In Chapter 1, you studied many common geometric shapes and learned ways to describe a shape using its attributes. In this chapter, you will further investigate how to describe a complex shape by developing ways to accurately determine its angles, area, and perimeter. You will also use transformations from Chapter 1 to uncover special relationships between angles within a shape.

Throughout this chapter you will be asked to solve problems, such as those involving area or angles, in more than one way. This will require you to "see" shapes in multiple ways and to gain a broader understanding of problem solving.

In this chapter, you will learn:

• the relationships between pairs of angles formed by transversals and the angles in a triangle

• the sum of the interior angles of a polygon

• how the measure of the interior and exterior angles of a regular polygon are related to the number of sides of the polygon

2.1

What Is an Angle?Pg. 3

Angle Notation

Today you are going to discover how the rotation of a ray creates a new object. Then you are going to learn the proper way of naming an angle and how it is measured. You will then copy and bisect an angle with a compass.

2.1 – BUILDING AN ANGLEExamine the ray below. Copy it with tracing paper.

a. Rotate the angle 90 counter-clockwise. What shape is created? How is it measured?

A right angle degrees

b. What markings do we add to this shape to show its measure? Add this to the picture above.

d. Rotate the following ray 90 counter-clockwise twice. What shape does it appear to make? What is its measure?

Straight line 180

e. Rotate the following ray 90 counter-clockwise four times. What shape does it appear to make? What is its measure?

Full circle 360

2.2 – USING A PROTRACTORAngles can be measured in degrees using a protractor. Find the center of the protractor and place it at the starting point of the ray. Line up one ray with 0 and determine how many degrees the other ray is. Angles that are less than 90 are called acute. An obtuse angle is when a ray is rotated more than 90, but less than 180. Right angles are exactly 90 and straight angles are exactly 180. Circular angles are 360. Find the measure and name the type of angle for each one below.

45 120 20

acute obtuse acute

Find the measure and name the type of angle for each one below.

2.3 – NAMING ANGLESIn order to clearly communicate relationships between angles, you will need a convenient way to refer to and name them. Examine the diagram of angle at right.

m A m B m C

There are 6!!!

mC’AC

mCAC’

c. Maria asked Audrey to be more specific. She explained, "One of my angles is At the same time, she marked her two angles with the same marking at right to indicate that they have the same measure. Name her other angle. Be sure to use three letters so there is no confusion about which angle you mean.

'."BAB

d. Does the order in which you label the three letters matter? For example, is BAB' the same as B'AB? What about and BAB' and ABB'

Order matters

The letter in the middle must be angle you are talking about

e. When should you use three letters to name an angle? When is it ok to only use one?

When more than one angle is touching

When the angle is alone

2.4– ANGLE MEASURESUse the following picture to find the angle measures.

a. MLN = 60

b. NLP =70

c. NLQ = 120

d. MLQ = 180

2.5 – COPYING AN ANGLEYou have learned how to use a compass and a straightedge to copy a line segment. But how can you use these tools to copy an angle? Find X below. With your team, discuss how you can construct a new angle (Y) that is congruent to X.

Y

a. Start by drawing a segment with endpoint Y.

b. With your compass point at X, draw an arc that intersects both sides of X.

c. Now draw an arc with the same radius and with center Y.

d. How can you use your compass to measure the "width" of X? Line up your compass with both points of intersection from the beginning arc on part (b). Measure that "width". Then copy that length over to the arc on part (c).

e. Draw in a line from point Y through the intersection you created on part (d). Y

2.6 – COPYING AN ANGLECreate a copy of Z.

Z

2.6 – COPYING AN ANGLECreate a copy of Z.

Z

2.7 – ADDING ANGLESCreate an angle that is the measure of A + B using a compass.

A

B

2.7 – ADDING ANGLESCreate an angle that is the measure of A + B using a compass.

A

B

2.8 – ANGLE BISECTORSIn the construction of a segment midpoint you were able to cut the segment into two equal pieces. In this construction you are going to cut the angle down the center creating two equal angles.

a. With your compass point at X, draw an arc that intersects both sides of X.

b. Label the two points of intersection Y and Z.

Z

Y

c. Using the same measure on the compass, put the point of the compass on Y and leave a mark in the middle of the angle. Then do the same for point Z. These two marks should intersect.

d. Draw in a ray that connects point X with the point of intersection from part (c).

Z

Y

2.9 – BISECT AN ANGLEBisect Z.

Z

A

B

2.9 – BISECT AN ANGLEBisect Z.

Z

A

B