1. Chapter 1
Basics of Geometry

2. Chapter Objectives Using Inductive Reasoning to identify patterns.
Identify collinear characteristics.
Utilize Distance and Midpoint Formulas
Label acute, obtuse, right, and straight angles.
Identify angle measures
Apply Angle Addition Postulate
Compare Complimentary v Supplementary angles
Define angle and segment bisectors
Identify the basics of perimeter, circumference, and area

3. Patterns And Inductive Reasoning
Lesson 1.1
Patterns
And
Inductive Reasoning

4. Lesson 1.1 Objectives
Identify patterns in numbers and shape sequences.
Use inductive reasoning.
Define Geometry.

5. Geometry is…
Geometry is a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids; the study of properties of given elements that remain invariant under specified transformations.
Basically what that means is geometry is the study of the laws that govern the patterns and elements of mathematics.
Definition from Merriam-Webster Online Dictionary.

6. Inductive Reasoning
Inductive Reasoning is the process in which one looks for patterns in samples and makes conjectures of how the pattern will work for the entire population.
A conjecture is an unproven statement based on observations.
A conjecture is math’s version of a hypothesis, or educated guess.
The education comes from the observation.

7. Using Inductive Reasoning
Much of the reasoning in Geometry consists of three stages
Look for a Pattern. Look at examples and organize any ideas of a pattern into a diagram or table.
Make a Conjecture. Use the examples to try to identify what step was taken to get from element to element in the pattern.
Verify the Conjecture. Use logical reasoning to verify the conjecture is true for all cases.

8. Example 1
Identify the next member of the group:
1 , 4 , 7 , 10
1 , 4 , 9 , 16 13 25

9. Counterexamples
A counterexample is one example that shows a conjecture is false.
Therefore to prove a conjecture is true, it must be true for all cases.
Conjecture: Every month has at least 30 days.
Counterexample: February has 28 (or 29).

10. Goldbach’s Conjecture
In the early 1700s a Prussian mathematician named Goldbach noticed that many even numbers greater than 2 can be written as the sum of two primes.
4 = 2 + 2
This conjecture is unproven for all cases, but has been proved for all even numbers up to 4 x 1014.
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7
400,000,000,000,000

11. Homework 1.1 In-class Homework Due Tomorrow 1-11
Page 6-9
Homework
12-26 ev, 27, ev, 47, 48, even
Due Tomorrow

12. Points, Lines, and Planes
Lesson 1.2
Points, Lines, and Planes

13. Lesson 1.2 Objectives Define the basic terms of geometry.
Sketch the basic components of geometrical figures.

14. Start-Up Give your definition of the following
Point
Line
Plane
Not an airplane!
These terms are actually said to be undefined, or have no formal definition.
However, it is important to have a general agreement on what each word means.

15. Point A A point has no dimension. It is usually represented as a dot.
Meaning it takes up no space.
It is usually represented as a dot.
When labeling we designate a capital letter as a name for that point.
We may call it Point A. A

16. Line A line extends in one dimension.
Meaning it goes straight in either a vertical, horizontal, or slanted fashion.
It extends forever in two directions.
It is represented by a line with an arrow on each end.
When labeling, we use lower-case letters to name the line.
Or the line can be named using two points that are on the line.
So we say Line n, or AB n A B

17. Plane A C M B A plane extends in two dimensions.
Meaning it stretches in a vertical direction as well as a horizontal direction at the same time.
It also extends forever.
It is usually represented by a shape like a tabletop or a wall.
When labeling we use a bold face capital letter to name the plane.
Plane M
Or the plane can be named by picking three points in the plane and saying Plane ABC.

18. Collinear The prefix co- means the same, or to share. Linear means
So collinear means that points lie on the same line. A B C
We say that points A, B, and C are collinear.

19. Coplanar Coplanar points are points that lie on the same plane. A C MB C M
So points A, B, and C are said to be coplanar.

20. Line Segment Consider the line AB.
It can be broken into smaller pieces by merely chopping the arrows off.
This creates a line segment or segment that consists of endpoints A and B.
This is symbolized as AB A B

21. Ray
A ray consists of an initial point where the figure begins and then continues in one direction forever.
It looks like an arrow.
This is symbolized by writing its initial point first and then naming any other point on the ray,
Or we can say ray AB. AB A B

22. Opposite Rays
If C is between A and B on a line, then ray CA and ray CB are opposite rays.
Opposite rays are only opposite if they are collinear. A C B

23. Intersections of Lines and Planes
Two or more geometric figures intersect if they have one or more points in common.
If there is no point or points shown, they the figures do not intersect.
The intersection of the figures is the set of points the figures have in common.
Two lines intersect at one point.
Two planes intersect at one line. A m n

24. Homework 1.2 In-class Homework Due Tomorrow 1-8
Page 13-15
Homework
9-42 every 3rd, 44 – 51, ev, 68-75
Due Tomorrow

25. Segments and Their Measures
Lesson 1.3
Segments and Their Measures

26. Lesson 1.3 Objectives Define what a postulate is.
Use segment postulates.
Utilize the Distance Formula.
Identify congruent segments.

