Download 1.4: Measuring Segments and Angles

1.4: Measuring Segments and
Angles
Vocabulary 1.4
•The numerical location of a point on a number line.
•Coordinate
•On a number line length AB = AB = |B - A|
•Length
•Congruent Segments •Sets of points that are of the same length. Symbol is: 
•The location of the middle of a segment. The midpoint
divides a segment into two equal halves.
•On a number line, midpoint of AB = 1/2 (B+A)
•Midpoint
The length of a segment on a number line is determined using The Ruler
Postulate:
The points on a line can be put in one-to-one correspondence with the real
number line so that the distance between any two points is the absolute
value of the difference of the corresponding numbers.
A
-8
B
-6
-4
-2 -1
0
C
2
E
D
4
6
8
The Segment Addition Postulate: If three points A,B,and C are collinear and
B is between A and C, then AB + BC = AC
Measuring Segments and Angles
GEOMETRY LESSON 1-4
Find which two of the segments XY, ZY, and ZW
are congruent.
Use the Ruler Postulate to find the length of each segment.
XY = | –5 – (–1)| = | –4| = 4
ZY = | 2 – (–1)| = |3| = 3
ZW = | 2 – 6| = |–4| = 4
Because XY = ZW, XY
ZW.
1-4
Measuring Segments and Angles
GEOMETRY LESSON 1-4
If AB = 25, find the value of x. Then find AN and NB.
Use the Segment Addition Postulate to write an equation.
AN + NB = AB
Segment Addition Postulate
(2x – 6) + (x + 7) = 25
Substitute.
3x + 1 = 25
3x = 24
x=8
AN = 2x – 6 = 2(8) – 6 = 10
NB = x + 7 = (8) + 7 = 15
Simplify the left side.
Subtract 1 from each side.
Divide each side by 3.
Substitute 8 for x.
AN = 10 and NB = 15, which checks because the sum of the segment lengths
equals 25.
1-4
Measuring Segments and Angles
GEOMETRY LESSON 1-4
M is the midpoint of RT. Find RM, MT, and RT.
Use the definition of midpoint to write an equation.
RM = MT
Definition of midpoint
5x + 9 = 8x – 36
Substitute.
5x + 45 = 8x
Add 36 to each side.
45 = 3x
Subtract 5x from each side.
15 = x
Divide each side by 3.
RM = 5x + 9 = 5(15) + 9 = 84
MT = 8x – 36 = 8(15) – 36 = 84
Substitute 15 for x.
RT = RM + M T = 168
RM and MT are each 84, which is half of 168, the length of RT.
1-4
The Protractor Postulate
Let OA and OB be opposite rays in a plane, OA, OB, and all the rays with endpoint
O that can be drawn on one side of AB can be paired with the real numbers
from 0º to 180º so that:
a. OA is paired with 0º and OB is paired with 180º.
b. If OC is paired with x and OD is paired with y, then mCOD = |x-y|º
D
C
y 77º
mCOD = |x-y|
= | 51 - 77 |
= | -26 |
= 26º
x 51º
180º
B
O
A
0º
Vocabulary 1.4, cont.
•Angle
•Right Angle
•Obtuse Angle
•Acute Angle
•Straight Angle
•Congruent Angles
•Formed by two rays with the same endpoint.
•The rays: sides
•Common endpoint: the vertex
•Name: FAD , FBC, 1
•Measures exactly 90º
•Measure is GREATER than 90º FAD
•Measure is LESS than 90º ADE
•Measure is exactly 180º ---this is a line FAB
•Angles with the same measure.
1
2
Measure Angles
Use a Protractor
1
m 1 = 40º
The Angle Addition Postulate
Find
m AOB,
m BOC and
m AOC
B
A
C
O
D
m AOB = 60º
m BOC = | 60 - 120 |º
= 60 º
and m AOC = 120 º
The Angle Addition Postulate says that as long as AOB and BOC do not
overlap, then mAOC = m AOB +m BOC = 120
Measuring Segments and Angles
GEOMETRY LESSON 1-4
Name the angle below in four ways.
The name can be the number between the sides of the angle:
The name can be the vertex of the angle:
3.
G.
Finally, the name can be a point on one side, the vertex, and a point
on the other side of the angle: AGC, CGA.
1-4
Measuring Segments and Angles
GEOMETRY LESSON 1-4
Find the measure of each angle. Classify each as acute, right,
obtuse, or straight.
Use a protractor to measure each angle.
m 1 = 110
Because 90 < 110 < 180,
m 2 = 80
Because 0 < 80 < 90,
1 is obtuse.
2 is acute.
1-4
Measuring Segments and Angles
GEOMETRY LESSON 1-4
Suppose that m
1 = 42 and m
ABC = 88. Find m
2.
Use the Angle Addition Postulate to solve.
m
1+m
2=m
ABC Angle Addition Postulate.
42 + m
2 = 88
Substitute 42 for m
m
2 = 46
Subtract 42 from each side.
1-4
1 and 88 for m
ABC.
Re Cap:
•The numerical location of a point on a number line.
•Coordinate
•On a number line length AB = AB = |B - A|
•Length
•Congruent Segments •Sets of points that are of the same length. Symbol is: 
•The location of the middle of a segment. The midpoint
divides a segment into two equal halves.
•On a number line, midpoint of AB = 1/2 (B+A)
•Midpoint
The length of a segment on a number line is determined using The Ruler
Postulate:
The points on a line can be put in one-to-one correspondence with the real
number line so that the distance between any two points is the absolute
value of the difference of the corresponding numbers.
A
-8
B
-6
-4
-2 -1
0
C
2
E
D
4
6
8
The Segment Addition Postulate: If three points A,B,and C are collinear and
B is between A and C, then AB + BC = AC
Recap 2.
•Angle
•Right Angle
•Obtuse Angle
•Acute Angle
•Straight Angle
•Congruent Angles
•Formed by two rays with the same endpoint.
•The rays: sides
•Common endpoint: the vertex
•Name: FAD , FBC, 1
•Measures exactly 90º
•Measure is GREATER than 90º FAD
•Measure is LESS than 90º ADE
•Measure is exactly 180º ---this is a line FAB
•Angles with the same measure.
1
2
Re Cap 3
Find
m AOB,
m BOC and
m AOC
B
A
C
O
D
m AOB = 60º
m BOC = | 60 - 120 |º
= 60 º
and m AOC = 120 º
The Angle Addition Postulate says that as long as AOB and BOC do not
overlap, then mAOC = m AOB +m BOC = 120
Extra Practice
GEOMETRY LESSON 1-4
Use the figure below for Exercises 1-3.
Use the figure below for Exercises 4–6.
1. If XT = 12 and XZ = 21, then TZ = 7.
9
2. If XZ = 3x, XT = x + 3, and TZ = 13,
find XZ.
24
3. Suppose that T is the midpoint of XZ.
If XT = 2x + 11 and XZ = 5x + 8,
find the value of x.
14
4. Name 2 two different ways.
DAB, BAD
5. Measure and classify 1, 2,
and BAC.
90°, right; 30°, acute; 120°, obtuse
6. Which postulate relates the measures
of 1, 2, and BAC?
Angle Addition Postulate
1-4