1. Reasoning and Proofs
Chapter 2

2. Conditional Statements
I can write conditional and bi-conditional statements.

3. Conditional Statements
Vocabulary (page 35 in Student Journal)
conditional: an if-then statement
hypothesis: the part of the statement following if
conclusion: the part of the statement following then
negation: the opposite of a statement
converse: exchange the hypothesis and the conclusion of a conditional statement

4. Conditional Statements
inverse: negate both the hypothesis and the conclusion of a conditional statement
contrapositive: negate both the hypothesis and the conclusion of the converse statement
equivalent statements: statements with the same truth value
biconditional: a single statement that combines a true conditional statement with its true converse, which uses the phrase if and only if
truth value: determining if the conditional statement is true or false

5. Conditional Statements
Example (page 38 in Student Journal)
#3) Let p be “Quadrilateral ABCD is a rectangle” and let q be “the sum of the angle measures is 360 degrees.” Write conditional, converse, inverse, and contrapositive statements, then decide if each statement is true or false.

6. Conditional Statements
Solution
Conditional: If quadrilateral ABCD is a rectangle, then the sum of its angle measures is 360 degrees. (true)
Converse: If the sum of the angle measures is 360 degrees, then quadrilateral ABCD is a rectangle. (false)
Inverse: If quadrilateral ABCD is not a rectangle, then the sum of its angle measures is not 360 degrees. (false)
Contrapositive: If the sum of the angle measures is not 360 degrees, then quadrilateral ABCD is not a rectangle. (true)

7. Conditional Statements
Additional Example (space on pages 36 and 37 in Student Journal)
Write the following statement as a conditional statement.
a) dolphins are mammals

8. Conditional Statements
Solution
a) If an animal is a dolphin, then it is a mammal.

9. Conditional Statements
Additional Examples (space on pages 36 and 37 in Student Journal)
Determine the truth value of the conditional statement.
Note: If we can find one example (counterexample) for which the hypothesis is true but the conclusion is false then the truth value of the conditional statement is false.
b) If you live in a country that borders the United States, then you live in Canada.
c) If 2 angles form a linear pair, then they are supplementary.

10. Conditional Statements
Solutions
b) false, Mexico borders the United States as well
c) true, for angles to form a linear pair they must be supplementary

11. Conditional Statements
Additional Example (space on pages 36 and 37 in Student Journal)
Determine if the converse of the conditional below is true and write a biconditional statement if possible.
Note: We can write a biconditional statement by joining the 2 parts of the conditional statement (hypothesis and conclusion) with the phrase if and only if.
d) If 2 angles have equal measures, then the angles are congruent.

12. Conditional Statements
Solution
d) converse: If 2 angles are congruent, then the angles have equal measures. (true)
biconditional: Two angles have equal measure if and only if the angles are congruent.

13. Conditional Statements
Additional Example (space on pages 36 and 37 in Student Journal)
Write the biconditional statement below as 2 conditional statements that are converses.
e) Two numbers are reciprocals if and only if their product is 1.

14. Conditional Statements
Solution
e) If 2 numbers are reciprocals, then their product is 1.
If the product of 2 numbers is 1, then the numbers are reciprocals.

15. Inductive and Deductive Reasoning
I can use reasoning to solve problems.

16. Inductive and Deductive Reasoning
Vocabulary (page 41 in Student Journal)
conjecture: an unproven statement based on observation
inductive reasoning: using a pattern for a specific situation to then make a conjecture for a general situation
counterexample: a specific case that shows a conjecture is false
deductive reasoning: the process of reasoning logically from given statements or facts to a conclusion

17. Inductive and Deductive Reasoning
Core Concepts (page 42 in Student Journal)
Law of Detachment
If the hypothesis of a true conditional statement is true, then the conclusion is true.
Law of Syllogism
Allows us to state a conclusion from 2 true conditional statements when the conclusion of 1 statement is the hypothesis of another statement.

18. Inductive and Deductive Reasoning
Examples (page 42 and 43 in Student Journal)
Describe the pattern. The find the next 2 terms in the sequence.
#1) 20, 19, 17, 14, 10, …
Make a conjecture about the given quantity.
#5) the sum of 2 negative integers

19. Inductive and Deductive Reasoning
Solutions
#1) down 1, down 2, down 3, down 4, …, 5, -1
#5) is always a negative integer

20. Inductive and Deductive Reasoning
Example (page 43 in Student Journal)
Find a counterexample to show the conjecture is false.
#8) Line k intersects plane P at point Q on the plane. Plane P is perpendicular to line k.

