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. 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 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.

2.1

What Is an Angle?Pg. 4

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 – What Is An Angle?_ ____Angle Notation

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.

box

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°

Acute

Right

Obtuse

Straight

Angle more than 0°, but less than 90°

Angle more than 90°, but less than 180°

Angle that measures 90°

Angle that measures 180°

A

R

O

S

m∠A =50°

m∠R =90°

m∠O =130°

m∠S =180°

CircularAngle that measures 360°

C

e C

2.2 – NAMING ANGLESFind the angle measure. Then determine if they are acute, right, obtuse, or straight.

.  ________

 

.  ________

 

.  ________

a m AOC

b m DOB

c m BOE

∠

∠

∠

80 acute

65 acute

90 right

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

60°a.∠MLN

70°b. ∠NLP =

120°c. ∠NLQ

180°d . ∠MLQ

2.4 – 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. 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.

2.5 –ANGLE MEASURESFind the indicated measure.

= 81 + 42 = 123°

= 90 – 26 = 64°

= 180 – 70 = 110°

2.6 –ANGLE ADDITIONUse the angle relationship to solve for x. Then find the indicated angle measure.

8x+7 + 4x-1 = 78

12x+6 = 78-6 -6

12x = 72x = 6

6

4(6)-1

23°

11x-7 + 5x-3 = 118

16x-10 = 118 +10 +10

16x = 128x = 8

8

11(8)-7

81°

2.7 – 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 below. With your team, discuss how you can construct a new angle ( ) that is congruent to .

∠X

∠X∠Y

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 .

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

∠X

d. How can you use your compass to measure the "width" of ? 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

∠X

2.8 – COPYING AN ANGLECreate a copy of

Z

∠Z.

2.8 – COPYING AN ANGLECreate a copy of

Z

∠Z.

2.9 – ADDING ANGLESCreate an angle that is the measure of using a compass.

A

B

∠A ∠B

2.9 – ADDING ANGLESCreate an angle that is the measure of using a compass.

A

B

∠A ∠B

2.10 – 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 .

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

Z

Y

∠X

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

2.11 – BISECT AN ANGLEBisect

Z

∠Z.

2.11 – BISECT AN ANGLEBisect

Z

A

B

∠Z.