2.2 Definitions and Biconditional Statements. Geometry Mr. Peebles 03/20/13
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1 2.2 Definitions and Biconditional Statements Geometry Mr. Peebles 03/20/13
2 Geometry Bell Ringer Write the Contrapositive of the following conditional statement: If the polygon has three sides, then it s a triangle.
3 Geometry Bell Ringer Write the Contrapositive of the following conditional statement: If the polygon has three sides, then it s a triangle. Answer: If it s NOT a triangle, then the polygon does not have three sides.
4 Standard/Objectives Daily Learning Target (DLT) Wednesday March 20, 2013 I can recognize, use, and write biconditional statements in real life.
5 Assignment Due Today: Pp (1-17 Odds, 31, 37, 39, 54-58, 64-66)
6 Biconditional Statements A bi-conditional statement can either be true or false it has to be one or the other. To be true, BOTH the conditional statement and its converse must be true. This means that a true biconditional statement is true both forward and backward. All definitions can be written as true bi-conditional statements. Biconditional statements always contains the phrase if and only if.
7 Example 1 The biconditional statement below can be rewritten as a conditional statement and its converse. Conditional statement: If three lines are coplanar, then they lie in the same plane. Converse: If three lines lie in the same plane, then they are coplanar. Hint: Are the conditional and converse statements true? If so, write the biconditional.
8 Example 1 The biconditional statement below can be rewritten as a conditional statement and its converse. Conditional statement: If three lines are coplanar, then they lie in the same plane. Converse: If three lines lie in the same plane, then they are coplanar. Answer: Three lines are coplanar if and only if they lie in the same plane.
9 Example 2: Analyzing Biconditional Statements Consider the following statement: x = 3 if and only if x 2 = 9. Is this a biconditional statement?
10 Example 2: Analyzing Biconditional Statements Consider the following statement: x = 3 if and only if x 2 = 9. Is this a biconditional statement? The statement is biconditional because it contains the phrase if and only if. Is the statement true? Conditional statement: If x = 3, then x 2 = 9. Converse: x 2 = 9, then x = 3. The first part of the statement is true, but what about -3? That makes the second part of the statement false.
11 Example 3: Writing a Biconditional Statement Each of the following is true. Write the converse if each statement and decide whether the converse is true or false. If the converse is TRUE, then combine it with the original statement to form a true biconditional statement. If the statement is FALSE, then state a counterexample. If two points lie in a plane, then the line containing them lies in the plane.
12 Example 3: Writing a Biconditional Statement Converse: If a line containing two points lies in a plane, then the points lie in the plane. The converse is true. It can be combined with the original statement to form a true biconditional statement written below: Biconditional statement: Two points lie in a plane if and only if the line containing them lies in the plane.
13 Example 4: Writing a Biconditional Statement Conditional: If a number ends in 0, then the number is divisible by 5. Converse: If a number is divisible by 5; then the number ends in 0. Can you write a biconditional statement from the information given from the conditional and converse statements?
14 Example 4: Writing a Biconditional Statement Conditional: If a number ends in 0, then the number is divisible by 5. Converse: If a number is divisible by 5; then the number ends in 0. The converse isn t true. What about 25? Knowing how to use true biconditional statements is an important tool for reasoning in Geometry. For instance, if you can write a true biconditional statement, then you can use the conditional statement or the converse to justify an argument.
15 Example 5: Writing a Postulate as a Biconditional Statement The second part of the Segment Addition Postulate is the converse of the first part. Combine the statements to form a true biconditional statement. If B lies between points A and C, then AB + BC = AC. Converse: If AB + BC = AC; then B lies between points A and C. Now combine these statements into one bi-conditional statement.
16 Example 5: Writing a Postulate as a Biconditional Statement Answer: Point B lies between points A and C if and only if AB + BC = AC.
17 Assignment: pp (1-6 and 38-40)
18 Geometry Exit Quiz 5 Points Write the Contrapositive of the following conditional statement: If all sides are proportional, then the shapes are similar.
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