Transcript Slide 1

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
As a group, 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,
or B
A “Rabbit Ear” antenna is a
physical model of an angle
Angle
• An angle consists of
two different rays
(sides) that share a
common endpoint
(vertex).
– Angle ABC, ABC,
or B
Example 1
How many angles can
be seen in the
diagram?
Name all the angles.
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.
C Comes Before S…
m1  m2  90
m3  m4  90
m5  m6  180
m7  m8  180
Example 1a
1. Given that <1 is a complement of <2 and
m<1 = 68°, find m<2.
2. Given that <3 is a supplement of <4 and
m<3 = 56°, find m<4.
Example 2
Let <A and <B be complementary angles
and let m<A = (2x2 + 35)° and m<B =
(x + 10)°. What is (are) the value(s) of x?
What are the measures of the angles?
Linear Pairs of Angles
Linear Pairs of Angles
• Two adjacent angles
form a linear pair if
their noncommon
sides are opposite
rays.
• The angles in a linear
pair are
supplementary.
Vertical Angles
Vertical Angles
• Two nonadjacent
angles are vertical
angles if their sides
form two pairs of
opposite rays.
• Vertical angles are
formed by two
intersecting lines.
• GSP
Example 3
Identify all of the linear pairs of angles and all
of the vertical angles in the figure.
Example 4: SAT
y
z
In the figure  5 and  4 , what is the value
x
x
of x?
x
y
z
How To Use a Protractor
The measure of this
angle is written:
mABC  34
Example 3
What is the measure
of DOZ?
D
G
25
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.
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.
Angle Bisector
An angle bisector is a
ray that divides an
angle into two
congruent angles.
Example 5
In the diagram, YW
bisects XYZ, and
mXYW = 18°. Find
mXYZ.
X
W
Y
Z
Example 6
In the diagram, OE bisects angle LON. Find the
value of x and the measure of each angle.
Use Midpoint and Distance Formulas
Objectives:
1. To define midpoint and segment bisector
2. To use the Midpoint and Distance
Formulas
3. To construct a segment bisector with a
compass and straightedge
Midpoint
The midpoint of a segment is the point on the
segment that divides, or bisects, it into two
congruent segments.
Segment Bisector
A segment bisector is a point, ray, line, line
segment, or plane that intersects the segment at
its midpoint.
Example 1
Find DM if M is the midpoint of segment DA,
DM = 4x – 1, and MA = 3x + 3.
A
M
D
Example 3
Segment OP lies on a real number line with
point O at –9 and point P at 3. Where is
the midpoint of the segment?
O
- 10
P
-5
0
5
What if the endpoints of segment OP were at
x1 and x2?
In the Coordinate Plane
nts on the
ate Plane
A: (-6.00, -2.00)
B: (6.00, 4.00)
4
B
Midpoint: (0.00, 1.00)
2
Midpoint
-5
5
-2
A
We could extend the
previous exercise by
putting the segment in
the coordinate plane.
Now we have two
dimensions and two
sets of coordinates.
Each of these would
have to be averaged
to find the coordinates
of the midpoint.
The Midpoint Formula
If A(x1,y1) and B(x2,y2)
are points in a
coordinate plane,
then the midpoint M
of AB has
coordinates
 x1  x2 y1  y2 
,


2 
 2
The Midpoint Formula
The coordinates of
the midpoint of a
segment are
basically the
averages of the xand y-coordinates of
the endpoints
Example 4
Find the midpoint of the segment with
endpoints at (-1, 5) and (3, 8).
Example 5
The midpoint C of IN has coordinates (4, -3).
Find the coordinates of point I if point N is
at (10, 2).
Example 6
Use the Midpoint Formula multiple times to
find the coordinates of the points that
divide AB into four congruent segments.
A
B
Parts of a Right Triangle
Which segment is the longest in any right
triangle?
The Pythagorean Theorem
In a right triangle, if a and b are
the lengths of the legs and c
is the length of the
hypotenuse, then c2 = a2 + b2.
Example 7
How high up on the
wall will a twentyfoot ladder reach if
the foot of the
ladder is placed five
feet from the wall?
The Distance Formula
Sometimes instead of finding a segment’s
midpoint, you want to find it’s length. Notice
how every non-vertical or non-horizontal
segment in the coordinate plane can be
turned into the hypotenuse of a right triangle.
Example 8
Graph AB with A(2, 1) and B(7, 8). Add
segments to your drawing to create right
triangle ABC. Now use the Pythagorean
Theorem to find AB.
Distance Formula
In the previous problem, you found the
length of a segment by connecting it to a
right triangle on graph paper and then
applying the Pythagorean Theorem. But
what if the points are too far apart to be
conveniently graphed on a piece of
ordinary graph paper? For example, what
is the distance between the points (15, 37)
and (42, 73)? What we need is a formula!
The Distance Formula
To find the distance between
points A and B shown at the
right, you can simply count
the squares on the side AC
and the squares on side BC,
then use the Pythagorean
Theorem to find AB. But if
the distances are too great
to count conveniently, there
is a simple way to find the
lengths. Just use the Ruler
Postulate.
B
8
6
4
2
A
C
5
The Distance Formula
You can find the horizontal
distance subtracting the xcoordinates of points A and
B: AC = |7 – 2| = 5. Similarly,
to find the vertical distance
BC, subtract the ycoordinates of points A and
B: BC = |8 – 1| = 7. Now you
can use the Pythagorean
Theorem to find AB.
B
8
6
4
2
A
C
5
Example 9
Generalize this result
and come up with a
formula for the
distance between
any two points
(x1, y1) and (x2, y2).
B
8
(x 2, y 2)
6
4
2
(x 1, y 1)
(x 2, y 1)
A
C
5
The Distance Formula
If the coordinates of
points A and B are
(x1, y1) and (x2, y2),
then
AB 
x2  x12  y2  y12
Example 10
To the nearest tenth of a unit, what is the
approximate length of RS, with endpoints
R(3, 1) and S(-1, -5)?
Example 11
A coordinate grid is placed over a map. City
A is located at (-3, 2) and City B is located
at (4, 8). If City C is at the midpoint
between City A and City B, what is the
approximate distance in coordinate units
from City A to City C?
Example 12
Points on a 3-Dimensional
coordinate grid can be
located with coordinates of
the form (x, y, z). Finding
the midpoint of a segment
or the length of a segment
in 3-D is analogous to
finding them in 2-D, you
just have 3 coordinates
with which to work.
Example 12
Find the midpoint and
the length of the
segment with
endpoints (2, 5, 8) and
(-3, 1, 2).