math 7 geometry 03 angles and angle measurements
TRANSCRIPT
The session shall begin shortly…
TXT, pp. 395-400, 406-416, 421-428, 432-435
An angle the union of two different rays that
have the same initial point (or common
endpoint). The rays are the sides of the angle.
The initial point is called the vertex.
Angles
vertex
sidesA
B
C
Angles & Angle Measurements
Naming Angles
A
B
C
Notation: We denote an angle with
three points and “” symbol.
Naming:
Use the three points, with the vertex
always in the middle. The angle can
also be named with just the vertex point
as long as it is a “stand alone angle”.
The angle shown can be named as:
BAC, CAB, A
Angles & Angle Measurements
Use the diagram below to answer the following questions.
a. Name the vertex.
R
b. Name the sides of the angle.
c. Name the angle in three different ways.
SRT, TRS, R
R
S
T
RS and RT
Check your understanding
Measure of an Angle
A
B
C
The MEASURE of an angle is the amount
of rotation a side makes with respect to
the other side. The unit is DEGREES ()
Notation: We put the letter “m” in
front of the angle name to denote its
measure.
The measure of the angle shown can be
written as:
mBAC, mCAB, mA
Angles & Angle Measurements
Example: Here are some examples of angles
and their degree measurements.
Measure of an Angle
Angles & Angle Measurements
ProtractorA device used to measure angles.
Measure of an Angle
Angles & Angle Measurements
Measure of an Angle
Angles & Angle Measurements
A
B
C
What is the
measure of this
angle?
ANSWER: 50
Measure of an Angle
Angles & Angle Measurements
A
B
C
What is the
measure of this
angle?
ANSWER: 110
Measure of an Angle
Angles & Angle Measurements
A
B
C
What is the
measure of this
angle?
ANSWER: 90
Measure of an Angle
Angles & Angle Measurements
A
B
C
What is the
measure of this
angle?
ANSWER: 70
Reading a Protractor
Date: 01/13/2014 at 15:51:52
From: Donya Ina
Subject: Reading a Protractor
Anak, alam kong may protractor ka at marunong kang mag-measure ng angle. At least may pakinabang ‘yan kesa kung anu-ano ang ipino-post mo sa FB. Pero tanong ko lang: which side of a protractor scale do you use?
Paki-explain. Lab U!
Angles & Angle Measurements
Measure of an Angle
Date: 01/13/2014 at 15:52:53
From: Anak
Subject: Re: Reading a Protractor
Donya Ina, I think that the easiest way to decide which scale to use is to look at the angle you're measuring. If it is greater than a right angle (more than ninety degrees) then use the scale that has numbers greater than 90. If the angle is obviously acute (measures less than 90 degrees), use the scale which has numbers less than 90. Of course you have to line up the protractor correctly with the center point on the vertex of the angle and one of the rays of the angle coincide with the line on the protractor.
I-push mo lang ‘yan!
Angles & Angle Measurements
Measure of an Angle
Wala Lang…
The Angle
Measurement
postulate
To every angle, there corresponds a unique
measure r in degrees such that r is between
0 and 180.
Measure of an Angle
Angles & Angle Measurements
Measure of an Angle
B
A
C
In half-plane H1 through the endpoint of
ray BC lying in the edge of the half-
plane there is exactly one ray BA, with A
in H1, such that angle ABC formed by the
two rays has a given measure between 0
and 180 degrees.
Angles & Angle Measurements
The Angle construction Postulate (or Protractor
Postulate)
WHAT?!
What do the two postulates mean?
1] All angles are measured between 00 and 1800.
2] They can be measured with a protractor.
3] The measurement is the absolute values of the
numbers read on the protractor.
4] The values of 0 and 180 on the protractor
were arbitrarily selected.
Measure of an Angle
Angles & Angle Measurements
Two angles are congruent if they have the same
measure.
Symbol: “” (is congruent to)
Congruent Angles
B
A
C
Angles & Angle Measurements
Y
X
Z
ABC and XYZ are congruent, so we write
ABC XYZ which means mABC = mXYZ
Straight angle
m A= 180°
Obtuse angle
90 < m A < 180°
Acute angle
0< m A < 90°
Classifying Angles
Angles are classified as acute, right, obtuse, and straight, according to their measures. Angles have measures greater than 0°and less or equal to 180°.
A A
Right angle
m A = 90°
A A
Angles & Angle Measurements
An acute angle is an angle measuring between 0
and 90 degrees.
Classifying Angles
Angles & Angle Measurements
An obtuse angle is an angle measuring between
90 and 180 degrees.
Classifying Angles
Angles & Angle Measurements
Classifying Angles
Angles & Angle Measurements
A right angle is an angle measuring exactly 90
degrees. It is formed by two perpendicular
rays with a common endpoint.
Denotes 90
degrees
Classifying Angles
Angles & Angle Measurements
A straight angle is an angle measuring exactly
180 degrees. It is formed by two opposite rays
with a common endpoint.
