1. Unit 2: Deductive Reasoning
Pre-AP Geometry 1
Unit 2: Deductive Reasoning 1

2. 2.1 If-then statements, converse, and biconditional statements
Pre-AP Geometry 1 Unit 2
2.1 If-then statements, converse, and biconditional statements
p. 33 in text 2

3. Conditional Statements
A statement with two parts (hypothesis and conclusion)
Also known as Conditionals
If-then form
A way of writing a conditional statement that clearly showcases the hypothesis and conclusion p→q
Hypothesis-
The “if” part of a conditional statement
Represented by the letter “p”
Conclusion
The “then” part of a conditional statement
Represented by the letter “q” 3

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. 4

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. 5

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. 6

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

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. 8

9. 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. 9

10. 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.
p if and only if q
All definitions can be written as biconditional statements 10

11. Example Give the converse of the statement.
If the converse and the statement are both true, then rewrite as a biconditional statement
If it is Thanksgiving, then there is no school.
If an angle measures 90º, then it is a right angle.

12. Quiz- Get out a piece of paper and answer the following questions:
Underline the hypothesis and circle the conclusion. Then, write the converse of the statement. If the converse and the statement are true, rewrite as a biconditional statement. If not, give a counterexample.
1. If a number is divisible by 10, then it is divisible by 5.
2. If today is Friday, then tomorrow is Saturday.
3. If segment DE is congruent to segment EF, then E is the midpoint of segment DF.

13. Assignment
Lesson 2.1
P. 35 #2-30 even

15. 2.2: Properties from Algebra p. 37
Pre-AP Geometry 1 Unit 2
2.2: Properties from Algebra
p. 37
p. 33 in text 15

16. Properties of equality
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 16

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

18. Two-column proof
A way of organizing a proof in which the statements are made in the left column and the reasons (justification) is in the right column
Given: Information that is given as fact in the problem.

19. 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

20. Reasoning with Properties from Algebra
Example 2
2(x – 3) = 6x + 6
Given 20

21. Reasoning with Properties from Algebra
Determine if the equations are valid or invalid, and state which algebraic property is applied
(x + 2)(x + 2) = x2 + 4
x3x3 = x6
-(x + y) = x – y
Invalid.
Valid
Invalid 21

22. Quiz
Complete proof # 11 and # 12 on p. 40
Homework: p. 41 #