1. Unit 2: Reasoning and Proof
Geometry 1
Unit 2: Reasoning and Proof

2. 2.1 Conditional Statements
Geometry 1 Unit 2
2.1 Conditional Statements

3. Conditional Statements
A statement with two parts
If-then form
A way of writing a conditional statement that clearly showcases the hypothesis and conclusion
Hypothesis-
The “if” part of a conditional Statement
Conclusion
The “then” part of a conditional Statement

4. Conditional Statements
Examples of Conditional Statements
If today is Saturday, then tomorrow is Sunday.
If it’s a triangle, then it has a right angle.
If x2 = 4, then x = 2.
If you clean your room, then you can go to the mall.
If p, then q.
The first statement is true.
The second statement is false, triangles do not have to have a right angle. This was put together with a subject (a triangle) and predicate (it has a right angle)
The third statement is false, x could equal -2.

5. Conditional Statements
Example 1
Circle the hypothesis and underline the conclusion in each conditional statement
If you are in Geometry 1, then you will learn about the building blocks of geometry
If two points lie on the same line, then they are collinear
If a figure is a plane, then it is defined by 3 distinct points
Statements do not have to be true. The last one is clearly false.

6. Conditional Statements
Example 2
Rewrite each statement in if…then form
A line contains at least two points
When two planes intersect their intersection is a line
Two angles that add to 90° are complementary
If a figure is a line, then it contains at least two points
If two planes intersect, then their intersection is a line.
If two angles add to equal 90°, then they are complementary.

7. Conditional Statements
Counterexample
An example that proves that a given statement is false
Write a counterexample
If x2 = 9, then x = 3

8. Conditional Statements
Example 3
Determine if the following statements are true or false.
If false, give a counterexample.
If x + 1 = 0, then x = -1
If a polygon has six sides, then it is a decagon.
If the angles are a linear pair, then the sum of the measure of the angles is 90º.

9. Conditional Statements
Negation
In most cases you can form the negation of a statement by either adding or deleting the word “not”.
In most cases you can form the negation of a statement by either adding or deleting the word “not”.

10. Conditional Statements
Examples of Negations
Statement:
Negation :
Statement: Mr. Ross is not more than 6 feet tall.
Negation: Mr. Ross is more than 6 feet tall
I am doing my homework.
Negation:

11. Conditional Statements
Example 4
Write the negation of each statement. Determine whether your new statement is true or false.
Stanfield is the largest city in Arizona.
All triangles have three sides.
Dairy cows are not purple.
Some VGHS students have brown hair.

12. Conditional Statements
Converse
Formed by switching the if and the then part.
Original
If you like green, then you will love my new shirt.
If you love my new shirt, then you like green.

13. Conditional Statements
Inverse
Formed by negating both the if and the then part.
Original
If you like green, then you will love my new shirt.
If you do not like green, then you will not love my new shirt.

14. Conditional Statements
Contrapositive
Formed by switching and negating both the if and then part.
Original
If you like green, then you will love my new shirt.
If you do not love my new shirt, then you do not like green.

15. Conditional Statements
Write in if…then form.
Write the converse, inverse and contrapositive of each statement.
I will wash the dishes, if you dry them.
A square with side length 2 cm has an area of 4 cm2.

16. Conditional Statements
Point-line postulate:
Through any two points, there exists exactly one line
Point-line converse:
A line contains at least two points
Intersecting lines postulate:
If two lines intersect, then their intersection is exactly one point

17. Conditional Statements
Point-plane postulate:
Through any three noncollinear points there exists one plane
Point-plane converse:
A plane contains at least three noncollinear points
Line-plane postulate:
If two points lie in a plane, then the line containing them lies in the plane
Intersecting planes postulate:
If two planes intersect, then their intersection is a line

18. 2.2: Definitions and Biconditional Statements
Geometry 1 Unit 2
2.2: Definitions and Biconditional Statements

19. Definitions and Biconditional Statements
Can be rewritten with “If and only if”
Only occurs when the statement and the converse of the statement are both true.
A biconditional can be split into a conditional and its converse.

20. Definitions and Biconditional Statements
Example 1
An angle is right if and only if its measure is 90º
A number is even if and only if it is divisible by two.
A point on a segment is the midpoint of the segment if and only if it bisects the segment.
You attend school if and only if it is a weekday.
Conditional: If an angle is a right angle, then its measure is 90º.
Converse: If an angle’s measure is 90º, then it’s a right angle.
Conditional: If a number is even, then it is divisible by two.
Converse: If a number is divisible by two, then it is even.
Conditional: If a point on a segment is the midpoint of the segment, then it bisects the segment.
Converse: If a point on a segment bisects the segment, then it is the midpoint of the segment.