A
Determine whether each of the following is
acute, right, or obtuse.
1.
2.
3.
4.
5.
??
acute obtuseright
obtuseacute
Check your understanding
Complementary Angles
Two angles are called complementary angles if the sum of
their degree measurements equals 90 degrees.
We say that one is the COMPLEMENT of the other.
Example:
These two angles are complementary.
Angle Pairs
Angles & Angle Measurements
These two angles can
be "pasted" together to
form a right angle!
Angle Pairs
Angles & Angle Measurements
Supplementary AnglesTwo angles are called supplementary angles if the sum
of their degree measurements equals 180 degrees.
We say that one is the SUPPLEMENT of the other.
Example:
These two angles are supplementary.
Angle Pairs
Angles & Angle Measurements
These two angles can be
"pasted" together to form a
straight line!
Angle Pairs
Angles & Angle Measurements
1. Find the complement of 65.
Examples. (See solution on the board)
Answer : 25
Answer : 40
Angle Pairs
Angles & Angle Measurements
2. Two angles are supplementary. One measures 140.
What is the measure of the other angle?
Answer : 30
3. The supplement of an angle is 30 degrees more than
twice its complement. What is the angle?
A. Determine the complement of
1. 48 2. 7942
Check your understanding
11
B. Determine the supplement of
1. 48 2. 123132 57
Angle Pairs
Angles & Angle Measurements
Adjacent Angles
Adjacent angles are two angles with a common vertex
and a common side but no common interior points.
A
B
C
DAdjacent angles:
BAC and DAB
Angle Pairs
Angles & Angle Measurements
Angle Addition PostulateThe sum of the measures of two adjacent angles is
equal to the measure of the angle formed by the two
nonadjacent rays.
A
B
C
D mBAC + mDAB
= mDAC
Angle Pairs
Angles & Angle Measurements
Bisector of an AngleAn ANGLE BISECTOR is a ray in the interior of an angle
that divides the angle into two congruent adjacent angles
A
B
C
D
If ray AB is a bisector of
mDAC, then
BAC DAB
This means:
mBAC = mDAB
= ½ mDAC
Angle Pairs
Angles & Angle Measurements
Linear PairsTwo adjacent angles form a linear pair if their
uncommon sides form opposite rays
B
D
C
A
ABD and mDBC are
linear pairs
Angle Pairs
Angles & Angle Measurements
Linear Pair PostulateLinear pairs are supplementary.
B
D
C
A
Since ABD and mDBC
are linear pairs, then
they are supplementary;
that is,
mABD + mDBC =
180
Examples: Find x (see solution on the board).
2. 3.x
55 165x
x =35x=15
Angle Pairs
Angles & Angle Measurements
1.
55
x
P
U
S
H mPUH =140
x = 85
Examples: (see solution on the board).
Angle Pairs
Angles & Angle Measurements
4.
PS
H
a. If mHUP = 120, what is mSUH?
U
is a bisector of HUPUS
60
mHUS = 74
b. If mSUP = 74, what is mHUS and mPUH?
mPUH = 148
Vertical Angles
VERTICAL angles are two
nonadjacent angles
formed by two
intersecting lines.
Angle Pairs
Angles & Angle Measurements
AED and BEC are vertical angles
AEB and DEC are vertical angles
Vertical Angle TheoremVertical angles are
congruent
Angle Pairs
Angles & Angle Measurements
AED and BEC are vertical
angles, so AED BEC or
mAED = mBEC
AEB and DEC are also vertical
angles, so AEB DEC or
mAEB = mDEC
Vertical AnglesNote that the adjacent
angles form linear pairs.
Angle Pairs
Angles & Angle Measurements
The ff. are linear pairs:
AEB and BEC
AED and AEB
AED and CED
DEC and BEC
Example
1.
Angle Pairs
Angles & Angle Measurements
Example:
2. Find the value of x, y, and z.
58x
x = 58
y = 122
z = 122y
z
Angle Pairs
Angles & Angle Measurements
52
B
A
F E
D
C
60
G
68
68
60
52
Example 3. Find the measures of the numbered angles.
3 2
1
4
Angle Pairs
Angles & Angle Measurements
Perpendicular LinesLines that intersect at right
angles.
Angle Pairs
Angles & Angle Measurements
In the figure, line m is
perpendicular to line n.
We write: m n
m
n
Perpendicular BisectorA line, ray, or line segment
that passes through the
midpoint of a segment.
Angle Pairs
Angles & Angle Measurements
In the figure, line m is a
perpendicular bisector of segment
AB. It means:
• m is perpendicular to line AB
• m passes through the midpoint C,
which means AC = CB
m
A BC
D
20
C
J
D
EF
G
H
70
9070
20
90
Example:
Given the figure shown on
the right. Find the
measures of the following:
a) DCE
b) GCH
c) ECF
d) DCJ
e) JCH
Angle Pairs
Angles & Angle Measurements
Summing it up!Angles & Angle Measurements
Thank you!