21. Inductive and Deductive Reasoning
Solution
#8) line k can intersect plane P at any angle

22. Inductive and Deductive Reasoning
Examples (page 43 in Student Journal)
Use the Law of Detachment, if possible, to determine what you can conclude.
#9) If a triangle has equal side lengths, then each interior angle measure is 60 degrees. Triangle ABC has equal side lengths.
Use the Law of Syllogism to write a new conditional statement, if possible.
#12) If x > 1, the 3x > 3. If 3x > 3, then (3x)2 > 9.

23. Inductive and Deductive Reasoning
Solutions
#9) Triangle ABC has each interior angle measure of 60 degrees.
#12) If x > 1, the (3x)2 > 9.

24. Inductive and Deductive Reasoning
Additional Example (space on pages 41 and 42 in Student Journal)
What can you conclude from the given information?
Note: In some cases we can use both the Law of Detachment and the Law of Syllogism to form a conclusion.
a) If a river is more than 4000 miles long, then it is longer than the Amazon. If a river is longer than the Amazon, then it is the longest river in the world. The Nile River is 4132 miles long.

25. Inductive and Deductive Reasoning
Solution
a) the Nile River is the longest river in the world

26. Postulates and Diagrams
I can identify postulates using diagrams and sketch and interpret diagrams.

27. Postulates and Diagrams
Core Concepts (pages 46 and 47 in Student Journal)
Two Point Postulate (Postulate 2.1)
Through any 2 points, there exists exactly 1 line.
Line-Point Postulate (Postulate 2.2)
A line contains at least 2 points.
Line Intersection Postulate (Postulate 2.3)
If 2 lines intersect, then their intersection is exactly 1 point

28. Postulates and Diagrams
Three Point Postulate (Postulate 2.4)
Through any 3 noncollinear points, there exists exactly 1 plane.
Plane-Point Postulate (Postulate 2.5)
A plane contains at least 3 noncollinear points.
Plane-Line Postulate (Postulate 2.6)
If 2 points lie in a plane, then the line containing them lies in the plane.
Plane Intersection Postulate (Postulate 2.7)
If 2 planes intersect, then their intersection is a line.

29. Postulates and Diagrams
Examples (pages 47 and 48 in Student Journal)
State the postulate illustrated in the diagram.
#2)
Use the diagram to give an
example of the postulate.
#6) Plane Intersection Postulate

30. Postulates and Diagrams
Solutions
#2) Plane-Line Postulate
#6) Plane P and Plane Q intersect in line k

31. Postulates and Diagrams
Examples (page 48 in Student Journal)
Sketch a diagram of the description.
#7) ray RS bisects segment KL at point R
Use the diagram to determine if
you can assume the statement.
#10) Points C and D are collinear.

32. Postulates and Diagrams
Solutions
#7) sample answer:
#10) yes

33. Algebraic Reasoning
I can use algebraic properties to to help solve equations.

34. Algebraic Reasoning Core Concepts (pages 51 and 52 in Student Journal)
Properties of Equality (for real numbers a, b and c)
Addition: if a = b, then a + c = b + c
Subtraction: if a = b, then a - c = b - c
Multiplication: if a = b, then ac = bc
Division: if a = b, then a/c = b/c
Substitution: if a = b, then b can replace a
Distributive: a(b + c) = ab + ac
Reflexive: a = a
Symmetric: if a = b, then b = a
Transitive: if a = b and b = c, then a = c

35. Algebraic Reasoning Example (page 53 in Student Journal)
Solve the equation. Justify each step.
Note: We can use deductive reasoning when solving an equation by justifying each step with a postulate, property or definition.
#3) -5(2u + 10) = 2(u – 7)

36. Algebraic Reasoning Solution #3) -5(2u + 10) = 2(u – 7) given
-10u – 50 = 2u – 14 Distributive Property
-50 = 12u – 14 Addition Property
-36 = 12u Addition Property
-3 = u Division Property
u = Symmetric Property

37. Proving Statements about Segments and Angles
I can prove mathematical statements through two-column proofs.

38. Proving Statements about Segments and Angles
Vocabulary (page 56 in Student Journal)
proof: an argument that uses deductive reasoning to logically show why a conjecture is true
two-column proof: lists each statement in the left column and the justification on the right
theorem: a conjecture or statement that can be proven true

39. Proving Statements about Segments and Angles
Core Concepts (page 56 in Student Journal)
Properties of Segment Congruence (Theorem 2.1)
Reflexive: segment AB is congruent to segment AB
Symmetric: if segment AB is congruent to segment CD, then segment CD is congruent to segment AB
Transitive: if segment AB is congruent to segment CD and segment CD is congruent to segment EF, then segment AB is congruent to segment EF
These theorems also apply to angles (Theorem 2.2).

