- Classify angles.
- Apply the Protractor Postulate and the Angle Addition Postulate.
- Label the following geometric figure. What is it called?
- Find a,XY and YZ.
B is between A and C on AC¯¯¯¯¯¯¯¯. If AB=4 and BC=9, what is AC?
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
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.
Example 1: How many angles are in the picture below? Label each one.
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.
Just like the Ruler Postulate for line segments, there is a Protractor Postulate for angles.
Example 3: What is the measure of the angle shown below?
Example 4: What is the measure of the angle shown below?
Angles can be grouped into four different categories.
Perpendicular: When two lines intersect to form four right angles.
Line l is perpendicular to line m.
Line AC is perpendicular to line DE.
Example 6: Name the angle and determine what type of angle it is.
Because it opens wider than a right angle, and less than a straight angle it is obtuse.
Drawing an Angle
1. Start by drawing a horizontal line across the page, 2 in long.
2. Place an endpoint at the left side of your line.
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 1-3: Copying an Angle with a Compass and Ruler
2. With the point (non-pencil 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/construct-anglesame.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.
Example 9: Write all equal angle and segment statements.
Angle Addition Postulate
Like the Segment Addition Postulate, there is an Angle Addition Postulate.
- Questions 1-10 use the definitions, postulates and theorems from this section.
- Questions 11-16 are similar to Investigation 1-2 and Examples 7 and 8.
- Questions 17 and 18 are similar to Investigation 1-3.
- Questions 19-22 are similar to Examples 2-5.
- Question 23 is similar to Example 9.
- Questions 24-28 are similar to Examples 10 and 11.
- Questions 29 and 30 are similar to Example 12.
For questions 1-10, 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¯¯¯¯¯¯¯¯ and BC¯¯¯¯¯¯¯¯ are the sides.
- The m in front of m∠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 11-16, draw the angle with the given degree, using a protractor and a ruler. Also, state what type of angle it is.
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 19-22, 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.
- Make a sketch.
- Find m∠QOP
- Find m∠QOT
- Find m∠ROQ
- Find m∠SOP
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