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

14. 2.2: Properties from Algebra p. 37
Pre-AP Geometry 1 Unit 2
2.2: Properties from Algebra
p. 37
p. 33 in text 14

15. 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 15

16. 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

17. 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.

18. 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

19. Reasoning with Properties from Algebra
Example 2
2(x – 3) = 6x + 6
Given 19

20. 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 20

21. Warmup
With a partner, Complete proof # 11 and 12 on p. 40

22. Lesson 2.3 Pre-AP Geometry
Proving Theorems
Lesson 2.3
Pre-AP Geometry 22

23. Proofs Geometric proof is deductive reasoning at work.
Throughout a deductive proof, the “statements” that are made are specific examples of more general situations, as is explained in the "reasons" column.
Recall, a theorem is a statement that can be proved. 23

24. Vocabulary Definition of a Midpoint
The point that divides, or bisects, a segment into two congruent segments.
If M is the midpoint of AB, then AM is congruent to MB
Bisect
To divide into two congruent parts.
Segment Bisector
A segment, line, or plane that intersects a segment at its midpoint. 24

25. Midpoint Theorem
If M is the midpoint of AB, then AM = ½AB and MB = ½AB 25

26. Proof: Midpoint Formula
Given: M is the midpoint of Segment AB
Prove: AM = ½AB; MB = ½AB
Statement
1. M is the midpoints of segment AB
2. Segment AM= Segment MB,
or AM = MB
3. AM + MB = AB
4. AM + AM = AB, or 2AM = AB
5. AM = ½AB 
6. MB = ½AB
Reason
1. Given
2. Definition of midpoint
3. Segment Addition Postulate
4. Substitution Property
(Steps 2 and 3)
5. Division Prop. of  Equality
6. Substitution Property.
(Steps 2 and 5) 26

27. The Midpoint Formula
The Midpoint Formula If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the midpoint of segment AB has coordinates: 27

28. The Midpoint Formula Application:
Find the midpoint of the segment defined by the points A(5, 4) and B(-3, 2). 28

29. Midpoint Formula Application:
Find the coordinates of the other endpoint B(x, y) of a segment with endpoint C(3, 0) and midpoint M(3, 4). 29

30. Vocabulary Definition of an Angle Bisector
A ray that divides an angle into two adjacent angles that are congruent.
If Ray BD bisects angle ABC, then ABD is congruent to DBC 30

31. Angle Bisector Theorem
If BX is the bisector of ∠ABC, then the measure of ∠ABX is one half the measure of ∠ABC and the measure of ∠XBC one half of the ∠ABC. A X C B 31

32. Proof: Angle Bisector Theorem
Given: BX is the bisector of ∠ABC.
Prove: m ∠ABX = ½ m ∠ABC; m ∠XBC = ½m ∠ABC
Statement
Reason
1. BX is the bisector of ∠ABC
1. Given
2. m∠ABX + m∠XBC = m∠ABC
2. Angle addition postulate
3. m∠ ABX = m∠ XBC
3. Definition of bisector of an angle
4. m∠ ABX + m∠ ABX = m∠ ABC; 2 m∠ ABX = m∠ ABC
4. Substitution property
5. m∠ ABX = ½ m∠ ABC;
m∠ XBC = ½ m∠ ABC
5. Division property 32

33. Reasons used in proofs
Given
Definitions
Postulates
Theorems

34. 2.4: Special Pairs of Angles
Page 50
Pre-AP Geometry 1

35. Angle Pair Relationships
Complementary Angles
Two angles that have a sum of 90º
Each angle is a complement of the other.
Non-adjacent complementary Adjacent angles complementary angles

36. Angle Pair Relationships
Supplementary Angles
Two angles that have a sum of 180º
Each angle is a supplement of the other.

37. Angle Pair Relationships
Example 1
Given that G is a supplement of H and mG is 82°, find mH.
Given that U is a complement of V, and mU is 73°, find mV.

38. Angle Pair Relationships
Example 2
T and S are supplementary.
The measure of T is half the measure of S. Find mS. 38

39. Angle Pair Relationships
Example 3
D and E are complements and D and F are supplements. If mE is four times mD, find the measure of each of the three angles.

40. Theorem 2-3 Vertical angles are congruent
Given: angle 1 and angle 2 are vertical angles
Prove∠1≅ ∠2 3 2 1
Statement
Reasons 1. 2. 3. 4.

41. Angle pair relationships
Find x and the measure of each angle. ∠A
32°
2x + 10

42. 2.5: Perpendicular Lines
Page 56
Pre-AP Geometry 1

43. Perpendicular lines Two lines that intersect to form right angles
We use the symbol ⊥ to show that lines are perpendicular. Line AB ⊥ Line CD C A B D

44. Perpendicular lines theorems
Theorem 2-4: If two lines are perpendicular, then they form congruent adjacent angles
Theorem 2-5: If two lines form congruent adjacent angles, then the lines are perpendicular
Theorem 2-6: If the exterior sides of two adjacent angles are perpendicular, then the angles are complementary.

45. p. 60 Pre-AP Geometry 1 September 11, 2008
Unit 2.6: Planning a proof
p. 60
Pre-AP Geometry 1
September 11, 2008

46. Parts of a proof Statement of the theorem you are trying to prove
A diagram to illustrate given information
A list of the given information
A list of what you are trying to prove
A series of Statements and Reasons that lead from the given information to what you are trying to prove.

47. Example proof of theorem 2-7
If 2 angles are supplements of congruent angles, then the two angles are congruent.
Given: ∠2 ≅ ∠4
∠1 and ∠2 are supplementary
∠3 and ∠4 are supplementary
Prove: ∠1 ≅ ∠3 1 2 3 4
Statement
Reason
1. ∠1 and ∠2 are supplementary
∠3 and ∠4 are supplementary
1. Given
2. m ∠1 +m ∠2 =180; m ∠3 + m∠4 =180
2. Definition of supp. ∠’s
3. m ∠1 +m ∠2 = m ∠3 + m∠4
3. Substitution property
4. ∠2 ≅ ∠4
4.given
5. ∠1 ≅ ∠3
5. Subtraction property of equality

48. Theorem 2-8:
If two angles are complements of congruent angles, then the two angles are congruent.
Prove theorem 2-8. Use the proof from theorem 2-7 (p. 61) to help. You may do this with a partner. Due at end of hour. Make sure you include all 5 parts (p. 60).