27. Definition of a Postulate
A postulate is a rule that is accepted without a proof.
They may also be called an axiom.
Basically we do not need to know the reason for the rule when it is a postulate.
Postulates are used together to prove other rules that we call theorems.

28. Postulate 1: Ruler Postulate
The points on a line can be matched to real numbers called coordinates.
The distance between the points, say A and B, is the absolute value of the difference of the coordinates.
Distance always positive. 4 8 A B

29. Length Finding the distance between points A and B is written asAB
Writing AB is also called the length of line segment AB.

30. Betweenness
When three points lie on a line, we can say that one of them is between the other two.
This is only true if all three points are collinear.
We would say that B is between A and C. A B C

31. Postulate 2: Segment Addition Postulate
If B is between A and C, then
AB + BC = AC.
Also, the opposite is true.
If AB + BC = AC, then B is between A and C. AB BC A B C AC

32. Lesson 1.3A Homework In-class Homework Due Tomorrow Quiz Monday
1, 3, 4-8 ev, 11
p21-22
Homework
13-33
Due Tomorrow
Quiz Monday
Lessons

33. Lesson 1.3
Part II

34. Segment Addition Postulate Review
Identify the unknown lengths given that BD=4, AE=17, AD=7, and BC=CD BC 2 AB 3 AC 5 DE 10 A B C D E

35. Distance Formula To find the distance on a graph between two points
We use the Distance Formula
AB =
(x2 – x1)2 + (y2 – y1)2
Distance can also be found using the Segment Addition Postulate, which simply adds up each segment of a line to find the total length of the line.

36. Example 2
Using the Distance Formula, find the length of segment OK with endpoints
O(2,6)
K(5,10)
(x2 – x1)2 + (y2 – y1)2
(5 – 2)2 + (10 – 6)2
9 + 16 25
= 5

37. Example 3 This is one part of the problem for #34
Find the distance between points A and C.
A(-4,7)
C(3,-2)
(x2 – x1)2 + (y2 – y1)2
(3 – -4)2 + (-2 – 7)2
72 + (-9)2
130

38. Congruent Segments
Segments that have the same length are called congruent segments.
This is symbolized by =.
Hint: If the symbols are there, the congruent sign should be there.
LE = NT
LE = NT
If you want to state two segments are congruent, then you write
If you want to state two lengths are equal, then you write

39. Lesson 1.3B Homework In-Class Homework Due Tomorrow Quiz Tomorrow
4-8 ev
p22-24
Homework
37-56, ev
skip 44,47,52-54
Due Tomorrow
Quiz Tomorrow
Lesson

40. Angles and Their Measures
Lesson 1.4
Angles and Their Measures

41. Lesson 1.4 Objectives Use the angle postulates.
Identify the proper name for angles.
Classify angles as right, obtuse, acute, or straight.
Measure the size of an angle.

42. What is an Angle?
An angle consists of two different rays that have the same initial point.
The rays form the sides of the angle.
The initial point is called the vertex of the angle.
Vertex can often be thought of as a corner.

43. Naming an Angle All angles are named by using three points
Name a point that lies on one side of the angle.
Name the vertex next.
The vertex is always named in the middle.
Name a point that lies on the opposite side of the angle.
So we can call It WON
Or NOW W N O

44. Congruent Angles
Congruent angles are angles that have the same measure.
To show that we are finding the measure of an angle
Place a “m” before the name of the angle. m
WON =
NOW m
WON
NOW =
Equal Measures
Congruent Angles

45. Types of Angles Acute Right Obtuse Straight Looks like Measure <90
=90
>90
=180

46. Other Parts of an Angle
The interior of an angle is defined as the set of points that lie between the sides of the angle.
The exterior of an angle is the set of points that lie outside of the sides of the angle.
Exterior
Interior

47. Postulate 4: Angle Addition Postulate
The Angle Addition Postulate allows us to add each smaller angle together to find the measure of a larger angle.
What is the total?
49o
32o
17o

48. Adjacent Angles
Two angles are adjacent angles if they share a common vertex and side, but have no common interior points.
Basically they should be touching, but not overlapping. C A T R
CAT and TAR are adjacent.
CAR and TAR
are not adjacent.

49. Homework Lesson 1.4 In-Class Homework Due Tomorrow 1-16
p29-32
Homework
18-48 ev, 50-53, ev
skip 30,32,34
Due Tomorrow

50. Segments and Angle Bisectors
Lesson 1.5
Segments
and
Angle Bisectors

51. Lesson 1.5 Objectives Identify a segment bisector
Identify an angle bisector
Utilize the Midpoint Formula

52. Congruence marks are used to show that
segments are congruent. If there is more than one pair of congruent segments, then each pair should get a different number of congruence marks.
Midpoint
The midpoint of a segment is the point that divides the segment into two congruent segments.
The midpoint bisects the segment, because bisect means to divide into two equal parts.
A segment bisector is a segment, ray, line, or plane that intersects the original segment at its midpoint. H T O J Y
Now we can say
line HT is a
segment bisector of segment JY.
We say that O is the midpoint
of line segment JY.

