Transcript 1.4 Angles and Their Measures

INTRO TO ANGLE
MEASUREMENT
Measuring Angles
• Angles are measured using a protractor, which
looks like a half-circle with markings around
its edges.
• Angles are measured in units called degrees
• 45 degrees, for example, is symbolized like
this:
• Every angle on a protractor measures more
than 0 degrees and less than or equal to 180
degrees.
2
A Protractor
3
• The smaller the opening between the two
sides of an angle, the smaller the angle
measurement.
• The largest angle measurement (180
degrees) occurs when the two sides of
the angle are pointing in opposite
directions.
• To denote the measure of an angle we
write an “m” in front of the symbol for
the angle.
4
• Here are some common angles and
their measurements.
1
2
4
3
5
Types of Angles
• An acute angle is an angle that measures
less than 90 degrees.
• A right angle is an angle that measures
exactly 90 degrees.
• An obtuse angle is an angle that measures
more than 90 degrees.
acute
right
obtuse
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• A straight angle is an angle that measures
180 degrees. (It is the same as a line.)
• When drawing a right angle we often mark its
opening as in the picture below.
right angle
straight angle
7
Adjacent Angles
• Adjacent Angles
share a RAY and a
VERTEX but no
INTERIOR POINTS.
• Angles x and y do
not share a ray.
• <DOC is adjacent to
<COB, but it is not
adjacent to <DOB.
Can you tell why?
(think about Point C)
Angles and Their Parts
• An angle consists of two
different rays that have the
same initial point. The rays
are the sides of the angle.
The initial point is the
vertex of the angle.
• Point M is on the INTERIOR
of <CAB.
• Point Q is on the
EXTERIOR of <CAB.
• Point A is the VERTEX.
• Points A, C, and B sit ON
the angle.
•Q
•M
Note:
• The measure of  A is denoted by m A.
The measure of an angle can be
approximated using a protractor, using
units called degrees(°). For instance,
 BAC has a measure of 50°, which can
be written as
B
m BAC = 50°.
A
C
Reading a protractor.
A
O
What is the measure of
angle BOA?
113°
B
Angle Addition Postulate
• If C is in the interior
of <ABD, then
m<ABC + m<CBD =
m<ABD.
• In other words, little
angle + little angle =
big angle.
•
Angle Addition Post., Continued
• m<CAB + m<DAC =
m<DAB, so…
• m<CAB + 53° = 64°
• m<CAB = 11°
• Find the m<DAB.
• m<DAC + m<CAB =
m<DAB.
• 35° + 30° = 65°
• If m<ABM = (3x + 2)° and m<MBC = (5x-4)° and
m<ABC = 102°, find x and the angle measures.
• Angle Addition Postulate:
• m<ABM + m<MBC = m<ABC (use subst.)
• 3x + 2 + 5x – 4 = 102
• 8x – 2 = 102
• 8x = 104
• x = 13; m<ABM = 41°; m<MBC = 61°