1.3: Angles and Measurement
Learning Objectives
 Classify angles.
 Apply the Protractor Postulate and the Angle Addition Postulate.
Review Queue
 Label the following geometric figure. What is it called?
 Find
a,XY andYZ . 
B is betweenA andC onAC¯¯¯¯¯¯¯¯ . IfAB=4 andBC=9 , what isAC ?
Know What? Back to the building blocks. Every block has its own dimensions, angles and measurements. Using a protractor, find the measure of the three outlined angles in the “castle” to the right.
Two Rays = One Angle
In #1 above, the figure was a ray. It is labeled
Angle: When two rays have the same endpoint.
Vertex: The common endpoint of the two rays that form an angle.
Sides: The two rays that form an angle.
Label It  Say It 


Angle 

Angle 
The vertex is
Example 1: How many angles are in the picture below? Label each one.
Solution: There are three angles with vertex
So, the three angles can be labeled,
Protractor Postulate
We measure a line segment’s length with a ruler. Angles are measured with something called a protractor. A protractor is a measuring device that measures how “open” an angle is. Angles are measured in degrees, and labeled with a
There are two sets of measurements, one starting on the left and the other on the right side of the protractor. Both go around from
Example 2: Measure the three angles from Example 1, using a protractor.
Solution: Just like in Example 1, it might be easier to measure these three angles if we separate them.
With measurement, we put an
Just like the Ruler Postulate for line segments, there is a Protractor Postulate for angles.
Protractor Postulate: For every angle there is a number between
In other words, you do not have to start measuring an angle at
Example 3: What is the measure of the angle shown below?
Solution: This angle is lined up with
Example 4: What is the measure of the angle shown below?
Solution: This angle is not lined up with
Inner scale:
Outer scale:
Example 5: Use a protractor to measure
Solution: Lining up one side with
Classifying Angles
Angles can be grouped into four different categories.
Straight Angle: An angle that measures exactly
Right Angle: An angle that measures exactly
This halfsquare marks right, or
Acute Angles: Angles that measure between
Obtuse Angles: Angles that measure between
Perpendicular: When two lines intersect to form four right angles.
Even though all four angles are
The symbol for perpendicular is
Label It  Say It 


Line 

Line 
Example 6: Name the angle and determine what type of angle it is.
Solution: The vertex is
Because it opens wider than a right angle, and less than a straight angle it is obtuse.
Example 7: What type of angle is
Solution:
Drawing an Angle
Investigation 12: Drawing a
1. Start by drawing a horizontal line across the page, 2 in long.
2. Place an endpoint at the left side of your line.
3. Place the protractor on this point, such that the bottom line of the protractor is on the line and the endpoint is at the center. Mark
4. Remove the protractor and connect the vertex and the
This process can be used to draw any angle between
Example 8: Draw a
Solution: Following the steps from above, your angle should look like this:
Now that we know how to draw an angle, we can also copy that angle with a compass and a ruler. Anytime we use a compass and ruler to draw geometric figures, it is called a construction.
Compass: A tool used to draw circles and arcs.
Investigation 13: Copying an Angle with a Compass and Ruler
1. We are going to copy the
2. With the point (nonpencil side) of the compass on the vertex, draw an arc that passes through both sides of the angle. Repeat this arc with the line we drew in #1.
3. Move the point of the compass to the horizontal side of the angle we are copying. Place the point where the arc intersects this side. Open (or close) the “mouth” of the compass so that you can draw an arc that intersects the other side and the arc drawn in #2. Repeat this on the line we drew in #1.
4. Draw a line from the new vertex to the arc intersections.
To watch an animation of this construction, see http://www.mathsisfun.com/geometry/constructanglesame.html
Marking Angles and Segments in a Diagram
With all these segments and angles, we need to have different ways to label equal angles and segments.
Angle Markings
Segment Markings
Example 9: Write all equal angle and segment statements.
Solution:
Angle Addition Postulate
Like the Segment Addition Postulate, there is an Angle Addition Postulate.
Angle Addition Postulate: If
Example 10: What is
Solution: Using the Angle Addition Postulate,
Example 11: What is
Solution:
Example 12: Algebra Connection If
Solution:
Know What? Revisited Using a protractor, the measurement marked in the red triangle is
Review Questions
 Questions 110 use the definitions, postulates and theorems from this section.
 Questions 1116 are similar to Investigation 12 and Examples 7 and 8.
 Questions 17 and 18 are similar to Investigation 13.
 Questions 1922 are similar to Examples 25.
 Question 23 is similar to Example 9.
 Questions 2428 are similar to Examples 10 and 11.
 Questions 29 and 30 are similar to Example 12.
For questions 110, determine if the statement is true or false.
 Two angles always add up to be greater than
90∘ . 
180∘ is an obtuse angle. 
180∘ is a straight angle.  Two perpendicular lines intersect to form four right angles.
 A construction uses a protractor and a ruler.
 For an angle
∠ABC,C is the vertex.  For an angle
∠ABC,AB¯¯¯¯¯¯¯¯ andBC¯¯¯¯¯¯¯¯ are the sides.  The
m in front ofm∠ABC means measure.  Angles are always measured in degrees.
 The Angle Addition Postulate says that an angle is equal to the sum of the smaller angles around it.
For 1116, draw the angle with the given degree, using a protractor and a ruler. Also, state what type of angle it is.

55∘ 
92∘ 
178∘ 
5∘ 
120∘ 
73∘  Construction Copy the angle you made from #12, using a compass and a ruler.
 Construction Copy the angle you made from #16, using a compass and a ruler.
For 1922, use a protractor to determine the measure of each angle.
 Interpret the picture to the right. Write down all equal angles, segments and if any lines are perpendicular.
In Exercises 2429, use the following information:
 Make a sketch.
 Find
m∠QOP  Find
m∠QOT  Find
m∠ROQ  Find
m∠SOP
Algebra Connection Solve for

m∠ADC=56∘

m∠ADC=130∘
Review Queue Answers
1.
2.
3. Use the Segment Addition Postulate,
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