or rays. If you take any two lines or rays, will you form at least one angle?

 

Angles and Lines

A line segment is a portion of a line with two endpoints. A ray is a portion of a line with one endpoint. Line segments are named by their endpoints and rays are named by their endpoint and another point. In each case, a segment or ray symbol is written above the points. Below, the line segment is \begin{align*}\overline{AB}\end{align*} and the ray is \begin{align*}\overrightarrow{CD}\end{align*}.

When two rays meet at their endpoints, they form an angle. Depending on the situation, an angle can be named with an angle symbol and by its vertex or by three letters. If three letters are used, the middle letter should be the vertex. The angle below could be called \begin{align*}\angle B\end{align*} or \begin{align*}\angle ABC\end{align*} or \begin{align*}\angle CBA\end{align*}. Use three letters to name an angle if using one letter would not make it clear what angle you are talking about.

Angles are measured in degrees. You can use a protractor or geometry software to measure angles. Remember that a full circle has \begin{align*}360^\circ\end{align*}.

 

An angle that is exactly \begin{align*}90^\circ\end{align*} (one quarter of a circle) is called a right angle. A right angle is noted with a little square at its vertex. An angle that is more than \begin{align*}90^\circ\end{align*} but less than \begin{align*}180^\circ\end{align*} is called an obtuse angle. An angle that is less than \begin{align*}90^\circ\end{align*}  is called an acute angle. An angle that is exactly \begin{align*}180^\circ\end{align*} (one half of a circle) is called a straight angle.

 

Two angles are complementary if the sum of their measures is \begin{align*}90^\circ\end{align*}. Two angles are supplementary if the sum of their measures is \begin{align*}180^\circ\end{align*}. Two angles that together form a straight angle will always be supplementary. When two lines intersect, many angles are formed, as shown below.

In the diagram above \begin{align*}\angle AEC\end{align*} and \begin{align*}\angle AED\end{align*} are adjacent angles because they are next to each other and share a ray. They are also supplementary because together they form a straight angle. \begin{align*}\angle AEC\end{align*} and \begin{align*}\angle DEB\end{align*} are called vertical angles. You can show that vertical angles will always have the same measure.

Solve the following problems

Explain why you must use three letters to identify any of the angles in the diagram below.

All angles in this diagram have a vertex of \begin{align*}E\end{align*}. Therefore, \begin{align*}\angle E\end{align*} is ambiguous because it could refer to many different angles. Use three letters with \begin{align*}E\end{align*} as the middle letter to be clear about which angle you are referring to.

\begin{align*}\angle ABC\end{align*} and \begin{align*}\angle DEF\end{align*} are complementary angles with \begin{align*}m\angle ABC =40^\circ\end{align*}. What is \begin{align*}m\angle DEF\end{align*}?

The “\begin{align*}m\end{align*}” in front of the angle symbol is read as “the measure of”. \begin{align*}m\angle ABC\end{align*} means “the measure of angle \begin{align*}ABC\end{align*}”. Because the two angles are complementary, their measures must add to \begin{align*}90^\circ\end{align*}. Therefore, \begin{align*}m\angle DEF =50^\circ\end{align*}.

Let \begin{align*}m\angle AEC=x^\circ\end{align*}. Show that \begin{align*}m\angle DEB\end{align*} must also equal \begin{align*}x^\circ\end{align*}.

If \begin{align*}m\angle AEC=x^\circ\end{align*}, then \begin{align*}m\angle AED=(180-x)^\circ\end{align*} because \begin{align*}\angle AEC\end{align*} and \begin{align*}\angle AED\end{align*} form a straight angle and are therefore supplementary. Similarly, \begin{align*}m\angle DEB=[180-(180-x) ]^\circ=(180-180+x)^\circ=x^\circ\end{align*}. This is how you can be confident that vertical angles will always have the same measure.

Examples 

Example 1

Earlier, you were asked will you form at least one angle if you take any two lines or rays. 

As long as the lines or rays intersect, at least one angle will be formed. If the lines (or rays) are parallel, and therefore don't intersect, then no angles will be formed.

 

Example 2

Name the angle below and classify it by its size.

 

\begin{align*}\angle F\end{align*} or \begin{align*}\angle DFE\end{align*} or \begin{align*}\angle EFD\end{align*}. It is an acute angle.

Example 3

Estimate the measure of the angle from #1. Use a protractor to confirm your answer.

Remember that exactly half of a right angle is \begin{align*}45^\circ\end{align*}. This angle looks to be more than half of a right angle. You might guess that it is approximately \begin{align*}55^\circ\end{align*}. Using a protractor, you can see that it is about \begin{align*}60^\circ\end{align*}.

 

Source: CK-12 Foundation. Author: Kaitlyn Spong and Laura Guerin
License: CC BY-NC 3.0

Example 4

What are two lines that form a right angle called?

Perpendicular lines.

Review

1. What's the difference between a line segment, a line, and a ray?

2. Draw an example of a right angle.

3. Draw an example of an obtuse angle.

4. Draw an example of an acute angle.

5. Why are two angles that make a straight angle always supplementary?

6. If \begin{align*}m\angle ABC=(2x+4)^\circ\end{align*}, \begin{align*}m\angle DEF=(3x-5)^\circ\end{align*}, and \begin{align*}\angle ABC\end{align*} and \begin{align*}\angle DEF\end{align*} are complementary, what are the measures of the angles?

7. If \begin{align*}m\angle ABC=(2x+4)^\circ\end{align*}, \begin{align*}m\angle DEF=(3x-5)^\circ\end{align*}, and \begin{align*}\angle ABC\end{align*} and \begin{align*}\angle DEF\end{align*} are supplementary, what are the measures of the angles?

Use the diagram below for #8-#12.

 

8. Give an example of vertical angles.

9. Give an example of a straight angle.

10. Give an example of supplementary angles.

11. If \begin{align*}m\angle ABC=70^\circ\end{align*}, find \begin{align*}m\angle ABF\end{align*}.

12. If \begin{align*}m\angle ABC=70^\circ\end{align*}, find \begin{align*}m\angle FBG\end{align*}.

13. What do you remember about perpendicular lines?

Use the angle below for #14-#15.

 

14. Name the angle and classify it based on its size.

15. Estimate the measure of the angle. Use a protractor to confirm your answer.

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 1.2. 

Vocabulary

Acute Angle

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

Adjacent Angles

Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'next-to'.

Angle

A geometric figure formed by two rays that connect at a single point or vertex.

Complementary angles

Complementary angles are a pair of angles with a sum of 90^{\circ}.

line segment

A line segment is a part of a line that has two endpoints.

Obtuse angle

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

Ray

A ray is a part of a line that has one endpoint and continues indefinitely in one direction.

Right Angle

A right angle is an angle equal to 90 degrees.

Supplementary angles

Supplementary angles are two angles whose sum is 180 degrees.

Vertical Angles

Vertical angles are a pair of opposite angles created by intersecting lines.

Image Attributions

  1. [1]^ Source: CK-12 Foundation. Author: Kaitlyn Spong and Laura Guerin; License: CC BY-NC 3.0

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