1. Reasoning in Geometry § 1.1 Patterns and Inductive Reasoning
§ 1.2 Points, Lines, and Planes
§ 1.3 Postulates
§ 1.4 Conditional Statements and Their Converses
§ 1.5 Tools of the Trade
§ 1.6 A Plan for Problem Solving

2. 2) Solve the equation. Check your answer.
5 Minute-Check
1) Both answers can be calculated Which one is right? What makes it right? What makes the other one incorrect?
2) Solve the equation. Check your answer.
3) If a dart is thrown at the circle to the right,
what is the probability that it will land in a yellow sector? The odds?

3. Patterns and Inductive Reasoning
What You'll Learn
You will learn to identify patterns and use inductive reasoning.
If you were to see dark, towering clouds approaching, you might want to take cover.
Your past experience tells you that a thunderstorm is likely to happen.
When you make a conclusion based on a pattern of examples or past events, you are using inductive reasoning.

4. Patterns and Inductive Reasoning
You can use inductive reasoning to find the next terms in a sequence.
Find the next three terms of the sequence:
3, 6, ,
24, , ,
X 2
X 2
X 2
X 2
X 2
Find the next three terms of the sequence:
7, 8, ,
16,
23, 32
+ 1
+ 3
+ 5
+ 7
+ 9

5. Patterns and Inductive Reasoning
Draw the next figure in the pattern.

6. Patterns and Inductive Reasoning
A _________ is a conclusion that you reach based on inductive reasoning.
conjecture
In the following activity, you will make a conjecture about rectangles.
1) Draw several rectangles on your grid paper.
2) Draw the diagonals by connecting each corner with its opposite corner. Then measure the diagonals of each rectangle.
3) Record your data in a table
Diagonal 1
Diagonal 2
Rectangle 1
7.5 inches
d1 = 7.5 in.
d2 = 7.5 in.
Make a conjecture about the diagonals of a rectangle

7. Patterns and Inductive Reasoning
A conjecture is an educated guess.
Sometimes it may be true, and other times it may be false.
How do you know whether a conjecture is true or false?
Try different examples to test the conjecture.
If you find one example that does not follow the conjecture, then the conjecture is false.
Such a false example is called a _____________.
counterexample
Conjecture: The sum of two numbers is always greater than either number.
Is the conjecture TRUE or FALSE ?
Counterexample: = - 2
- 2 is not greater than 3.

8. Patterns and Inductive Reasoning
End of Lesson

9. Find the next three terms of each sequence.
5 Minute-Check
Find the next three terms of each sequence. 1.
59, 63, 67 2.
15.5, , 3.
Draw the next figure in the pattern shown below. 4.
Find a counterexample for this statement:
“The sum of two numbers is always greater than either addend.”
= and 2 < 4
5) If a dart is thrown at the circle to the right,
what is the probability that it will land in a shaded sector? The odds?

10. Points, Lines, and Planes
What You'll Learn
You will learn to identify and draw models of points, lines, and planes, and determine their characteristics.
Geometry is the study of points, lines, and planes and their relationships. Everything we see contains elements of geometry.
Georges Seurat, Sunday Afternoon on the Island of LeGrande Jatte,
Even the painting to the right is made entirely of small, carefully placed dots of color.

11. Points, Lines, and Planes
A ____ is the basic unit of geometry.
point A
POINT:
A point has no ____.
size
Points are named using capital letters. B
The points at the right are named point A and point B.

12. Points, Lines, and Planes
A ____is a series of points that extends without end in two directions.
line
LINE:
A line is made up of an ______ _______ of points.
infinite number
The ______ show that the line extends without end in both directions.
arrows
A line can be named with a single lowercase script letter or by two points on the line.
The line below is named line AB, line BA, or line l.
The symbol for line AB is A B l

13. Points, Lines, and Planes
1) Name two points on line m. R T m S
Possible answers:
point R and point S
point R and point T
point S and point T
2) Give three names for the line.
Possible answers:
or line m
NOTE: Any two points on the line or the script letter can be used to name it.

