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1-6 Measuring Angles. Objectives: 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

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1 6 measuring angles
1-6 Measuring Angles

Objectives:

  • 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


Angle sides vertex
Angle, Sides, Vertex

QBT

1

B

TBQ

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

1

Names:

T


Naming angles
Naming angles

X

W

1

2

Y

Z

  • 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.


Interior and exterior
Interior and exterior

exterior

W

Y

Z

interior

A

B

An angle separates a plane into three parts:

interior

  • the ______, which is the set of points

  • between the sides of the angle

exterior

2) the ______, which is the set of points

outside the angle

angle itself

3) the _________

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.


Measuring angles
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


Drawing an angle with a protractor
Drawing an Angle with a Protractor

1) Draw AB

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

C

A

B

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

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


Classifying angles
Classifying Angles

A

A

obtuse angle 90 < m A < 180

acute angle 0 < m A < 90

A

right angle m A = 90

A

straight angle m A = 180


Congruent angles
Congruent Angles

1

2

  • 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)


Hands on measurement of angles
Hands-on Measurement of Angles

A

B

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

O

C

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

45°

75°

30°

  • For each angle, find

    • mAOC

    • mCOB

    • mAOB.


Angle addition postulate
Angle Addition Postulate

A

B

45°

75°

30°

O

C

  • For any angle AOC, if B is in the interior of AOC, thenmAOB + mBOC = mAOC.


P 38 txtbk ex 3
p. 38 TxtBk Ex. 3

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

T

W

125°

50°

R

S

  • m∠RST +m∠TSW = m∠RSW

  • 50+m∠TSW = 125

  • m∠TSW = 125 – 50 = 75


Identifying angle pairs adjacent angles
Identifying Angle Pairs – Adjacent Angles

J

2

common side

R

M

1

1 and 2 are adjacent

with the same vertex R and

N

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

A) share a common side

B) have the same vertex

C) have no interior points in common

D) are coplanar


Identifying angle pairs adjacent angles1
Identifying Angle Pairs: Adjacent Angles

B

2

1

1

2

G

N

L

1

The side of 1 is ____

J

2

The side of 2 is ____

Determine whether 1 and 2 are adjacent angles.

No. They have a common vertex B, but

_____________

no common side

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

No. They do not have a common vertex nor ____________

a common side


Identifying angle pairs vertical angles
Identifying Angle Pairs: Vertical Angles

  • Vertical Angles

Two angles are vertical if and only if they are two nonadjacent

angles formed by a pair of intersecting lines.

Vertical angles:

1 and 3

1

4

2

2 and 4

3

Vertical

Angles


Identifying angle pairs complementary angles
Identifying Angle Pairs: Complementary Angles

E

D

A

60°

30°

F

B

C

  • 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)


Complementary angles examples
Complementary Angles: Examples

I

75°

15°

H

P

Q

40°

50°

H

S

U

V

60°

T

30°

Z

W

Some examples of complementary angles are shown below.

mH + mI = 90

mPHQ + mQHS = 90

Remember:

Complementary angles can form a

Corner (which measures 90°).

mTZU + mVZW = 90


Identifying angle pairs supplementary angles
Identifying Angle Pairs: Supplementary Angles

D

C

130°

50°

E

B

F

A

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

mB+ mE = 50 + 130 = 180


Supplementary angles examples
Supplementary Angles: Examples

I

75°

105°

H

Q

130°

50°

H

S

P

U

V

60°

120°

60°

Z

W

T

Some examples of supplementary angles are shown below.

mH + mI = 180

mPHQ + mQHS = 180

Remember:

Supplementary angles can form a linear pair or

Straight line (which measures 180°)

mTZU + mUZV = 180

and

mTZU + mVZW = 180


Linear pair
Linear Pair

Q

130°

50°

H

S

P

  • 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


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