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Chapter 2 Geometric Reasoning

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1 Chapter 2 Geometric Reasoning
By April Stephens and Jackie Foley

2 2-1 Using Inductive Reasoning to Make Conjectures
Inductive Reasoning- the process of reasoning that a rule or statement is true because specific cases are true Conjecture- a statement you believe to be true based on inductive reasoning Counterexample- an example in which your conjecture is not true You must prove your conjecture to show that it is always true. To prove that it is false, only one counterexample is needed.

3 Examples Which number comes next in the pattern? 1, 2, 4, 8, _
The answer is 16. You can form a conjecture based on the pattern that the numbers double each time. 1x2=2, 2x2=4, and 4x2=8. 8x2=16, therefore the next number is 16.

4 Example 2 See the chart below. Find a counterexample to disprove the conjecture that July is the rainiest month.

5 2-2 Conditional Statements
Conditional Statement- a statement that can be written in the form “if p, then q.” Hypothesis- the part p of a conditional statement following the word if Conclusion- the part q of a conditional statement following the word then Logically Equivalent Statements- related conditional statements that have the same truth value

6 How to Write Statements
If you have the following statement - the midpoint M of a segment bisects the segment - then you can form a conditional statement by identifying the different parts. The hypothesis of the statement is “M is the midpoint of a segment.” The conclusion is “M bisects the segment.” You can rewrite this to make the conditional statement, “If M is the midpoint of a segment, then M bisects the segment.”

7 Definitions Truth Value- whether the statement is true or false
A conditional statement is false only if the hypothesis is true and the conclusion is false Converse- the statement formed by exchanging the hypothesis and conclusion Inverse- the statement formed by negating the hypothesis and the conclusion Contrapositive- the statement formed by both exchanging and negating the hypothesis and conclusion

8 Examples Identify the hypothesis and the conclusion of the following conditional statement. If a person is at least 16 years old, then the person can drive a car. Hypothesis: a person is 16 years old. Conclusion: they can drive a car Give the converse and inverse of the following conditional statement. If a patient is ill, then their heart rate is monitored. Converse- If a patient’s heart rate is monitored, then they are ill. Inverse- if a patient is not ill, then their heart rate is not monitored.

9 2-3 Using Deductive Reasoning to Verify Conjectures
Deductive Reasoning- the process of using logic to draw conclusions from given facts, definitions, and properties To prove a conjecture is true, you must use deductive reasoning. Law of Detachment- If if p then q is a true statement and p is true, then q is true. Law of Syllogism- If if p then q and if q then r are true, then if p then r is true.

10 Examples What can you conclude from the following statements?
At Bell High School, students must take Biology before they take Chemistry. Anthony is in Chemistry. Therefore, Anthony has taken Biology. The sum of all angle measures in a triangle equal 180*. Two of the angles equal 50* and 70*. Therefore, the third angle has a measure of 60*.

11 2-4 Biconditional Statements and Definitions
Biconditional Statement – a statement that can be written in the form p if and only if q This means that if q is untrue, then p cannot be true and vice versa. Definition – a statement that describes a mathematical object and can be written as a true biconditional statement Polygon – a closed plane figure formed by three or more line segments Triangle – three-sided polygon Quadrilateral – four-sided polygon

12 Examples What is the conditional statement and converse within the following biconditional statement? A student is a sophomore if and only if they are in the tenth grade. Conditional statement – If a student is a sophomore, then they are in the tenth grade. Converse – If a student is in the tenth grade, then they are a sophomore.

13 Examples What is the converse and the biconditional statement that can be created from this condtitional statement? If today is Saturday or Sunday, then it is the weekend. Converse – If it is the weekend, then today is Saturday or Sunday. Biconditional – Today is Saturday or Sunday if and only if it is the weekend.

14 2-5 Algebraic Proof Proof – an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true Properties of Equality 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 and c ≠ 0, then a/c = b/c Reflexive – a = a Symmetric – if a = b, then b = a Transitive – if a = b and b = c, then a = c Substitution – if a = b, then b can be substituted for a in any expression

15 Examples Solve the following equations using proofs. 2x + 5 = 3x + 2
First, subtract 2 from each side to get 2x + 3 = 3x. Then subtract 2x from each side to get x = 3. Identify the property that justifies each statement. m<1=m<2, and m<2=m<3, so m<1=m<3. This is using the transitive property.

16 2-6 Geometric Proof Theorem – any statement that you can prove
Two-column proof – a chart in which you list the steps of the proof in the left column and write the matching reason for each step in the right column

17 Theorems Linear Pair Theorem – if two angles form a linear pair, then they are supplementary Congruent Supplements Theorem – if two angles are supplementary to the same angle, then the two angles are congruent Right Angle Congruence Theorem – all right angles are congruent Congruent Complements Theorem – if two angles are complementary to the same angle, then the two angles are congruent

18 Examples Write a two-column proof using the following statements and illustration. Given – X is midpoint of AY, Y is midpoint of XB. Prove – AX=YB By the definition of midpoint, AX=XY, and XY=YB. Using the Transitive Property, we can prove that AX=YB. A X Y B

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