Transcript Measuring Segments and Angles

Focus
Graph and label the following points on a coordinate grid.
P(-1, -1), Q(0, 4), R(-3, 5), S(2, 5), and T(3, -4)
1. Name three noncollinear points.
2. Name three collinear points.
3. Name two intersecting lines
Lesson 4: Measuring Segments and Angles
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Lesson 1-4
Measuring
Segments
and Angles
Lesson 4: Measuring Segments and Angles
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The Ruler Postulate (1-5)
The Ruler Postulate: Points on a line can be paired with the real
numbers in such a way that:
• Any two chosen points can be paired with 0 and 1.
• The distance between any two points on a number line is the
absolute value of the difference of the real numbers corresponding
to the points.
Formula: Take the absolute value of the difference of the two
coordinates a and b: │a – b │
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Ruler Postulate : Example
Find the distance between P and K.
G
H
I
J
K
L
M
N
O
P
-5
Note:
Q
R
S
5
The coordinates are the numbers on the ruler or number line!
The capital letters are the names of the points.
Therefore, the coordinates of points P and K are 3 and -2 respectively.
Substituting the coordinates in the formula │a – b │
PK = | 3 - -2 | = 5
Remember : Distance is always positive
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Between
Definition:
X is between A and B if AX + XB = AB.
X
A
X
B
AX + XB = AB
A
B
AX + XB > AB
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The Segment Addition Postulate
Postulate: If C is between A and B, then AC + CB = AB.
Example: If AC = x , CB = 2x and AB = 12, then, find x, AC
and CB.
B
2x
C
A x
Step 1: Draw a figure
12
Step 2: Label fig. with given info.
AC + CB = AB
x + 2x = 12
Step 3: Write an equation
Step 4: Solve and find all the answers
3x = 12
x = 4
Lesson 4: Measuring Segments and Angles
x = 4
AC = 4
CB = 8
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Congruent Segments
Definition: Segments with equal lengths. (congruent symbol: 
Congruent segments can be marked with dashes.
If numbers are equal the objects are congruent.
)
B
A
C
D
AB: the segment AB ( an object )
AB: the distance from A to B ( a number )
Correct notation:
AB = CD
AB  CD
Incorrect notation:
AB  CD
AB = CD
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Midpoint
Definition: A point that divides a segment into
two congruent segments
ab
If DE  EF , then E2 is the midpoint of
Formulas:
DF .
D
F
E
On a number line, the coordinate of the midpoint of a segment
whose endpoints have coordinates a and b is
.
(x , y )
In a coordinate plane,
the coordinates of the midpoint of a
segment whose endpoints have coordinates ( x1 , y1 ) and
2
is
 x1  x 2 y1  y 2 
,


2
2


2
.
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Midpoint on Number Line Example
Find the coordinate of the midpoint of the segment PK.
G
H
I
J
K
L
M
N
O
P
Q
-5
R
S
5
ab
2

3  (  2)
2

1
 0.5
2
Now find the midpoint on the number line.
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Segment Bisector
Definition: Any segment, line or plane that divides a segment into two
congruent parts is called segment bisector.
A
F
F
A
B
E
E
AB bisects DF .
D
B
D
F
A
E
D
AB bisects DF .
Plane M bisects DF .
B
AB bisects DF .
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Angle and Points
ray
vertex

ray
Angles can have points in the interior, in the exterior or on the
angle.
A
E
D
B
C
Points A, B and C are on the angle. D is in the interior and E is in the exterior.
B is the vertex.
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Naming an angle: (1) Using 3 points
(2) Using 1 point
(3) Using a number – next slide
Using 3 points: vertex must be the middle letter
This angle can be named as
ABC
or  C B A
Using 1 point: using only vertex letter
* Use this method is permitted when the vertex point is the vertex
of one and only one angle.
Since B is the vertex of only this angle, this can
also be called  B .
B
Lesson 4: Measuring Segments and Angles
A
C
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Naming an Angle - continued
Using a number: A number (without a degree symbol) may be
used as the label or name of the angle. This
A
number is placed in the interior of the angle near
its vertex. The angle to the left can be named
B
2
C
as  2 .
* The “1 letter” name is unacceptable when …
more than one angle has the same vertex point. In this case, use
the three letter name or a number if it is present.
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Example

K is the vertex of more than one angle.
Therefore, there is NO
There is
K
in this diagram.
 L K M ,  P K M , and  L K P
T here is also  2 and  3 but there is no  5 !!!
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4 Types of Angles
Acute Angle: an angle whose measure is less than 90.
Right Angle: an angle whose measure is exactly 90 .
Obtuse Angle: an angle whose measure is between
90 and 180.
Straight Angle: an angle that is exactly 180 .
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Measuring Angles

Just as we can measure segments, we can also measure angles.

We use units called degrees to measure angles.
•
A circle measures _____
360º
?
•
A (semi) half-circle measures _____
?
•
?
A quarter-circle measures _____
90º
•
One degree is the angle measure of 1/360th of a circle.
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Adding Angles
A
B
36 °
m1 + m2 = mADC also.
22 °
Therefore, mADC = 58.
1
C
2
D
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Angle Addition Postulate
Postulate: The sum of the two smaller angles will always equal
the measure of the larger angle.
M
K
MRK
KRW
MRW
W
R
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Example: Angle Addition
K is interior to MRW, m  MRK = (3x), m KRW = (x + 6) and
mMRW = 90º. Find mMRK.
First, draw it!
K
M
W
3x
x+6
R
3x + x + 6 = 90
4x + 6 = 90
– 6 = –6
4x = 84
x = 21
Are we done?
mMRK = 3x = 3•21 = 63º
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Angle Bisector
An angle bisector is a ray in the interior of an angle that splits the
angle into two congruent angles.
Example: Since 4   6, UK is an angle bisector.
41 ° K
41 °
5
3
j
4
6
U
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Congruent Angles
Definition: If two angles have the same measure, then they are
congruent.
Congruent angles are marked with the same number of “arcs”.
The symbol for congruence is 
Example:
3   5.
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5
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Example

Draw your own diagram and answer this question:
If ML is the angle bisector of PMY and mPML = 87,
then find:
mPMY = _______

mLMY = _______


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