53. ( ) Midpoint Formula , We can also find the midpoint of segment AB by
using its endpoints in…
The Midpoint Formula
A(1,2)
B(7,10) ( ) ,
Midpoint of AB =
(x1 + x2)
(y1 + y2) 2 2
This gives the coordinates of the midpoint, or point that is halfway between A and B.

54. Example 4
This is an example of how to determine the midpoint knowing the two endpoints.
(x1 + x2)
(y1 + y2) ( ) , 2
A(1,2)
B(7,10) ( ) , 2
(1 + 7)
(2 + 10) ( ) , 2 8 12 ( ) , 4 6

55. Example 5
This is an example of how to find an endpoint knowing the midpoint and the other endpoint.
Say the midpoint is (8,5) and one endpoint is (4,9). ( ) ,
So, use each coordinate from the midpoint formula to solve for x2.
Remember that each coordinate from the midpoint was found from…
(x1 + x2)
(y1 + y2) 2
(x1 + x2) 2 =
(4 + x2) 2 =
(y1 + y2) 2 =
(9 + y2) 2 =
x 2 8
x 2 5
4 + x2 =
So the coordinates for the other endpoint are
(12,1) 10
9 + x2 = 16 -9 -4 y2 = x2 = 12 1

56. Short Cut to Find Endpoint
Say the midpoint is (8,5) and one endpoint is (4,9).
Remember that the midpoint is half way between the endpoints.
Add 4 to x
(4,9)
(8,5)
Minus 4 from y
Add 4 to x
(12,1)
Minus 4 from y

57. Angle Bisector
An angle bisector is a ray that divides an angle into two adjacent angles that are congruent.
To show that angles are congruent, we use congruence arcs.

58. Homework 1.5 In-Class Homework Due Tomorrow 1-2, 4-13
p38-42
Homework
18-32 ev, 38-54, ev
Due Tomorrow

59. Angle Pair Relationships
Lesson 1.6
Angle Pair Relationships

60. Lesson 1.6 Objectives Identify vertical angle pairs.
Identify linear pairs.
Differentiate between complementary and supplementary angles.

61. Vertical Angles
Two angles are vertical angles if their sides form two pairs of opposite rays.
Basically the two lines that form the angles are straight.
To identify the vertical angles, simply look straight across the intersection to find the angle pair.
Hint: The angle pairs do not have to be vertical in position.
Vertical Angle pairs are always congruent! 1 3 2 4 1 4 2 3

62. Linear Pair
Two adjacent angles form a linear pair if their non-common sides are opposite rays.
Simply put, these are two angles that share a straight line.
Since they share a straight line, their sum is…
180o 1 2

63. Complementary v Supplementary
Complementary angles are two angles whose sum is 90o.
Complementary angles can be adjacent or non-adjacent.
Supplementary angles are two angles whose sum is 180o.
Supplementary angles can be adjacent or non-adjacent.

64. Homework 1.6 In-Class Homework Due Tomorrow Test Tuesday 1, 4-7
p47-50
Homework
8-40 ev, ev, ev
Due Tomorrow
Test Tuesday

65. Intro to Perimeter, Circumference and Area
Lesson 1.7
Intro to Perimeter,
Circumference and
Area

66. Lesson 1.7 Objectives
Find the perimeter and area of common plane figures.
Establish a plan for problem solving.

67. Perimeter and Area of a Rectangle
Recall that the perimeter of a figure is the sum of the lengths of the sides.
A rectangle has two pairs of opposite sides that are congruent.
l + w + l + w =
P = 2l + 2w
Recall the area of a figure is the measure of space inside the figure.
This is found by taking the length of the rectangle times the width of the rectangle.
l • w =
A = lw

68. Perimeter and Area of a Square
Since a square has four congruent sides, the formula is quite simple…
s + s + s + s =
P = 4s
Since a square is also a rectangle, we can find the area by multiplying the length times the width
s • s =
A = s2 s

69. Perimeter and Area of a Triangle
The perimeter can be found by adding the three sides together.
P = a + b + c
If the third side is unknown, use the Pythagorean Theorem to solve for the unknown side.
a2 + b2 = c2
Where a,b are the two shortest sides and c is the longest side.
The area of a triangle is half the length of the base times the height of the triangle.
The height of a triangle is the perpendicular length from the base to the opposite vertex of the triangle.
A = ½bh a b c h

70. Circumference and Area of a Circle
The perimeter of a circle is called the circumference.
It is found by taking the diameter times .
C = d or 2r
r is the radius, which is half the diameter.
The area is found by taking pi times the radius squared.
A = r2
d=diameter
r=radius

71. Homework In-Class Homework Due Tomorrow Test Tuesday 1-8
p55-58
Homework
10-32 ev, ev, ev
Due Tomorrow
Test Tuesday
December 18th