1. Segment Measure and Coordinate Graphing
§ 2.1 Real Numbers and Number Lines
§ 2.2 Segments and Properties of Real Numbers
§ 2.3 Congruent Segments
§ 2.4 The Coordinate Plane
§ 2.5 Midpoints

2. 1. Find the next three terms of the sequence 12, 17, 23, 30, … ,
5 Minute-Check
1. Find the next three terms of the sequence , 17, 23, 30, … ,
38, 47, 57
2. Name the intersection of planes ABC and CDE in the figure. A B D E F C
3. How does a ray differ from a line?
A ray extends in only one direction and has an endpoint.
A line extends in two directions.
4. Find the perimeter and area of a rectangle with length of 10 centimeters and width of 4 centimeters.

3. Real Numbers and Number Lines
What You'll Learn
You will learn to find the distance between two points on a number line.
Vocabulary
1) Whole Numbers
2) Natural Numbers
3) Integers
4) Rational Numbers
5) Terminating Decimals
6) Nonterminating Decimals
7) Irrational Numbers
8) Real Numbers
9) Coordinate
10 Origin
11) Measure
12) Absolute Value

4. Real Numbers and Number Lines
Numbers that share common properties can be classified or grouped into sets.
Different sets of numbers can be shown on number lines.
This figure shows the set of _____________ .
whole numbers 2 1 4 3 6 5 7 9 8 10
The whole numbers include 0 and the natural, or counting numbers.
The arrow to the right indicate that the whole numbers continue _________.
indefinitely

5. Real Numbers and Number Lines
This figure shows the set of _______ .
integers 2 1 4 3 -5 5 -4 -2 -3 -1
negative integers
positive integers
The integers include zero, the positive integers, and the negative integers. The arrows indicate that the numbers go on forever in both directions.

6. Real Numbers and Number Lines
A number line can also show ______________.
rational numbers 2 1 -1 -2
A rational number is any number that can be written as a _______, where a and b are integers and b cannot equal ____.
fraction
zero
The number line above shows some of the rational numbers between -2 and 2.
In fact, there are _______ many rational numbers between any two integers.
infinitely

7. Real Numbers and Number Lines
Rational numbers can also be represented by ________.
decimals
Decimals may be __________ or _____________.
terminating
nonterminating
0.375
0.49
terminating decimals.
nonterminating decimals.
The three periods following the digits in the nonterminating decimals indicate that there are infinitely many digits in the decimal.

8. Real Numbers and Number Lines
Some nonterminating decimals have a repeating pattern.
repeats the digits 1 and 7 to the right of the decimal point.
A bar over the repeating digits is used to indicate a repeating decimal.
Each rational number can be expressed as a terminating decimal or a nonterminating decimal with a repeating pattern.

9. Real Numbers and Number Lines
Decimals that are nonterminating and do not repeat
are called _______________.
irrational numbers
and
appear to be irrational numbers

10. Real Numbers and Number Lines
____________ include both rational and irrational numbers.
Real numbers 2 1 -1 -2
The number line above shows some real numbers between -2 and 2.
Postulate 2-1
Number Line
Postulate
Each real number corresponds to exactly one point on a number line.
Each point on a number line corresponds to exactly one real number

11. Real Numbers and Number Lines
The number that corresponds to a point on a number line is called the _________ of the point.
coordinate
On the number line below, __ is the coordinate of point A. 10
The coordinate of point B is __ -4
Point C has coordinate 0 and is called the _____.
origin x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 B C A

12. Real Numbers and Number Lines
The distance between two points A and B on a number line is found by using the Distance and Ruler Postulates.
Postulate 2-2
Distance
Postulate
For any two points on a line and a given unit of measure, there is a unique positive real number called the measure of the distance between the points. A B
measure
Postulate 2-3
Ruler
Postulate
Points on a line are paired with real numbers, and the measure of the distance between two points is the positive difference of the corresponding numbers. B A a b
measure = a – b

13. Real Numbers and Number Lines
The measure of the distance between B and A is the positive difference 10 – 2, or 8. x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 B A 1 2 4 3 5 7 6 8 9 10 11
Another way to calculate the measure of the distance is by using ____________.
absolute value

14. Real Numbers and Number Lines
Use the number line below to find the following measures. A B C F x -2 2 -1 -3 1 1 2 3

15. Real Numbers and Number Lines
Use the number line below to find the following measures. A D E F x -2 2 -1 -3 1 1 2 3

16. Real Numbers and Number Lines
Traveling on I-70, the Manhattan exit is at mile marker 313. The Hays exit is mile marker What is the distance between these two towns?