40. Proving Statements about Segments and Angles
Example (page 58 in Student Journal)
Complete the proof.
#2)

41. Proving Statements about Segments and Angles
Solutions
#2) 2. m<AEB + m<BEC = 90 degrees
3. Angle Addition Postulate
4. Transitive Property of Equality
6. m<AED + 90 degrees = 180 degrees
7. Subtraction Property of Equality

42. Proving Statements about Segments and Angles
Example (page 58 in Student Journal)
Write a 2-column proof.
#5) Given: M is the midpoint of segment RT
Prove: MT = RS + SM

43. Proving Statements about Segments and Angles
Solution
#5) Statements Reasons
1. M is the midpoint of 1. given
segment RT
2. segment RM is congruent
to segment MT 2. def. of midpoint
3. RM = MT 3. def. of congruence
4. RM = RS + SM 4. segment add. post.
5. MT = RS + SM 5. substitution prop.

44. Proving Statements about Segments and Angles
Additional Example (space on pages 56 and 57 in Student Journal)
a) Prove that vertical angles are congruent using a two-column proof (Angle 1 and angle 3 are vertical angles).

45. Proving Statements about Segments and Angles
Solution
a) Statements Reasons
1. angle 1 and angle 3 are
vertical angles given
2. angle 1 and angle 2 are
supplementary, angle 2 and
angle 3 are supplementary 2. def. linear pair
3. measure of angle 1 +
measure of angle 2 = 180,
measure angle 2 +
measure angle 3 = def. suppl.

46. Proving Statements about Segments and Angles
Solution (continued)
Statements (cont.) Reasons (cont.)
4. measure of angle 1 +
measure of angle 2 =
measure of angle 2 +
measure of angle trans. equal.
5. measure angle 1 =
measure angle subtrac. equal.
6. angle 1 is congruent to
angle def. congruent

47. Proving Geometric Relationships
I can use flowcharts and paragraphs to prove geometric relationships.

48. Proving Geometric Relationships
Vocabulary (page 61 in Student Journal)
flowchart proof: uses boxes and arrows to show a flow of a logical argument
paragraph proof: a proof written in sentences in a paragraph

49. Proving Geometric Relationships
Core Concepts (pages 61 and 62 in Student Journal)
Right Angles Congruence Theorem (Theorem 2.3)
All right angles are congruent.
Congruent Supplements Theorem (Theorem 2.4)
If 2 angles are supplementary to the same angle (or congruent angles), then they are congruent.
Congruent Complements Theorem (Theorem 2.5)
If 2 angles are complementary to the same angle (or congruent angles), then they are congruent.

50. Proving Geometric Relationships
Linear Pair Postulate (Postulate 2.8)
If 2 angles form a linear pair, then they are supplementary.
Vertical Angles Congruence Theorem (Theorem 2.6)
Vertical angles are congruent.

51. Proving Geometric Relationships
Example (page 63 in Student Journal)
Complete the flowchart proof.
#1) Given: <ACB and <ACD are
supplementary, <EGF and <ACD are supplementary
Prove: <ACB is congruent to <EGF

52. Proving Geometric Relationships
Solution
#1) m<ACB + m<ACD =
m<EGF + m <ACD
def. supplementary
<EGF and <ACD are
supplementary
def. congruent

53. Proving Geometric Relationships
Additional Example (space on pages 61 and 62 in Student Journal)
a) Prove the Congruent Supplements Theorem using a paragraph proof (Angles 1 and 3 are supplementary and angles 2 and 3 are supplementary).

54. Proving Geometric Relationships
Solution
a) Angles 1 and 3 are supplementary because it is given. The measure of angle 1 + the measure of angle 3 = 180 by the definition of supplementary. Angles 2 and 3 are supplementary because it is given. The measure of angle 2 + the measure of angle 3 = 180 by the definition of supplementary. The measure of angle 1 + the measure of angle 3 = the measure of angle 2 + the measure of angle 3 by the transitive property of equality. The measure of angle 1 = the measure of angle 2 by the subtraction property of equality. Angle 1 is congruent to angle 2 by the definition of congruent.