congruent angles

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Transcript congruent angles

1.4: Measure and Classify Angles
1.
2.
3.
4.
Objectives:
To define, classify, draw, name, and
measure various angles
To use the Protractor and Angle Addition
Postulates
To construct congruent angles and angle
bisectors with compass and straightedge
To convert angle measurement between
degrees and radians
Vocabulary
In your notes, define
each of these without
your book. Draw a
picture for each word
and leave a bit of
space for additions
and revisions.
Angle
Vertex
Sides
Acute 
Obtuse 
Right 
Straight 
Congruent ’s
Angle
Vertex
A
Sides
B
C
• An angle consists of two
different rays (sides)
that share a common
endpoint (vertex).
• Angle ABC, ABC,
CBA or B
• The vertex is the key
A “Rabbit Ear” antenna is a
physical model of an angle
Angle
• Each angles creates
2 regions: INTERIOR
and EXTERIOR.
Example 1
How many angles can
be seen in the
diagram? 3
Name all the angles.
< WXY, < YXZ, <WXZ
W
Y
X
Z
How Big is an Angle?
Is the angle between
the two hands of the
wristwatch smaller
than the angle
between the hands of
the large clock?
– Both clocks read 9:36
Click me to learn more
about measuring angles
Measure of an Angle
The measure of an angle is
the smallest amount of
rotation about the vertex
from one side to the other,
measured in degrees.
• Can be any value between
0 and 180
• Measured with a protractor
Classifying Angles
Surely you are familiar with all of my angular
friends by now.
How To Use a Protractor
Click the button to
see how to measure
an angle.
The measure of this
angle is written:
mABC  34
Example 3
What is the measure
of DOZ?
D
G
25
650
O
40
Z
Example 3
You basically used the
Angle Addition
Postulate to get the
measure of the
angle, where
mDOG + mGOZ
= mDOZ.
D
G
25
O
40
Z
Angle Addition Postulate
If P is in the interior of RST, then
mRST = mRSP + mPST.
Example 4
Given that mLKN = 145°,
find mLKM and
mMKN.
Set up an equation
and solve
X = 23
<LKM = 56
<MKN = 89
CHECK: 56 + 89 = 145
M
L
2x+10
4x-3
K
N
Congruent Angles
• Two angles are congruent angles if
they have the same measure.
Add the appropriate
markings to your picture.
Congruent Angles
Draw angle A in your notebook. How could
you copy that angle to another part of your
paper using only a
compass and a
straightedge?
Congruent Angles
1. Draw angle A.
Congruent Angles
2. Draw a ray with endpoint A’.
Congruent Angles
3. Put point of compass on A and draw an
arc that intersects both sides of the angle.
Label these points
B and C.
Congruent Angles
4. Put point of compass on A’ and use the
compass setting from Step 3 to draw a
similar arc on the ray.
Label point B’ where
the arc intersects
the ray.
Congruent Angles
5. Put point of compass on B and pencil on
C. Make a small arc.
Congruent Angles
6. Put point of compass on B’ and use the
compass setting from Step 5 to draw an
arc that intersects the
arc from Step 4.
Label the
new point
C’.
Congruent Angles
7. Draw ray A’C’.
Congruent Angles
Click on the
button to
watch a
video of the
construction.
Angle Bisector
An angle bisector is a
ray that divides an
angle into two
congruent angles.
Bisect an Angle
1. Draw an acute angle and label the vertex
A.
Bisect an Angle
2. Using vertex A as the center, draw an arc
intersecting both sides of your angle. Label the
intersections B and C.
Bisect an Angle
3. Using the same compass setting, draw two
intersecting arcs in the interior of your angle,
one centered at B, the other centered at C.
Bisect an Angle
4. Label the intersection D.
Bisect an Angle
5. Draw a ray from vertex A through point D.
Angle Bisector: Video
Click on the
button to
watch a
video of the
construction.
Example 5
In the diagram, YW
bisects XYZ, and
mXYW = 18°. Find
mXYZ.
360
X
W
Y
Z
Example 6
In the diagram, OE bisects angle LON. Find the
value of x and the measure of each angle.
Set up and equation and solve.
X=3
Both angles are 38
Radians
You can also measure an angle in radians.
Radians are like the less well-known
greasy, nerdy-type who eats lots of pie.
Radians
One radian is the measure of the angle
formed by stretching the radius of a circle
around its circumference.
Example 7
How many radians would be the equivalent
to one full revolution around the unit circle?
How many radians would equal 180°?
Example 8
Use the conversion factor 180° =  radians
to convert the following angle measures.
1. Convert 27° into radians.
=?
180
27
27 π
180
3
20
3
2. Convert
rad into degrees.
4
 = 3/4π
180 ?
? = 135