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1-6 Measuring AnglesObjectives:• Define and name angles, sides, and rays• Use the Protractor Postulate for measuring

angles• Classify angles as acute, right, obtuse, or

straight• Use the Angle Addition Postulate• Define vertical angles, adjacent angles,

complementary angles, and supplementary angles

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Angle, Sides, Vertex

An angle is a figure formed by two rays that have a common endpoint.

The rays are the sides of the angle. (rays BT and BQ)

The common endpoint is called the vertex of the angle (point B). When naming an angle with 3 letters, the vertex must be the middle letter.

B Q

T

1

Names: QBT

1

B

TBQ

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Naming angles

• What are two other names for ∠ 1?

• ∠ XWY, ∠ YWX• Is ∠ W a good name for ∠

1?• No, it would not be clear

which angle ∠ W would be referring to.

W

Y

X

Z

1

2

4

An angle separates a plane into three parts:

1) the ______, which is the set of pointsbetween the sides of the angle2) the ______, which is the set of pointsoutside the angle3) the _________

interior

exterior

angle itself

exterior

interior

W

Y

Z

A

B

In the figure shown, point B and all other points in the blue region are in the interiorof the angle.

Point A and all other points in the greenregion are in the exterior of the angle.

Points Y, W, and Z are on the angle.

Interior and exterior

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Measuring Angles

• We measure an angle using a protractor.– Determine the amount of rotation between the two sides of an angle.

• For every angle, there is a unique positive number between 0 and 180 called the degree measure of the angle.

• Special angles:– 0°, 90°, 180°, 360°

• Simulation or hands-on for measuring angles:http://www.mathcasts.org/gg/student/angles/angles/angle_meas3.html

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Use a protractor to draw an angle having a measure of 135.

1) Draw AB

2) Place the center point of the protractor on A. Align the mark labeled 0 with the ray.

3) Locate and draw point C at the mark labeled 135. Draw AC.

C

A B

Drawing an Angle with a Protractor

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Classifying Angles

A

right angle m A = 90

acute angle 0 < m A < 90

A

obtuse angle 90 < m A < 180

A

A

straight angle m A = 180

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Congruent Angles

• Angles with the same measure

m 1 = m 2 (the measure of angle 1 equals the measure of angle 2)

1 ≅ 2 (Angle 1 is congruent to angle 2)(May also be indicated by arc on both angles)

1 2

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1) Draw an acute, an obtuse, or a right angle. Label the angle AOC.

A

CO

2) Draw and label a point B in the interior of the angle. Then draw OB.

B

3) For each angle, find • mAOC• mCOB• mAOB.

30°

45°

75°

Hands-on Measurement of Angles

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Angle Addition Postulate• For any angle AOC, if B is in the

interior of AOC, thenmAOB + mBOC = mAOC.

A

CO

B

30°

45°

75°

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p. 38 TxtBk Ex. 3

• What is m∠TSW if m∠RST = 50 and m∠RSW = 125 ?

R

W

T

S

• m∠RST + m∠TSW = m∠RSW• 50 + m∠TSW = 125• m∠TSW = 125 – 50 = 75

125°

50°

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Identifying Angle Pairs – Adjacent Angles

Adjacent (next to, joining) angles are angles that:

M

J

N

R1

2

1 and 2 are adjacent

with the same vertex R and

common side RM

A) share a common side

B) have the same vertex

C) have no interior points in common

D) are coplanar

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Determine whether 1 and 2 are adjacent angles.

No. They have a common vertex B, but _____________no common side

1 2

B

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G

Yes. They have the same vertex G and a common side with no interior points in common.

N

1

2J

L

No. They do not have a common vertex nor ____________a common side

The side of 1 is ____LN

JNThe side of 2 is ____

Identifying Angle Pairs: Adjacent Angles

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Identifying Angle Pairs: Vertical Angles

• Vertical AnglesTwo angles are vertical if and only if they are two nonadjacent angles formed by a pair of intersecting lines.

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34

Vertical angles:

1 and 3

2 and 4

Angles

Vertical

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Identifying Angle Pairs: Complementary Angles

• Two angles are complementary if and only if the sum of their degree measures is 90.

• Each angle is a complement of the other. (Angle B is the complement of angle E)

30°

A

BC

60°D

E

F

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15°H

75° I

Some examples of complementary angles are shown below.

mH + mI = 90

mPHQ + mQHS = 90

Remember:

Complementary angles can form a

Corner (which measures 90°).

50°H

40°Q

P

S

30°60°T

UV

WZ

mTZU + mVZW = 90

Complementary Angles: Examples

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Identifying Angle Pairs: Supplementary Angles

Two angles are supplementary if and only if the sum of their degree measures is 180.

50°

AB

C

130°

D

E F

mB+ mE = 50 + 130 = 180

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105°H

75° I

Some examples of supplementary angles are shown below.

mH + mI = 180

mPHQ + mQHS = 180

Remember:

Supplementary angles can form a linear pair or

Straight line (which measures 180°)

50°

H

130°

Q

P S

mTZU + mUZV = 180

60°120°

T

UV

W

Z

60° and

mTZU + mVZW = 180

Supplementary Angles: Examples

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Linear Pair

• A pair of adjacent angles whose noncommon sides that form opposite rays

• Hands-On:– On your paper, draw a linear pair– Measure each of the two angles and add the

measures

• Simulation:• http://www.geogebra.org/en/upload/files/english/Barbara_Perez/Linear_Angles.html

50°

H

130°

Q

P S