Bellringer
• Simplify each absolute value expression
1.
7 − 10
• 3
2.
−4 − 2
• 6
3.
• 2
−2 − (−4)
1.4
Measuring Segments and Angles
Postulate 1-5
Ruler
Postulate
• The points of a line can be put into oneto-one correspondence with the real
numbers so that the distance between
any two points is the absolute value of
the difference of the corresponding
numbers.
• Length of 𝐴𝐵 = 𝑎 − 𝑏
Congruent (≅)
Segments
• Two segments with the same length
Comparing
Segments
lengths
• Find AB and BC
A
B
C
D
E
• AB = −7 − (−4) = −3 = 3
• BC = −4 − (−1) = −3 = 3
So AB = BC or 𝐴𝐵 ≅ 𝐵𝐶
Postulate 1-6
Segment
Addition
Postulate
• If three points A, B, and C are collinear
and B is between A and C, then
AB + BC = AC
T
Using the Segment Addition Postulate
• If DT = 60, find the value of x. Then find DS and ST.
D
•
•
•
•
•
•
•
•
2x - 8
•
DS + ST = DT
(2x – 8) + (3x – 12) = 60 •
•
5x – 20 = 60
•
5x = 80
•
x = 16
Substitute x = 16
DS = 2x – 8 = 2(16) – 8 = 24
ST = 3x – 12 = 3(16) – 12 = 36
S
3x - 12
T
Segment Addition Postulate
Substitution
Simplify
Add 20 to each side
Divide each side by 5
Using the
Segment
Postulate
• If DT = 100. Find the value of x. Then find DS and ST.
D
• x = 15
• DS = 40
• ST = 60
4x - 20
S
2x + 30
T
Midpoint
• a point that divides a segment into two
congruent segments. A midpoint, or
any line, ray, or other segment through
a midpoint, is said to bisect the
segment.
• 𝐴𝑀 ≅ 𝑀𝐵
Finding • C is the midpoint of 𝐴𝐵. Find AC, CB, and AB
Lengths
A
•
•
•
•
AC = CB
2x +1 = 3x - 4
2x + 5 = 3x
5=x
•
•
•
•
• AC = 2x + 1 = 2(5) + 1 = 11
• CB = 3x – 4 = 3(5) – 4 =11
• AC + CB = AB
11 + 11 = 22
2x + 1
C
3x - 4
B
Definition of Midpoint
Substitution
Add 4 to each sides
Subtract 2x from each side
Finding
Lengths
• Z is the midpoint 𝑋𝑌, and XY = 30. Find XZ.
30
Z
X
• XZ =
𝑋𝑌
2
• XZ =
30
2
= 15
Y
Homework
• Pg. 29 #’s 1 – 4, 8-15
Bellringer
X
T
Z
• If XT = 12 and XZ = 21, then TZ = 9
1.4
CONTINUED
Measuring Segments and Angles
Angle(∠)
• Is formed by two rays with the
same endpoint. The rays are the
sides of the angle. The endpoint is
the vertex of the angle.
Naming
Angles
There are three ways to
name an angle
• Name ∠ a two other ways
• 1. ∠BAC
• 2. ∠CAB
• Name ∠ 𝜃 two other ways
• 1. ∠BCD
• 2. ∠DCB
1. Counterclockwise (∠CBA)
2. Clockwise (∠ABC)
3. The vertex (∠B)
Classifying Angles
• Def: Acute Angle : angle whose
measure is 0° < 𝑥 <90°.
• Def: Obtuse Angle: angle whose
measure is 90° < x < 180.
• Def: Right Angle (∟): angle whose
measure is 90°
• Def: Straight Angle: angle whose
measure is 180°.
Postulate 1-8
Angle
Addition
Postulate
• If point B is in the interior of ∠AOC, then
m ∠AOB + m ∠BOC = m ∠AOC
• If ∠AOB is a straight angle, then m ∠BOC + ∠COA = 180°
Using the
Angle
Addition
Postulate
• What is m∠COA if m∠BOC = 50 and the
m∠BOA = 125.
Using the
Angle
Addition
Postulate
• If m∠COA = 145, find the m∠𝐵𝑂𝐶.
Congruent
Angles
• Angles with the same measure. In
other words, if m∠1 = 𝑚∠2, then
∠1 ≅ ∠2.
Homework
• Pg. 30 #’s 16-28 all