17. Real Numbers and Number Lines
End of Lesson

18. 6. Find the next three terms of the sequence. 6, 12, 24, . . .
5 Minute-Check
Find the value or values of the variable that makes each equation true. 1.
g = 21 2.
x = 5 3.
y = 4 or y = -- 4 4.
z = -- 2 5. 6
6. Find the next three terms of the sequence. 6, 12, 24, . . .
48, 96, 192

19. Segments and Properties of Real Numbers
What You'll Learn
You will learn to apply the properties of real numbers to the measure of segments.
Vocabulary
1) Betweenness
2) Equation
3) Measurement
4) Unit of Measure
5) Precision

20. Segments and Properties of Real Numbers
Given three collinear points on a line, one point is always _______ the other two points.
between
Definition of
Betweenness
Point R is between points P and Q if and only if R, P, and Q are collinear and _______________.
PR + RQ = PQ P Q R PQ PR RQ
NOTE: If and only if (iff) means that both the statement and its converse are true. Statements that include this phrase are called biconditionals.

21. Segments and Properties of Real Numbers
Segment measures are real numbers. Let’s review some of the properties of real numbers relating to EQUALITY.
Properties of Equality for Real Numbers.
Reflexive Property
For any number a,
a = a
Symmetric Property
For any numbers a and b,
if a = b,
then b = a
Transitive Property
For any numbers a, b, and c,
if a = b and b = c
then a = c

22. Segments and Properties of Real Numbers
Segment measures are real numbers. Let’s review some of the properties of real numbers relating to EQUALITY.
Properties of Equality for Real Numbers.
Addition and Subtraction Properties
For any numbers a, b, and c, if a = b,
then a + c = b + c and
a – c = b – c
Multiplication and Division
Properties
For any numbers a, b, and c, if a = b,
then a * c = b * c and
a ÷ c = b ÷ c
Substitution Properties
For any numbers a and b, if a = b,
then a may be replaced by b in any equation.

23. Segments and Properties of Real Numbers
If QS = 29 and QT = 52, find ST. P Q S T
QS + ST = QT
QS + ST – QS = QT – QS
ST = QT – QS
ST = 52 – 29
= 23

24. Segments and Properties of Real Numbers
If PR = 27 and PT = 73, find RT. P Q R S T
PR + RT = PT
PR + RT – PR = PT – PR
RT = PT – PR
RT = 73 – 27
= 46

25. Segments and Properties of Real Numbers
End of Lesson

26. Refer to the figure below: Suppose AC = 49 and AB = 14.
5 Minute-Check
1. Points X, Y, and Z are collinear If XY = 32, XZ = 49, and YZ = 81, determine which point is between the other two. Y Z X
Refer to the figure below: Suppose AC = 49 and AB = 14. A C B
2. Find BC.
BC = AC - AB
BC =
BC = 35
3. Suppose D is 5 units to the right of C. What is AD?
AD = AC = 54

27. Congruent Segments
What You'll Learn
You will learn to identify congruent segments and find the midpoints of segments.
In geometry, two segments with the same length are called ________ _________
congruent segments
Definition of
Congruent
Segments
Two segments are congruent if and only if
________________________
they have the same length

28. Congruent Segments B A C P Q R S

29. Use the number line to determine if the statement is True or False.
Congruent Segments
Use the number line to determine if the statement is True or False.
Explain you reasoning. x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 R S T Y
Because RS = 4 and TY = 5,

30. These statements are called ________. theorems
Congruent Segments
Since congruence is related to the equality of segment measures, there are properties of congruence that are similar to the corresponding properties of equality.
These statements are called ________.
theorems
Theorems are statements that can be justified by using logical reasoning.
2 – 1
Congruence of segments is reflexive.
2 – 2
Congruence of segments is symmetric.
2 – 3
Congruence of segments is transitive.

