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

#### Watch This

http://www.youtube.com/watch?v=7iBc5bJdanI James Sousa: Angle Basics

#### Guidance

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
and the ray is
.

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
or
or
. 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 .

An angle that is exactly
(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
but less than
is called an
**
obtuse angle
**
. An angle that is less than
is called an
**
acute angle
**
. An angle that is exactly
(one half of a circle) is called a
**
straight angle
**
.

Two angles are
**
complementary
**
if the sum of their measures is
. Two angles are
**
supplementary
**
if the sum of their measures is
. 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
and
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.
and
are called
**
vertical
**
**
angles
**
. You can show that vertical angles will always have the same measure.

**
Example A
**

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

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

**
Example B
**

and are complementary angles with . What is ?

**
Solution:
**
The “
” in front of the angle symbol is read as “the measure of”.
means “the measure of angle
”. Because the two angles are complementary, their measures must add to
. Therefore,
.

**
Example C
**

Let . Show that must also equal .

**
Solution
**
: If
, then
because
and
form a straight angle and are therefore supplementary. Similarly,
. This is how you can be confident that vertical angles will always have the same measure.

**
Concept Problem Revisited
**

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.

#### Vocabulary

A
**
line segment
**
is a portion of a line with two endpoints.

A
**
ray
**
is a portion of a line with one endpoint.

When two rays meet at their endpoints, they form an
**
angle
**
.

An angle that is exactly
(one quarter of a circle) is called a
**
right angle
**
.

An angle that is more than
but less than
is called an
**
obtuse angle
**
.

An angle that is less than
is called an
**
acute angle
**
.

An angle that is exactly
(one half of a circle) is called a
**
straight angle
**
.

Two angles are
**
complementary
**
if the sum of their measures is
.

Two angles are
**
supplementary
**
if the sum of their measures is
.

When two lines intersect,
**
adjacent angles
**
are next to each other and share a ray.

**are across from one another and only share a vertex.**

*Vertical angles*#### Guided Practice

1. Name the angle below and classify it by its size.

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

3. What are two lines that form at a right angle called?

**
Answers:
**

1. or or . It is an acute angle.

2. Remember that exactly half of a right angle is . This angle looks to be more than half of a right angle. You might guess that it is approximately . Using a protractor, you can see that it is about .

#### Practice

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 , , and and are complementary, what are the measures of the angles?

7. If , , and and 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 , find .

12. If , find .

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.