14. Points, Lines, and Planes
Three points may lie on the same line. These points are _______ .
collinear
Points that DO NOT lie on the same line are __________ .
noncollinear R T S U V
1) Name three points that are collinear.
Possible answers:
points R, S, and point T
points U, S, and point V

15. Points, Lines, and Planes
Three points may lie on the same line. These points are _______ .
collinear
Points that DO NOT lie on the same line are __________ .
noncollinear R T S U V
1) Name three points that are noncollinear.
Possible answers:
points R, S, and point V
points R, T, and point U
points R, S, and point U
points R, T, and point V
points R, V, and point U
points S, T, and point V

16. Points, Lines, and Planes
Rays and line segments are parts of lines.
A ___ has a definite starting point and extends without end in one direction.
ray A B
RAY:
The starting point of a ray is called the ________.
endpoint
A ray is named using the endpoint first, then another point on the ray.
The ray above is named ray AB.
The symbol for ray AB is

17. Points, Lines, and Planes
Rays and line segments are parts of lines.
A ___________ has a definite beginning and end.
line segment
LINE SEGMENT:
A line segment is part of a line containing two endpoints and all points between them. A B
A line segment is named using its endpoints.
The line segment above is named segment AB or segment BA.
The symbol for segment AB is

18. Points, Lines, and Planes
1) Name two segments.
Possible Answers: D A B C U
2) Name a ray.
Possible Answers:

19. Points, Lines, and Planes
A _____ is a flat surface that extends without end in all directions.
plane
Points that lie in the same plane are ________.
coplanar
Points that do not lie in the same plane are ___________.
noncoplanar
PLANE:
For any three noncollinear points, there is only one plane that contains all three points. A M
A plane can be named with a single uppercase script letter or by three noncollinear points. B C
The plane at the right is named plane ABC or plane M.

20. Place points A, B, C, D, & E on a piece of paper as shown.
Hands On
Place points A, B, C, D, & E on a piece of paper as shown. B D
Fold the paper so that point A is on the crease. A
Open the paper slightly The two sections of the paper represent different planes. C E
Answers (may be others)
1) Name three points that are coplanar ______________________
A, B, & C
2) Name three points that are noncoplanar ______________________
D, A, & B
3) Name a point that is in both planes ______________________ A

21. Points, Lines, and Planes
End of Lesson

22. r 1) Name three points on line r D E C F
5-Minute Check
1) Name three points on line r D F r E C
D, E, F
2) Give three other names for line r
3) Name two segments that have point F as an endpoint.
4) Name three different rays.
5) Are points C, E, and F collinear or noncollinear?
noncollinear

23. Postulates
What You'll Learn
You will learn to identify and use basic postulates about points, lines, and planes.

24. Geometry is built on statements called _________. postulates
Postulates are statements in geometry that are accepted to be true.
Postulate 1-1: Two points determine a unique ___. Q P
line
There is only one line that contains Points P and Q T l m
Postulate 1-2: If two distinct lines intersect, then their intersection is a ____.
point
Lines l and m intersect at point T
Postulate 1-3: Three noncollinear points determine a unique _____. A B C
plane
There is only one plane that contains points A, B, and C.

