Reasoning and Proof in Geometry

Sect 2-1 Conditional Statements Conditional statement- a statement that has two parts, a hypothesis and a conclusion. Ex. All sharks have a boneless skeleton. If-then form of a conditional statement- the “if” part contains the hypothesis and the “then” part contains the conclusion. Ex. If a fish is a shark, then it has a boneless skeleton. Conditional statements may be true or false. A single counterexample shows the statement is not always true. Statement: If today is Monday, then tomorrow is Tuesday. Converse of a conditional is formed by switching the hypothesis and the conclusion. Ex. Converse: If tomorrow is Tuesday, then today is Monday. Negation- writing the negative of a statement Inverse of a conditional is formed by negating the hypothesis and the conclusion of the statement. Ex. Inverse: If today is not Monday, then tomorrow is not Tuesday. Contrapositive of a conditional is formed by negating the hypothesis and conclusion of the converse of a conditional statement. Ex. Contrapositive : If tomorrow is not Tuesday, then today is not Monday. When two statements are both true or both false, they are called logically equivalent statements. A conditional and its contrapositive are logically equivalent as are the converse and the inverse. Postulate 5: Through any two points there exists exactly one line. Postulate 6: A line contains at least two points. Postulate 7: If two lines intersect, then their intersection is exactly one point. Postulate 8: Through any three noncollinear points there exists exactly one plane. Postulate 9: A plane contains at least three noncollinear points. Postulate 10: If two points lie in a plane, then the line containing them lies in the plane. Postulate 11: If two planes intersect, then their intersection is a line. Sect 2-2 DEFINITIONS AND BICONDITIONAL STATEMENTS What is a good definition? A definition uses known words to describe a new word. perpendicular lines- two lines that intersect to form a right angle. All definitions in geometry can be interpreted “forward” and “backward.” The conditional and the converse are both true in a good definition. EX: Congruent angles have equal measures. Conditional: If two angles are congruent, then their measures are equal. Converse: If two angles have equal measures, then they are congruent. Is the following a good definition? A square has four right angles. Biconditional statement: a statement that contains the phrase “if and only if.” Writing a biconditional is equivalent to writing a conditional and its converse. A biconditional statement can be either true of false. To be true, both the conditional and the converse must be true. All definitions can be written as true biconditional statements. An angle is a right angle if and only if it measures 90. EX: Rewrite the biconditional statement as a conditional and its converse. Tom lives in Cincinnati if and only if he lives in Ohio. True or false? Geometry Sec. 2.3 Deductive Reasoning “p” is the hypothesis “q” is the conclusion 🡺 is “then” p🡺q is read “if p, then q” The converse is q🡺p p🡸🡺q is read “p if and only if q” or “p iff q” “~” is not We use ~ to negate a hypothesis and/or a conclusion The inverse would be ~p🡺~q The contrapositive would be ~q🡺~p Logical Argument – To use facts, definitions, and accepted properties (otherwise known as deductive reasoning) in a logical order Law of Detachment – If p🡺q is true and p is true, then q must be true Law of Syllogism – If p🡺q is true and q🡺r is true, then p🡺r is true Examples : let p be “x is an even number” and let q be “x2 is even” 1. Write p🡺q in words 2. Write q🡺p in words 3. Write ~p🡺~q in words 4. Write ~q🡺~p in words Is the argument valid 5. Jamal knows that if he misses the practice the day before the game, then he will not be a starting player in the game. Jamal misses practice on Tuesday so he concludes he will not start on Wednesday. 6. If two angles form a linear pair, then they are supplementary; angle 1 and angle 2 are a linear pair. Geometry Sec. 2.4 – Reasoning With Properties From Algebra Algebraic 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 and c≠0, then a/c = b/c. Reflexive Property – For any real number a, a=a. For any segment AB = AB. For any angle m∠1 = m∠1. Symmetric Property – For any real numbers a and b: if a=b, then b=a. If AB = BC, then BC = AB. If m∠1 = m∠2, then m∠2 = m∠1. Transitive Property – If a=b and b=c, then a=c. If AB = BC and BC = CD, then AB = CD. If m∠1 = m∠2 and m∠2 = m∠3, then m∠1 = m∠3. Substitution Property – If a=b, then a can replace b in any equation. Examples : 1. Solve and write the reasons for each step. 5x – 18 = 3x + 2 2. Solve and write the reasons for each step. 55z – 3(9z + 12) = -64 3. Solve for r and write each step. a = 220 – (10/7)r 4. Solve if a = 16 and write each step. 5. In the diagram AB = CD. Show AC = BD. 6. m∠1 + m∠2 = 66 m∠1 + m∠2 + m∠3 = 99 m∠3 = m∠1 m∠1 = m∠4 Find the m∠4 and write the reasons for each step. Geometry Sec. 2.5 Proving Statements About Segments Definitions : Theorem – A true statement that follows as a result of other true statements. A theorem needs to be proved. Two Column Proof – Has numbered steps on the left and corresponding reasons on the right. Paragraph Proof – A proof that can be written in paragraph form. Thm. 2.1 Properties of Segment Congruence Reflexive : For any segment AB, AB ≅ AB Symmetric : If AB ≅ CD, then CD ≅ AB. Transitive : If AB ≅ CD and CD ≅ FG, then AB ≅ FG. Examples : 1. Given : PQ ≅ XY Prove : XY ≅ PQ 2. Given : LK = 5, JK = 5, JK ≅ JL Prove : LK ≅ JL 3. Given : Q is the midpoint of PR. Prove : PQ = ½ PR and QR = ½ PR Sec. 2.6 – Proving Statements About Angles Theorem 2.2 – Properties of Angle Congruence Reflexive – For any angle, ∠A ≅ ∠A. Symmetric – If ∠A ≅ ∠B, then ∠B ≅ ∠A. Transitive – If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C. Theorem 2.3 – Right Angle Congruence Theorem All right angles are congruent. Theorem 2.4 – Congruent Supplements Theorem If two angles are supplementary to the same angle or congruent angles, then they are congruent. Theorem 2.5 – Congruent Complements Theorem If angles are complementary to the same angle or congruent angles, then they are congruent. Postulate – Linear Pair Postulate If two angles form a linear pair, then they are supplementary. Theorem 2.6 – Vertical Angle Theorem Vertical angles are congruent. Examples : 1. 2. 3.