21. Definitions and Biconditional Statements
Perpendicular lines
Two lines are perpendicular if they intersect to form a right angle
A line perpendicular to a plane
A line that intersects the plane in a point and is perpendicular to every line in the plane that intersects it
The symbol is read, “is perpendicular to.

22. Definitions and Biconditional Statements
Example 2
Write the definition of perpendicular as biconditional statement.

23. Definitions and Biconditional Statements
Example 3
Give a counterexample that demonstrates that the converse is false.
If two lines are perpendicular, then they intersect.

24. Definitions and Biconditional Statements
Example 4
The following statement is true. Write the converse and decide if it is true or false. If the converse is true, combine it with its original to form a biconditional.
If x2 = 4, then x = 2 or x = -2

25. Definitions and Biconditional Statements
Example 5
Consider the statement
x2 < 49 if and only if x < 7.
Is this a biconditional?
Is the statement true?
Yes No

26. Geometry 1 Unit 2
2.3 Deductive Reasoning

27. Deductive Reasoning Symbolic Logic
Statements are replaced with variables, such as p, q, r.
Symbols are used to connect the statements.

28. Deductive Reasoning Symbol Meaning ~ not Λ and V or → if…then
↔ if and only if

29. Deductive Reasoning Example 1
Let p be “the sum of the measure of two angles is 180º” and
Let q be “two angles are supplementary”.
What does p → q mean?
What does q → p mean?

30. Deductive Reasoning Example 2 p: Jen cleaned her room.
q: Jen is going to the mall.
What does p → q mean?
What does q → p mean?
What does ~q mean?
What does p Λ q mean?

31. Deductive Reasoning Example 3
Given t and s, determine the meaning of the statements below.
t: Jeff has a math test today
s: Jeff studied
t V s
s → t
~s → t

32. Deductive Reasoning Deductive Reasoning
Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical argument.

33. Deductive Reasoning Law of Detachment Given: p → q Given: p
When you have a true conditional statement and you know the hypothesis is true, you can conclude the conclusion is true.
Given: p → q
Given: p
Conclusion: q

34. Deductive Reasoning Example 4 Determine if the argument is valid.
If Jasmyn studies then she will ace her test.
Jasmyn studied.
Jasmyn aced her test.
Valid

35. Deductive Reasoning Example 5 Determine if the argument is valid.
If Mike goes to work, then he will get home late.
Mike got home late.
Mike went to work
Invalid

36. Deductive Reasoning Law of Syllogism Given: p → q Given: q → r
Given two linked conditional statements you can form one conditional statement.
Given: p → q
Given: q → r
Conclusion: p → r

37. Deductive Reasoning Example 6 Determine if the argument is valid.
If today is your birthday, then Joe will bake a cake.
If Joe bakes a cake, then everyone will celebrate.
If today is your birthday, then everyone will celebrate.
valid

38. Deductive Reasoning Example 7 Determine if the argument is valid.
If it is a square, then it has four sides.
If it has four sides, then it is a quadrilateral.
If it is a square, then it is a quadrilateral.

39. 2.4 Reasoning with Properties from Algebra
Geometry 1 Unit 2
2.4 Reasoning with Properties from Algebra 39

40. Reasoning with Properties from Algebra
Objectives
Review of algebraic properties
Reasoning
Applications of properties in real life

41. Reasoning with Properties from Algebra
Addition property
If a = b, then a + c = b + c
Subtraction property
If a = b, then a – c = b – c
Multiplication property
If a = b, then ac = bc
Division property
If a = b, then

42. Reasoning with Properties from Algebra
Reflexive property
For any real number a, a = a
Symmetric property
If a=b, then b = a
Transitive Property
If a = b and b = c, then a = c
Substitution property
If a = b, then a can be substituted for b in any equation or expression
Distributive property
2(x + y) = 2x + 2y

43. Reasoning with Properties from Algebra
Example 1
Solve 6x – 5 = 2x + 3 and write a reason for each step
Statement
Reason
6x – 5 = 2x + 3
Given
4x – 5 = 3
4x = 8
x = 2
Subtraction property of equality
Addition property of equality
Division property of equality