25. Postulates
Points A, B, and C are noncollinear. A
1) Name all of the different lines that can be drawn through these points. C B
2) Name the intersection of
Point C

26. Postulates
1) Name all of the planes that are represented in the figure.
There is only one plane that contains three noncollinear points. B A D C
plane ABC (side)
plane ACD (side)
plane ABD (back side)
plane BCD (bottom)

27. Postulates
Postulate 1-4: If two distinct planes intersect, then their intersection is a ___.
line
Plane M and plane N intersect in line DE. M N D E

28. Name the intersection of plane CDG and plane BCD.
Postulates
Name the intersection of plane CDG and plane BCD.
Name two planes that intersect in
planes ADF and CDF H E F G C B A D

29. Postulates
End of Lesson

30. 5-Minute Check
1) At which point or points do three planes intersect?
At each of the points A, B, C, and D.
2) Name the intersection of plane ABC and plane ACD.
3) Are there two planes in the figure that do not intersect? No
4) Name two planes that intersect in
Planes ABD and BCD. B A D C
5) How many points do and have in common?
One, (Point B)

31. Conditional Statements and Their Converses
What You'll Learn
You will learn to write statements in if-then form and write the converse of the statements.
In mathematics, you will come across many _______________.
if-then statements
For Example:
If a number is even,
then it is divisible by two.
If – then statements join two statements based on a condition:
A number is divisible by two only if the number is even.
Therefore, if – then statements are also called __________ __________ .
conditional statements

32. Conditional Statements and Their Converses
Conditional statements have two parts.
The part following if is the _________ .
hypothesis
The part following then is the _________ .
conclusion
If a number is even, then the number is divisible by two.
a number is even
the number is divisible by two.
Hypothesis:
Conclusion:

33. Conditional Statements and Their Converses Conditional Statement
How do you determine whether a conditional statement is true or false?
Conditional Statement
True or False
Why?
If it is the 4th of July (in the U.S.), then it is a holiday.
True
The statement is true because the conclusion follows from the hypothesis.
If an animal lives in the water, then it is a fish.
False
You can show that the statement is false by giving one counterexample.
Whales live in water, but whales are mammals, not fish.

34. Conditional Statements and Their Converses
There are different ways to express a conditional statement. The following statements all have the same meaning.
If you are a member of Congress, then you are a U.S. citizen.
All members of Congress are U.S. citizens.
You are a U.S. citizen if you are a member of Congress.
You write two other forms of this statement: “If two lines are parallel, then they never intersect.”
Possible answers:
All parallel lines never intersect.
Lines never intersect if they are parallel.

35. Conditional Statements and Their Converses
The ________ of a conditional statement is formed by exchanging the hypothesis and the conclusion.
converse
Conditional: If a figure is a triangle, then it has three angles.
a figure is a triangle
it has three angles
Converse: If _______________, then ________________.
NOTE: You often have to change the wording slightly so that the converse reads smoothly.
Converse: If the figure has three angles, then it is a triangle.

36. Conditional Statements and Their Converses
Write the converse of the following statements. State whether the converse is TRUE or FALSE.
If FALSE, give a counterexample:
“If you are at least 16 years old, then you can get a driver’s license.”
If ________________________, then _______________________.
you can get a driver’s license
you are at least 16 years old
TRUE!
“If today is Saturday, then there is no school.
FALSE!
If _______________, then ______________.
there is no school
today is Saturday
We don’t have school on New Years day which may fall on a Monday.

37. Conditional Statements and Their Converses
End of Lesson

38. If the power goes out, we will light candles.
5-Minute Check
If the power goes out, we will light candles.
1) Identify the hypothesis and conclusion of the statement.
Hypothesis: the power goes out
Conclusion: we will light candles
2) Write two other forms of the statement.
1) We will light candles if the power goes out.
2) Whenever the power goes out, we will light candles
3) Write the converse of the statement.
If we light candles, then the power has gone out.
4) Is the converse you wrote for # 3 (above) true?
NO! You could light candles for another reason, such as a birthday party.

39. What You'll Learn You will learn to use geometry tools.
Tools of the Trade
What You'll Learn
You will learn to use geometry tools.

40. As you study geometry, you will use some of the basic tools.
Tools of the Trade
As you study geometry, you will use some of the basic tools.
A __________ is an object used to draw a straight line.
straightedge
A credit card, a piece of cardboard, or a ruler can serve as a straightedge.
Determine whether the sides of the triangle are straight.
Place a straightedge along each side of the triangle.

