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## Lesson 2.1

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**Lesson 2.1**Conditional Statements You will learn to… * recognize and analyze a conditional statement * write postulates about points, lines, an planes using conditional statements**A conditional statement has two parts, a hypothesis and a**conclusion. p q If p, then q.**hypothesis (p)**If the team wins the game, then they will win the tournament. conclusion (q)**Write an if-then statement.**1. The intersection of two planes is a line. If two planes intersect, then their intersection is a line.**Write an if-then statement.**2. A line containing two given points lies in a plane if the two points lie in the plane. If two points lie in a plane, then the line containing them lies in the plane.**The converse is formed by switching the hypothesis and**conclusion. The converse is q p. If q, then p.**Write the converse of this if-then statement. Is it true or**false? 3. If mA = 125°, then A is obtuse. If A is obtuse, then mA = 125°. False**The negation of a statement is formed by negating the**statement. The negation is written ~ p.**Write the negation of this statement.**4. mA = 125° mA 125° 5.A is not obtuse A is obtuse**The inverse is formed by negating the hypothesis and the**conclusion. The inverse is ~ p ~ q. If ~ p, then ~ q.**Write the inverse of this if-then statement. Is it true or**false? 6. If mA = 125°, then A is obtuse. If mA 125°, then A is not obtuse. False**The contrapositive is formed by negating the hypothesis and**conclusion of the converse. The contrapositive is ~ q ~ p. If ~ q, then ~ p.**Write the contrapositive of this if-then statement. Is it**true or false? 7. If mA = 125°, then A is obtuse. If A is not obtuse, then mA 125°. True**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.**B**A C T 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 2 planes intersect, then their intersection is ___________. a line**Workbook**Page 23 (1-5)**Lesson 2.2**Biconditional Statements You will learn to… * recognize and use definitions * recognize and use biconditional statements**All definitions are biconditional.**All definitions can be interpreted “forward” and “backward.”**For example,**perpendicular lines are defined as two lines that intersect to form one right angle.**If two lines are perpendicular, then they intersect to form**one right angle. If two lines intersect to form one right angle, then they are perpendicular.**Two lines are perpendicular**if and only if they intersect to form one right angle. A biconditional statement contains the phrase “if and only if.”**A biconditional statement**is true when the original if-then statement AND its converse are both true.**1.Two angles are supplementary**if and only if the sum of their measures is 180°. If two angles are supplementary, then the sum of their measures is 180°. if-then statement: converse: If the sum of the measures of two angles is 180°, then they are supplementary.**2.If an angle is 135˚, then it is an obtuse angle.**converse: If an angle is obtuse, then its measure is 135°. Can we write a biconditional statement? counterexample?**3.If two angle measures add up to 90˚, then they are**complementary angles. converse: If two angles are complementary, then the sum of their measures is 90°. Can we write a biconditional statement? Two angles are complementary if and only if the sum of their measures is 90°.**Workbook**Page 25 (1-7)**Lesson 2.3**You will learn to… * use symbolic notation to represent logical statements * form conclusions by applying laws of logic Deductive Reasoning**Using these phrases, write the conditional statement.**If mB = 90˚, then B is a right angle. 1. p q If B is a right angle, then mB = 90˚ If mB ≠ 90˚, then B is not a right angle. 2. q p p: mB = 90˚ q: B is a right angle If B is not a right angle, then mB ≠ 90˚ 3. ~p ~ q mB = 90˚ if and only if B is a right angle. 4. ~ q ~ p 5. p q**facts**Deductive Reasoning uses facts to make a logical argument. definitions, properties, postulates, theorems, and laws of logic**p q**p conclusion must be true hypothesis is true Given facts q Law of Detachment Therefore: You can use these symbols when asked to explain your reasoning.**q**Therefore, I passed geometry. q p p Law of Detachment If I learn my facts, then I will pass geometry. I learned my facts.**p q**q r Given facts p r Law of Syllogism Therefore: You can use these symbols when asked to explain your reasoning.**p**Therefore, if I pass geometry, then I will get a cell phone. r p q q r Law of Syllogism If I pass geometry, then my dad will be happy. If my dad is happy, then I will get a cell phone.**6. Is this argument valid?**If 2 lines in a plane are parallel, then they do not intersect. p q Coplanar lines n and m are parallel. p Therefore, lines n and m do not intersect. q VALID – Law of Detachment**r p**p q 7. Is this argument valid? If 2 angles are supplementary, then the sum of their measures is 180˚. p q If 2 angles form a linear pair, then they are supplementary. r p Therefore, if 2 angles form a linear pair, then the sum of their measures is 180˚ r q VALID – Law of Syllogism**8. Is this argument valid?**If 2 angles are a linear pair, then the sum of their measures is 180˚. p q m1 + m2 = 180˚ q Therefore, 1 and 2 are a linear pair. p INVALID**r q**p q 9. Is this argument valid? If you live in Canada, then you live in North America. p q If you live in South Carolina, then you live in North America. r q Therefore, if you live in Canada, then you live in South Carolina p r INVALID**If you use this product,**then you will have even-toned skin. If you have even-toned skin, then If you use this product, then you will be beautiful. you will be beautiful.**Lesson 2.4**Properties of Equality and Congruence You will learn to… * use properties from algebra * use properties of length and measure to justify segment and angle relationships**Equality Properties**Addition Property Subtraction Property Multiplication Property Division Property Distributive Property Substitution Property Reflexive Property Symmetric Property Transitive Property**Transitive Property**If mA=mB and mB=10°, then mA=10° If XY = ST and ST = 10, then XY = 10**If 8x=16,**then x=2. Division Property