31. There is a unique point on every segment called the _______. midpoint
Congruent Segments
There is a unique point on every segment called the _______.
midpoint
On the number line below, M is the midpoint of
What do you notice about SM and MT? x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 S M T
SM = MT

32. Definition of Midpoint
Congruent Segments
Definition of Midpoint
A point M is the midpoint of a segment if and only if M is between S and T and SM = MT M S T
SM = MT
The midpoint of a segment separates the segment into two segments of _____ _____.
equal length
So, by the definition of congruent segments, the two segments are _________.
congruent

33. In the figure, B is the midpoint of . Find the value of x.
Congruent Segments
In the figure, B is the midpoint of
Find the value of x.
5x - 6 2x C B A
Check!
Since B is the midpoint:
AB = BC
AB = 5x – 6
= 5(2) – 6
= 10 – 6
= 4
Write the equation involving x:
5x – 6 = 2x
Solve for x:
5x – 2x – 6 = 2x – 2x
3x – = 0 + 6
3x = 6
x = 2
BC = 2x
= 2(2)
= 4

34. To bisect something means to separate it into ___ congruent parts. two
Congruent Segments
To bisect something means to separate it into ___ congruent parts.
two
The ________ of a segment bisects the segment because it separates the segment into two congruent segments.
midpoint
A point, line, ray, or plane can also bisect a segment. E G D B A C

35. Congruent Segments
End of Lesson

36. The lengths are the same, both are 4.
5 Minute-Check 1.
The lengths are the same, both are 4. 2.
d + 4 = 3d = 2d
2 = d
d + 4 3d S R Q 3.
True; segment congruence is symmetric. 4.
False; Points A, B, and C may not be collinear.
5. If a box has 5 red marbles, 5 blue marbles, and 5 green marbles, what is the probability of selecting either a blue or green marble?

37. The Coordinate Plane
What You'll Learn
You will learn to name and graph ordered pairs on a coordinate plane.
In coordinate geometry, grid paper is used to locate points.
The plane of the grid is called the coordinate plane. y x 5 -4 -2 1 3 -5 -1 4 -3 2

38. The horizontal number line is called the ______. x-axis
The Coordinate Plane
The horizontal number line is called the ______. y x 5 -4 -2 1 3 -5 -1 4 -3 2
x-axis
Quadrant II
(–, +)
Quadrant I
(+, +)
The vertical number line is called the ______.
y-axis O
The point of intersection of the two axes is called the _____.
Quadrant III
(–, –)
Quadrant IV
(+, –)
origin
The two axes separate the plane into four regions called _________.
quadrants

39. Completeness Property for Points in the Plane
The Coordinate Plane
An ordered pair of real numbers, called coordinates of a point, locates a point in the coordinate plane.
Each ordered pair corresponds to EXACTLY ________ in the coordinate plane.
one point
The point in the coordinate plane is called the graph of the ordered pair.
Locating a point on the coordinate plane is called _______ the ordered pair.
graphing
Postulate 2 – 4
Completeness Property for Points in the Plane
Each point in a coordinate plane corresponds to exactly one __________________________.
Each ordered pair of real numbers corresponds to exactly one __________________________.
ordered pair of real numbers
point in the coordinate plane

40. Graphing an ordered pair, (point): (x, y)
The Coordinate Plane
Graphing an ordered pair, (point): (x, y)
Graph point A at (4, 3)
The first number, 4, is called the
___________. y x 5 -4 -2 1 3 -5 -1 4 -3 2
x-coordinate
(4, 3)
It tells the number of units the point lies to the __________ of the origin.
left or right
(0, 0)
The second number, 3, is called the
___________.
y-coordinate
It tells the number of units the point lies _____________ the origin.
above or below
What is the coordinate of the origin?

41. Graphing an ordered pair, (point): (x, y)
The Coordinate Plane
Graphing an ordered pair, (point): (x, y)
Graph point B at (2, –3)
The first number, 2, is called the
___________. y x 5 -4 -2 1 3 -5 -1 4 -3 2
x-coordinate
It tells the number of units the point lies to the __________ of the origin.
left or right
The second number, –3, is called the
___________.
y-coordinate
(2, –3)
It tells the number of units the point lies _____________ the origin.
above or below

42. Name the points A, B, C, & D Point A(x, y) = (3, 2) B B A A
The Coordinate Plane
Name the points A, B, C, & D y x 5 -4 -2 1 3 -5 -1 4 -3 2
Point A(x, y) =
(3, 2)
(–3, 2) B B
(3, 2) A A
Point B(x, y) =
(–3, 2)
Point C(x, y) =
(–3, –2)
Point D(x, y) =
(3, –2)
(–3, –2) C C D
(3, –2) D

43. Consider these questions:
The Coordinate Plane
On a piece of grid paper draw lines representing the x-axis and the y-axis. y x 5 -4 -2 1 3 -5 -1 4 -3 2
Graph :
Point A(x, y) = A(2, 4)
Point B(x, y) = B(2, 0)
Point C(x, y) = C(2, –3)
Point D(x, y) = D(2, –5)
Consider these questions:
x = 2
1) What do you notice about the graphs of these points?
Write a general statement about ordered pairs that have the same x-coordinate.
They lie on a vertical line.
They lie on a vertical line that intersects the x-axis at the x-coordinate.
2) What do you notice about the x-coordinates of these points?
They are the same number.

