Which of the following values Which of the following values of x and y give the greatest of x and y give the greatest value of z if z=xvalue of z if z=x22-2y+106 ?-2y+106 ?

a. x=-5, y=11a. x=-5, y=11 b. x=2, y=0b. x=2, y=0

c. x=3, y=-2c. x=3, y=-2 d. x=5, y=8d. x=5, y=8

2

1.6 warm-up 2

Which of the following values Which of the following values of x and y give the greatest of x and y give the greatest value of z if z=xvalue of z if z=x22-2y+106 ?-2y+106 ?

a. x=-5, y=11a. x=-5, y=11 b. x=2, y=0b. x=2, y=0

c. x=3, y=-2c. x=3, y=-2 d. x=5, y=8d. x=5, y=8

2

1.6 warm-up

2.2 Biconditionals & Definitions2.2 Biconditionals & Definitions

You will develop an awareness You will develop an awareness of the structure of mathematical of the structure of mathematical systems, connecting definitions systems, connecting definitions and logical reasoning. You will and logical reasoning. You will be able to determine the validity be able to determine the validity of a conditional statement and of a conditional statement and

its converse.its converse.

Matt Groening PardekooperPardekooper

Before we begin, there are some terms you need to Before we begin, there are some terms you need to know.know.

Conditional: If A, then B.

Biconditional: When both the conditional and the converse are true.

A if and only if B

Converse: If B, then A.

Matt GroeningPardekooperPardekooper

p q q p

p q

Lets write a biconditional.

Matt GroeningPardekooperPardekooper

ConditionalConditional: If two lines are perpendicular, then they : If two lines are perpendicular, then they intersect to form right angles.intersect to form right angles.

ConverseConverse: If two lines intersect to form right angles, : If two lines intersect to form right angles, then they are perpendicular.then they are perpendicular.

BiconditionalBiconditional: Two lines are perpendicular, if and : Two lines are perpendicular, if and only if they intersect to form right angles.only if they intersect to form right angles.

DefinitionDefinition: Perpendicular lines are two lines that : Perpendicular lines are two lines that intersect to form right angles.intersect to form right angles.

DefinitionDefinition: Congruent angles are two angles that : Congruent angles are two angles that have the same measure..have the same measure..

Matt GroeningPardekooperPardekooper

Lets try a another biconditional.

ConditionalConditional: If two angles are congruent, then they : If two angles are congruent, then they have the same measure. have the same measure.

ConverseConverse: If two angles have the same measure, : If two angles have the same measure, then they are congruent.then they are congruent.

BiconditionalBiconditional: Two angles are congruent if and only : Two angles are congruent if and only if they have the same measure .if they have the same measure .

PardekooperPardekooper

Now, lets go the other way.

ConditionalConditional: If : If a line bisects a segmenta line bisects a segment, then , then the line the line intersects the segment only at its midpointintersects the segment only at its midpoint. .

ConverseConverse: If : If the line intersects the segment only at the line intersects the segment only at its midpointits midpoint, then , then a line bisects a segmenta line bisects a segment..

BiconditionalBiconditional: A line bisects a segment if and only the : A line bisects a segment if and only the line intersects the segment only at its midpoint. line intersects the segment only at its midpoint.

PardekooperPardekooper

Each conditional statement is true. Write the converse statement. If the converse is true, then write the biconditional.

1. If 2x-5=11, then x=8.

2. 2. If n=17, then |n| = 17.

3. If a figure has eight sides, then it is an octagon.

4. If a number is divisible by 20, then it is even.

5. In the United States, if it is July 4, then it is Independence Day.

6. If x = -10, then x2 = 100.

Given a definition, write the conditional, converse, and biconditional.

7. A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to a segment at its midpoint.

8. Parallel planes are planes that do not intersect.

9. A Tarheel is a person who was born in North Carolina.

10. A rectangle is a four-sided figure with at least one right angle.

2.2 Biconditional Assignment 1-19 all

Given a biconditional, write the conditional, converse, and the definition.

11. A whole number is a multiple of 5 if and only if its last digit is either a 0 or a 5.

12. Two lines are perpendicular if and only if they intersect to four right angles.

13. You live in Texas if and only if you live in the largest state in the contiguous United States.

Substitute for p, q, and r, and write each statement the way you would read it.

p: A and B are right angles

q: A and B are supplementary angles

r: A and B are adjacent angles.

14. p q 15. q p 16. p q

17. q p 18. p r 19. r q

PardekooperPardekooper

2.2 Biconditional Assignment 1-19 all

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2.2 Biconditional and DefinitionsGeometry

Given a biconditional, write the conditional, converse, and the definition.

11. A whole number is a multiple of 5 if and only if its last digit is either a 0 or a 5.

12. Two lines are perpendicular if and only if they intersect to four right angles.

13. You live in Texas if and only if you live in the largest state in the contiguous United States.

Substitute for p, q, and r, and write each statement the way you would read it.

p: ∠A and ∠B are right angles

q: ∠ A and ∠B are supplementary angles

r: ∠A and ∠B are adjacent angles.

14. p q 15. q p 16. p q

17. q p 18. p r 19. r q