Deductive Reasoning Algebraic Properties of Equality • Addition property: If a=b, then a+c = b+c. – Allowed to add same number on both sides • Subtraction property: If a=b, then a-c = b-c. – Allowed to add same number on both sides • Multiplication property: If a=b, then ac = bc. – Allowed to add same number on both sides • Division property: If a=b, and c≠0, then a/c = b/c. – Allowed to add same number on both sides Solve • Solve 5x – 18 = 3x + 2 and explain each step in writing. 5x – 18 = 3x + 2 2x – 18 = 2 Subtraction prop. of =. 2x = 20 Addition prop. of =. x = 10 Division prop. of =. More properties of equality • Reflexive property: For any real number a, a=a. • Symmetric property: If a=b, then b=a. • Transitive property: If a=b and b=c, then a=c. • Substitution property: If a=b, then a can be substituted for b in any equation or expression. Properties of Equality Segment Length Angle Measure Reflexive AB = AB m<A = m<A Symmetric If AB = CD, then If m<A = m<B, then CD = AB. m<B=m<A. Transitive If AB = CD and If m<A = m<B and CD = EF, then m<B=m<C, then AB=EF. m<A=m<C. Challenge • How is the product 4 · 6 related to 52? • How is the product 5 · 7 related to 62? • Make a conjecture about how the product of two positive inters n and n + 2 is related to the square of the integer between them. • Write a convincing argument to justify your conjecture. Conditional Statements Warmup • State whether each sentence is true or false. – If you live in Los Angeles, then you live in California. True – If you live in California, then you live in Los Angeles. False – If today is Wednesday, then tomorrow is Thursday. True – If tomorrow is Thursday, then today is Wednesday. True Conditional Statement • Conditional statement has two parts, hypothesis and a conclusion. • If _____________, then____________. hypothesis conclusion – Hypothesis is after “if” and the conclusion is after “then” Rewrite in If-Then form • A number divisible by 9 is also divisible by 3. – If a number is divisible by 9, then it is divisible by 3. • Two points are collinear if they lie on the same line. – If two points lie on the same line, then they are collinear. Writing a Counterexample • Write a counterexample to show that the following conditional statement is false. – If x2 = 16, then x = 4. – False, x could be negative four; x = (-4) Converse Example • Two points are collinear if they lie on the same line. – If two points are collinear, then they lie on the same line. Conditional Statement – If two points lie on the same line, then they are Converse collinear. • A statement can be altered by negation, that is, by writing the negative of the statement. – Statement: m<A = 30° – Negation: m < A ≠ 30° – Statement: <A is acute. – Negation: <A is not acute. Inverse Example • If two points lie on the same line, then they are collinear. Conditional • If two points do not lie on the same line, then they are not collinear. Inverse Contrapositive Example • If two points lie on the same line, then they are collinear. Conditional • If two points are not collinear, then they do not lie on the same line. Contrapositive • When two statements are both true or both false, they are called logically equivalent (have same truth value) statements. – A conditional statement is logically equivalent to its contrapositive. – The inverse and converse of any conditional statement are logically equivalent. • Write the – a) inverse – b) converse – c) contrapositive If there is snow on the ground, then flowers are not in bloom. a) If there is no snow on the ground, then flowers are in bloom. b) If flowers are not in bloom, then there is snow on the ground. c) If flowers are in bloom, then there is no snow on the ground. Definitions and Biconditional Statements Definition • Two lines are called perpendicular lines if and only if they intersect to form right angles. • A line perpendicular to a plane is a line that intersects the plane in a point and is perpendicular to every line in the plane that intersects it. Exercise • Decide whether each statement about the diagram is true. Explain your answer using the definitions you have learned. a. Points D, X, and B are collinear. A b. AC is perpendicular to DB. c. <AXB is adjacent to <CXD. .D X . B . C Biconditional Statement • When a conditional statement and its converse are both true, you can write them as a single biconditional statement. • A biconditional statement is a statement that contains the phrase “if and only if. ” • Any valid definition can be written as a biconditional statement. • Rewrite the biconditional as conditional statement and its converse.\ – Biconditional : Two angles are supplementary if and only if the sum of their measures is 180°. – Conditional: If two angles are supplementary, then the sum of their measures is 180°. – Converse: If the sum of two angles measure 180°, then they are supplementary. • State a counterexample that demonstrates that the converse of the statement is false. – If three points are collinear, then they are coplanar. – If an angle measures 48°, then it is acute. Warmup • Which statement about the diagram is not true? C . B A G H . D a. <GHE is adjacent to <CHD. b. BF is perpendicular to AH. F E c. <BGH and <BGA are supplementary. d. GHC EHD e. m<BGH = 90 .