2.2 Day 1: Date: Geometry

Transcription

1 2.2 Day 1: Date: Geometry A Conditional Statement is an statement. The is the part following if. The is the part following then. Ex 1). What are the hypothesis and the conclusion of the conditional statement? a) If a number is even, then it is divisible by 2. Hypothesis: Conclusion: b) If an angle measures 130, then the angle is obtuse. Hypothesis: Conclusion: Ex 2). How can you write the following statement as a conditional? a) Adjacent angles share a side. b) Dolphins are mammals. The of a conditional statement is either true or false. * To show the statement is true: show that every time the hypothesis is true, the conclusion is true. * To show that a statement is false: find counterexample for which the hypothesis is true, but the conclusion is false.

2 Ex 3). Is the conditional true or false? If it is false, find a counterexample. a) If you live in Copley, then you live in Ohio. b) If a number is divisible by 5, then it is odd. c) If a month 28 days, then it is February. d) If two angles form a linear pair, then they are supplementary. Homework: pg. 99 #1, 3 13, 14 28(e)

3 2.2 Day 2: Date: Geometry The of a statement p is the of the statement. The symbol is read not p. Statement How to Write It Example Symbols How to Read It Conditional Converse Inverse Contrapositive

4 Truth Values: Statement Example Truth Value Conditional Converse Inverse Contrapositive Ex 1). What are the converse, inverse, and contrapositive of each of the conditional statements below? What are the truth values of each? a) If a figure is a rectangle, then it is a parallelogram. b) If a vegetable is a carrot, then it contains beta carotene. Homework: pg. 103 #1, 3 7, 11 14, 18 25

5 2.3: Date: Geometry A is a single true statement that combines a true conditional and its true converse. You can write a biconditional by joining the two parts of each conditional with the phrase. Ex 1). What is the converse of the following true conditional? If the converse is also true, combine the statements as a biconditional. Conditional: If the sum of the measures of two angles is 90, then the angles are complementary. A biconditional combines p q and q p as. ex: A point is a midpoint if and only if it divides a segment into two congruent segments. Ex 2). What are the two conditional statements that form this biconditional? Two numbers are additive inverses if and only if their sum is 0. What makes a good definition? 1. A good definition uses terms. 2. A good definition is. 3. A good definition is. It can be written as a true.

6 Ex 3). Is this definition of an equilateral triangle reversible? If yes, write it as a true biconditional. An equilateral triangle is a triangle with 3 congruent sides. Ex 4). Which of the following is a good definition? [A] Dogs are animals with 4 paws. [B] Squares have 4 sides. [C] Tuesday is the day before Wednesday. [D] An acute angle has a small measure. Homework: pg. 108 #1 3, 8 26(e), 45 50

7 2.4: Date: Geometry is the process of reasoning from given statements or facts to make a. Law of : If the hypothesis of a conditional is, then the conclusion is. Ex 1). What can you conclude from the given true statement? a) Given: If it is raining outside, then there are clouds in the sky. It is raining today. b) Given: If Malcolm scores at least 85% on his final exam, then he will earn an A for the term. Malcolm scores a 90% on his final exam. c) Given: If two angles are vertical, then they are congruent. Angles 3 and 4 are vertical angles. Law of : Symbols: Examples: Ex 2). What can you conclude from the given information? a) Given: If a number is divisible by 12, then it is divisible by 6. If a number is divisible by 6, then it is divisible by 3.

8 b) Given: If a figure is a square, then the figure is a parallelogram. If a figure is a rectangle, then the figure is a parallelogram. Ex 3). What can you conclude from the given information? a) Given: If you live in Cincinnati, then you live in Ohio. If you live in Ohio, then you live in the United States. Ken lives in Cincinnati. b) If a river is more than 4000 mi long, then it is longer than the Amazon. If a river is longer than the Amazon, then it is the longest river in the world. The Nile is 4132 mi long. Homework: pg. 117 #1 3, 6 16(e), 19 24, 26, 28

9 2.5: Date: Geometry Ex 1). What is the value of x? Ex 2). Write a two-column proof: a) Given: 3(2x 8) = 6 Prove: x = 5 b) Given: 4(x 7) 2(3x + 4) = 2 Prove: x = 19 Important Properties: REFLEXIVE PROPERTY SYMMETRIC PROPERTY TRANSITIVE PROPERTY

10 Ex 3). What is the name of the property of equality or congruence that justifies going from the first statement to the second statement? a) 7x + 3 = 24 b) RS = ST and ST = TU 7x = 21 RS = TU c) 5x = 10 b) 3x 4 = x and x = = 5x 3(-2) 4 = -2 A is a convincing argument that uses deductive reasoning. A proof shows why a conjecture is true. A lists each statement on the left and the reason (justification) on the right. Each statement must follow logically from the steps it. Ex 4). Write a two-column proof for the following: Given: g = 2h g + h = k k = m Prove: m = 3h Ex 5). Write a two-column proof: Given: 3x+5 2 Prove: x = 3 = 7 Homework: pg. 124 #1 17

11 2.6: Date: Geometry A is a conjecture or statement that you prove Important Theorems: VERTICAL ANGLES THM CONGRUENT CONGRUENT SUPPLEMENTS THM COMPLEMENTS THM THEOREM 2-4 THEOREM 2-5 Ex 1). What is the value of x? a) b) Ex 2). Write a two column proof: Given: Prove:

12 Ex 3). Write a two-column proof. Given: Prove: m 1 = m 3 m AEC = m DEB A B C E D Ex 4). Write a two-column proof: Given: CD EF DE FG Prove: CE EG C D E F G G Ex 5). Write a two-column proof. Given: 4 is a right angle 1 2 Prove: 2 and 6 are complementary Homework: pg. 132 #1 3, 6 13, 16 17, 19