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## 2.1 Conditional Statements

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**2.1 Conditional Statements**Day 1 Part 1 CA Standards 1.0, 3.0**Warmup**• State whether each sentence is true or false. • If you live in Los Angeles, then you live in California. • If you live in California, then you live in Los Angeles. • If today is Wednesday, then tomorrow is Thursday. • If tomorrow is Thursday, then today is Wednesday. True False True True**Conditional Statement**• Conditional statement has two parts, hypothesis and a conclusion. • If _____________, then____________. hypothesis conclusion**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.**Converse**• Two points are collinear if they lie on the same line. • If two points are collinear, then they lie on the same line. • If two points lie on the same line, then they are collinear. Conditional Statement Converse**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**• If two points lie on the same line, then they are collinear. • If two points do not lie on the same line, then they are not collinear. Conditional Inverse**Contrapositive**• If two points lie on the same line, then they are collinear. • If two points are not collinear, then they do not lie on the same line. Conditional Contrapositive**When two statements are both true or both false, they are**called equivalent statements. • A conditional statement is equivalent to its contrapositive. • The inverse and converse of any conditional statement are equivalent.**Write the**• a) inverse • b) converse • c) contrapositive If there is snow on the ground, then flowers are not in bloom. • If there is no snow on the ground, then flowers are in bloom. • If flowers are not in bloom, then there is snow on the ground. • If flowers are in bloom, then there is no snow on the ground.**Point, Line, and Plane Postulates**• 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.**Review**• Write the converse of the conditional statement. • If x = 3, then y = 7. • If Carrie joins the softball team, then Mary will join. • If two angels are vertical, then their measures are equal.**2.2 Definitions and Biconditional Statements**Day 1 Part 2 CA Standards 1.0, 3.0**Definition**• Two lines are called perpendicular lines if they intersect to form a right angle. • 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. • Points D, X, and B are collinear. • AC is perpendicular to DB. • <AXB is adjacent to <CXD. . A . . D X B . C**Biconditional Statement**• Biconditional Statement • It is Saturday, only if I am working at the restaurant. • Conditional Statement • If it is Saturday, then I am working at the restaurant.**Consider the following statement**x = 3 if and only if x2 = 9. • Is this a biconditional statement? Yes • Is the statement true? No, because x also can be -3.**Rewrite the biconditional as conditional statement and its**converse. • 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.**Pg. 75 # 4 – 38 even**• Handout 2.2**2.3 Deductive Reasoning**Day 2 Part 1 CA Standards 1.0, 3.0**Warmup**• Rewrite the true statement in if-then form and write the converse. If the converse is true, combine it with the if-then statement to form a true biconditional statement. • The perimeter of a triangle is the sum of the lengths of its sides • Two angles measure 42 and 48 form a complementary angle.**Symbolic Notations**implies • Conditional Statement ( p q) • If the sun is out, then the weather is good. • Converse Statement ( q p) • If the weather is good, then the sun is out. p q q p**Exercise**• Let p be “the values of x is -5” and let q be “the absolute value of x is 5”. • Write p q in words. • Write q p in words. If the values of x is -5, then the absolute value of x is 5. If the absolute value of x is 5, then the values of x is -5.**Symbolic Notations**• Conditional Statement ( p q) • If the sun is out, then the weather is good. • Inverse Statement (~ p ~q) • symbol of negation (~) • If the sun is not out, then the weather is not good. p q ~q ~p**Law of Detachment**• Law of Detachment • If p q is a true conditional statement and p is true, then q is true. • Law of Syllogism • If p q and q r are true conditional statements, then p r is true.**Law of Syllogism**• If a bird is the fastest bird on land, then it is the largest of all birds. • If a bird is the largest of all birds, then it is an ostrich. • Combine two conditional statements together • If a bird is the fastest bird on land, then it is an ostrich.**Review**• Fill in the box with appropriate symbols and examples.**Review**• Rewrite the conditional statement in if-then form. • It must be true if you read it in a newspaper. If you read it in a newspaper, then it must be true.**Review**• Write the inverse, converse, and contrapositive of the conditional statement. • If you are indoors, then you are not caught in a rainstorm. Inverse: If you are not indoors, then you are caught in a rainstorm. Converse: If you are not caught in a rainstorm, then you are indoors. Contrapositive: If you are caught in a rainstorm, then you are not indoors.**Review**• Using p, q, r and s below, write the symbolic statements in words. • p: we go shopping. • q: we need a shopping list • r: we stop at the bank • s: we see our friends. • p q • ~p ~s If we go shopping, then we need a shopping list. If we do not go shopping, then we do not see our friends.**2.4 Reasoning with Properties from Algebra**Day 2 Part 2 CA Standard 3.0**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.**Solve**• Solve 5x – 18 = 3x + 2 and explain each step in writing. 5x – 18 = 3x + 2 2x – 18 = 2 2x = 20 x = 10 Subtraction p. of e. Addition p. of e. Division p. of e.**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.**Example**• Solve 55z – 3(9z + 12) = -64 and write a reason for each step.**Review**• Let p be “a shape is a triangle” and let q be “it has an acute angle”. • Write the contrapositive of p q. • Write the inverse of p q.**Review**• 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.**Pg. 91 # 8 – 20**• Pg. 92 # 26 – 42 • Pg. 99 # 15 – 23, 32**2.5 Proving Statements and Segments**Day 3 Part 1 CA Standard 2.0, 4.0, 16.0**Warmup**• Identify the property of equality. If m<1 = m<2, then m<2 = m<1.**Properties of Segment Congruence**• Segment congruence is reflexive, symmetric, and transitive. • Reflexive: For any segment AB, . • Symmetric: If , then . • Transitive: If , and , then**Symmetric Property of Segment Congruence**• Given: PQ XY • Prove: XY PQ Statements Reasons 1.PQ XY 1. Given 2.PQ = XY 2. Definition of congruent segments 3.XY = PQ 3. Symmetric property of equality 4.XY PQ 4. Definition of congruent segments P X Q Y**Another example**• Given: LK = 5, JK = 5, JK JL. • Prove: LK JL. Statements Reasons 1._________ 1. Given 2. _________ 2. Given 3. LK = JK 3. Transitive p. o. e 4. LK JK 4. ________________ 5. JK JL 5. Given 6. ________ 6. Transitive p o e LK = 5 JK = 5 Definition of congruent segments LK ≈ JL