Bell Work

1) Write the statement in “If/Then” Form. Then write the inverse, converse, and contrapositive:

• a) A linear pair of angles is supplementary.

• b) The measure of a right angle is 90°.

2) Find the measure of Angle ABD. What classification does it have?

A B E

C D35

7570

Outcomes• I will be able to: • 1) Define perpendicular lines

• 2) Recognize and use biconditional statements.

Agenda

• 1) Bell Work• 2) Outcomes• 3) Perpendicular definitions• 4) Biconditional definition

2.2 Definitions and Biconditional Statements

• In Lesson 1.2 we learned that a definition uses known words to describe a new word. Here are two examples.

Perpendicular Lines

• What does it mean to be perpendicular?

• Perpendicular Lines – Two lines are called perpendicular lines if they intersect to form a right angle(90°).

• Symbolically - mn

Signifies right angle

Perpendicular Lines

• Can a line be perpendicular to a plane?

• Perpendicular to a plane: A perpendicular to a plane is a line that intersects the plane in a point and is perpendicular to every line the plane that it intersects. The symbol is read “perpendicular to.”

Using Definitions• Using the diagram and definitions from Chapter

1, decide if the following statements are true

• True – all points lie on same line

• True – The lines form a right angle

• False – The angles do not share a side

Conditional Statements• Are all conditional statements always written in

“if-then” form?• Some conditional statements are written in

“only-if” form• Example: • It is Saturday, only if I am working at the store• Hypothesis Conclusion• Rewrite in “If/then” form• If it is Saturday, then I’m working at the store

Biconditional Statements• Biconditional Statement - a statement that contains

the phrase “if and only if.”• It is equivalent to writing a conditional statement, and

its converse• Example: Three points are coplanar if and only if they

lie on the same plane

• Conditional – If three points are coplanar, then they lie on the same plane

• Converse - If three points lie on the same plane, then they are coplanar

Example 2

If three lines lie in the same plane, then they are coplanar.

If three lines are coplanar, then they lie in the same plane.

Is this biconditional true?Are both the conditional statement and the converse true? Yes

Yes

Biconditional Statements• Are all biconditional statements true?• ***To be true, both the conditional and the converse

of a biconditional statement must be true• Ex 3: Consider the following statement: • x = 3 if and only if x² = 9 • a) Is this a biconditional statement?• Yes, it contains if and only if

• b) Is it true?• Conditional – If x = 3, then x² = 9.• Converse – If x² = 9, then x = 3.

White Board Practice

• Conditional Statement - If two points lie in a plane, then the line containing them lies in the plane.

• Write this as a biconditional statement• Biconditional – Two points lie in a plane, if

and only if the line containing them lies in the plane

• Is this true or false?

Example 4

• Each of the following statements is true. Write the converse of each statement and decide whether the converse is true or false. If the converse is true, combine it with the original statement to form a true biconditional statement. If the converse is false, state a counterexample.

• A) If two points lie in a plane, then the line containing them lies in the plane.

• B) If a number ends in 0, then the number is divisible by 5.

Using Biconditionals to help with Proofs

• 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.

Example 5

• The converse of the Angle Addition Postulate is true. Write the converse and combine it with the postulate to form a true biconditional statement.

• Angle Addition Postulate: If P is in the interior of ∠RST, then the m∠RSP + m∠PST = m∠RST. • Converse: If m∠RSP + m∠PST = m∠RST , then P is in

the Interior of ∠RST.

• Biconditional: P is in the interior of ∠RST if and only if m∠RSP + m∠PST = m∠RST .

Exit Quiz• Given the following conditional statement, write

as converse, inverse, contrapositive, and biconditional statements

• Determine if each is true or false, and explain why.

• Conditional – If I am at Herron, I am at school• Converse – • Inverse –• Contrapositive – • Biconditional -