1
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
2
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
3
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
5
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
6
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
7
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
8
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
9
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
10
Angle Addition Postulate• For any angle AOC, if B is in the
interior of AOC, thenmAOB + mBOC = mAOC.
A
CO
B
30°
45°
75°
11
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°
12
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
13
Determine whether 1 and 2 are adjacent angles.
No. They have a common vertex B, but _____________no common side
1 2
B
12
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
14
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.
12
34
Vertical angles:
1 and 3
2 and 4
Angles
Vertical
15
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
16
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
17
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
18
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
19
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
Top Related