41. A _______ is another useful tool. compass
Tools of the Trade
A _______ is another useful tool.
compass
A common use for a compass is drawing arcs and circles.
(an arc is part of a circle)

42. Use a compass to determine which segment is longer
Tools of the Trade
Use a compass to determine which segment is longer
1) Place the point of the compass on A and adjust the compass so that the pencil is on C.
2) Without changing the setting of the compass, place the point of the compass on B. The pencil point does not reach point D. Therefore, is longer. D C B A
In geometry, you will draw figures using only a compass and a straightedge.
These drawings are called ___________ .
constructions

43. Use a compass and straightedge to construct a six-sided figure.
Tools of the Trade
Use a compass and straightedge to construct a six-sided figure.
4) Use a straightedge to connect the points in order.
3) Move the compass point to the arc and draw another arc along the circle Continue doing this until there are six arcs.
1) Use the compass draw a circle.
2) Using the same compass setting, put the point on the circle and draw a small arc on the circle.

44. Constructing the Midpoint
You will learn to construct the midpoint of a line segment using only a straightedge and compass.
1) On your patty paper, draw two points.
2) Construct a line segment between the points
3) Fold the paper, and place one point on top of the other. This should produce a crease (fold mark) between the points.
4) Place the compass on one of the points and open it to over half way to the other point.
5) Repeat step 4 using the second point.
6) Connect the intersection of the two circles.

45. Tools of the Trade
End of Lesson

46. A Plan for Problem Solving
What You'll Learn
You will learn to solve problems that involve the perimeters and areas of rectangles and parallelograms.
Perimeter is the _____________________.
distance around an object
Perimeter is similar to ____________.
a line segment
Area is the _______________________________________________.
number of square units needed to cover an object’s surface
Area is similar to ______.
a plane

47. A Plan for Problem Solving
In this section you will learn to solve problems that involve the perimeters and areas of rectangles and parallelograms.
Perimeter is the ____________________.
distance around a figure
The perimeter is the ____ of the lengths of the sides of the figure.
sum
The perimeter of the room shown here is:
15 ft
+ 18 ft
+ 6 ft
+ 6 ft
+ 9 ft
+ 12 ft
= 66 ft

48. A Plan for Problem Solving
Some figures have special characteristics. For example, the opposite sides of a rectangle have the same length.
This allows us to use a formula to find the perimeter of a rectangle. (A formula is an equation that shows how certain quantities are related.)
(of a rectangle)

49. A Plan for Problem Solving
Find the perimeter of a rectangle with a length of 17 ft and a width of 8 ft.
17 ft
8 ft
(of a rectangle)
= 2(17 ft) + 2(8 ft)
= 2(17 ft + 8 ft)
= 34 ft + 16 ft
= 2(25 ft)
= 50 ft
= 50 ft

50. A Plan for Problem Solving
Another important measure is area.
The area of a figure is ____________________________________________.
the number of square units needed to cover its surface
The area of the rectangle below can be found by dividing it into 18 unit squares. 3 6
The area of a rectangle can also be found by multiplying the length and the width.

51. A Plan for Problem Solving
The area “A” of a rectangle is the product of the length l and the width w. l w
Find the area of the rectangle
14 in.
10 in.
The area of the rectangle is 140 square inches.
NOTE: units indicate area is being calculated

52. Plan for Problem Solving
Because the opposite sides of a parallelogram have the same length, the area of a parallelogram is closely related to the area of a ________.
rectangle
height
base
The area of a parallelogram is found by multiplying the ____ and the ______.
base
height
Base – the bottom of a geometric figure.
Height – measured from top to bottom, perpendicular to the base.

53. A Plan for Problem Solving
Find the area of the parallelogram:

54. §1.6 A Plan for Problem Solving
End of Lesson

55. §1.5 Tools of the Trade