44. Reasoning with Properties from Algebra
Example 2
2(x – 3) = 6x + 6
Given

45. Reasoning with Properties from Algebra
Determine if the equations are valid or invalid.
(x + 2)(x + 2) = x2 + 4
x3x3 = x6
-(x + y) = x – y
Invalid.
Valid
Invalid

46. Reasoning with Properties from Algebra
Geometric Properties of Equality
Reflexive property of equality
For any segment AB, AB = AB
Symmetric property of equality
If then
Transitive property of equality
If AB = CD and CD = EF, then, AB = EF

47. Reasoning with Properties from Algebra
Example 3
In the diagram, AB = CD. Show that AC = BD
A B
C D
Statement
Reason
AB = CD
AB + BC = BC + CD
AC = AB + BC
BD = BC + CD
AC = BD
1. Given
2. Addition property of equality
3. Segment addition postulate
4. Segment addition postulate
5. Substitution property of Equality

48. In the diagram, ABC = DBF. Show that  ABD = CBF
A C D B F
In the diagram, ABC = DBF. Show that  ABD = CBF
Statement
Reason

49. 2.5: Proving Statements about Segments
Geometry 1 Unit 2
2.5: Proving Statements about Segments

50. Proving Statements about Segments
Key Terms:
2-column proof
A way of proving a statement using a numbered column of statements and a numbered column of reasons for the statements
Theorem
A true statement that is proven by other true statements

51. Proving Statements about Segments
Properties of Segment Congruence
Reflexive
For any segment AB,
Symmetric
If , then
Transitive
If and ,then

52. Proving Statements about SegmentsJ K L
Example 1
In triangle JKL, Given: LK = 5, JK = 5, JK = JL Prove: LK = JL
Statement
Reason 1.
1. Given 2.
2. Given 3.
3. Transitive property of equality 4. 5.
5. Given 6.
6. Transitive property of congruence

53. Proving Statements about Segments
Duplicating a Segment
Tools
Straight edge: Ruler or piece of wood or metal used for creating straight lines
Compass: Tool used to create arcs and circles
A B
C D
Steps
Use a straight edge to draw a segment longer than segment AB
Label point C on new segment
Set compass at length of segment AB
Place compass point at C and strike an arc on line segment
Label intersection of arc and segment point D
Segment CD is now congruent to segment AB

54. 2.6: Proving Statements about Angles
Geometry 1 Unit 2
2.6: Proving Statements about Angles

55. Proving Statements about Angles
Properties of Angle Congruence
Reflexive
For any angle A,
Symmetric
Transitive

56. Proving Statements about Angles
Right Angle Congruence Theorem
All right angles are congruent.

57. Proving Statements about Angles
Congruent Supplements Theorem
If two angles are supplementary to the same angle, then they are congruent. 1 2 3

58. Proving Statements about Angles
Congruent Complements Theorem
If two angles are complementary to the same angle, then the two angles are congruent. 4 5 6

59. Proving Statements about Angles
Linear Pair Postulate
If two angles form a linear pair, then they are supplementary. 1 2

60. Proving Statements about Angles
Vertical Angles Theorem
Vertical angles are congruent 1 2 3 4

61. Proving Statements about Angles
Example 1
Given:
Prove: A 1 2 4 3 C B
Teacher edition page 110 example 1
Statement
Reason 1. 2. 3. 4.

62. Proving Statements about Angles
Example 2
Given:
Prove: 1 2 4 3
Teacher edition page 110 example 2
Statement
Reason 1. 2. 3. 4.

63. Proving Statements about Angles
Example 3
Given: are right angles
Prove: A D C B
Teacher edition page 110 example 3
Statement
Reason 1. 2. 3. 4.

64. Proving Statements about Angles
Example 4
Given:
m1 = 24º,
m3 = 24º
1 and 2 are complementary
3 and 4 are complementary
Prove: 2  4
Statement
Reason 1. 2. 3. 4.
Teacher edition page 111 example 4 1 2 3 4

65. Proving Statements about Angles
Example 5
In the diagram m1 = 60º and BFD is right. Explain how to show m4 = 30º.
Teacher edition page 111 example 5 1 2 3 4 A F E D C B

66. Proving Statements about Angles
Example 6
Given: 1 and 2 are a linear pair, 2 and 3 are a linear pair
Prove: 1  3
Statement
Reason 1. 2. 3.
Teacher edition page 111 example 6 1 2 3