44. On the same coordinate plane
The Coordinate Plane
On the same coordinate plane y x 5 -4 -2 1 3 -5 -1 4 -3 2
Graph :
Point W(x, y) = W(–4, –4)
Point X(x, y) = X(–2, –4)
Point Y(x, y) = Y(0, –4)
Point Z(x, y) = Z(3, –4)
y = – 4
Consider these questions:
1) What do you notice about the graphs of these points?
Write a general statement about ordered pairs that have the same y-coordinate.
They lie on a horizontal line.
They lie on a horizontal line that intersects the y-axis at the y-coordinate.
2) What do you notice about the y-coordinates of these points?
They are the same number.

45. If a and b are real numbers,
The Coordinate Plane
Theorem 2 – 4
If a and b are real numbers,
a vertical line contains all points (x, y) such that _____
and
a horizontal line contains all points (x, y) such that _____
x = a
y = b y x 5 -4 -2 1 3 -5 -1 4 -3 2
Graph the lines: x = –3
y = 2
(–3, 2)
Graph the point of intersection of these lines.

46. The Coordinate Plane
End of Lesson

47. 1) Name the coordinates of each point.
5 Minute-Check y x 5 -1 2 4 -2 1 -3 3
1) Name the coordinates of each point.
A = (2, 0)
B = (3, –2) A B
2) Graph point C at (0, –4). y x 5 -4 -2 1 3 -5 -1 4 -3 2
x = –4
(–4, 2)
3) Graph x = –4.
y = 2
4) Graph y = 2.
C(0, -4)
5) Graph and label the intersection of x = – and y = 2

48. What You'll Learn Midpoints
You will learn to find the coordinates of the midpoint of a segment. B A C
The midpoint of a line segment, , is the point C that ______ the segment.
bisects A C B -7 -6 -5 -4 -2 -3 -1 1 2 3 4 5 6 7
C = [3 + (-5)] ÷ 2
= (-2) ÷ 2
= -1

49. Midpoints
Theorem 2 – 5
On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is A B

50. Find the midpoint, C(x, y), of a segment on the coordinate plane.
Midpoints
Find the midpoint, C(x, y), of a segment on the coordinate plane.
Consider the x-coordinate: y x 10 -1 2 4 6 8 -2 3 7 1 5 9
x = 1
x = 9
It will be (midway) between the lines
x = and x = 9 A B
y = 7
C(x, y)
Consider the y-coordinate: y x
It will be (midway) between the lines
y = and y = 7
y = 3

51. On a coordinate plane, the coordinates of the midpoint of a
Midpoints
Theorem 2 – 6
On a coordinate plane, the coordinates of the midpoint of a
segment whose endpoints have coordinates (x1, y1) and (x2, y2) are O y x

52. Find the midpoint, C(x, y), of a segment on the coordinate plane.
Midpoints
Find the midpoint, C(x, y), of a segment on the coordinate plane. y x 10 -1 2 4 6 8 -2 3 7 1 5 9
x = 1
x = 9
A(1, 7)
B(9, 3)
y = 7
C(5, 5) y x
y = 3

53. Estimate the midpoint of AB.
Midpoints
Graph A(1, 1) and B(7, 9)
Draw AB y x 10 -1 2 4 6 8 -2 3 7 1 5 9
B(7, 9)
Estimate the midpoint of AB.
Check your answer using the midpoint formula. C
C(4, 5)
A(1, 1)

54. Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B.
Midpoints
Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B.
x-coordinate of B
y-coordinate of B y x 10 -1 2 4 6 8 -2 3 7 1 5 9
B(-1, 8)
Replace x1 with 7 and y1 with 2
B(x, y) is somewhere over there.
C(3, 5)
midpoint
A(7, 2)
Multiply each
side by 2
Add or subtract to isolate the variable

55. Midpoints
End of Lesson

56. A 2 1 4 3 6 5 7 9 8 10 2 1 4 3 6 5 7 9 8 10

57. 1 2 3 1 2 4 3 5 7 6 8 9 10 11