2. Which of the following values of x and y give the greatest value of z if z=x 2 -2y+106 ? a. x=-5, y=11b. x=2, y=0 c. x=3, y=-2d. x=5, y=8 2 1.6 warm-up 2

3. Which of the following values of x and y give the greatest value of z if z=x 2 -2y+106 ? a. x=-5, y=11b. x=2, y=0 c. x=3, y=-2d. x=5, y=8 2 1.6 warm-up

4. 2.2 Biconditionals & Definitions You will develop an awareness of the structure of mathematical systems, connecting definitions and logical reasoning. You will be able to determine the validity of a conditional statement and its converse. Matt Groening Pardekooper

5. Before we begin, there are some terms you need to 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. Pardekooper p q q p p q

6. Lets write a biconditional. Pardekooper Conditional: If two lines are perpendicular, then they intersect to form right angles. Converse: If two lines intersect to form right angles, then they are perpendicular. Biconditional: Two lines are perpendicular, if and only if they intersect to form right angles. Definition: Perpendicular lines are two lines that intersect to form right angles.

7. Definition: Congruent angles are two angles that have the same measure.. Pardekooper Lets try a another biconditional. Conditional: If two angles are congruent, then they have the same measure. Converse: If two angles have the same measure, then they are congruent. Biconditional: Two angles are congruent if and only if they have the same measure.

8. Pardekooper Now, lets go the other way. Conditional: If a line bisects a segment, then the line intersects the segment only at its midpoint. Converse: If the line intersects the segment only at its midpoint, then a line bisects a segment. Biconditional: A line bisects a segment if and only the line intersects the segment only at its midpoint.

9. Pardekooper 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 x 2 = 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

10. 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 q15. q p16. p q 17. q p18. p r19. r q Pardekooper 2.2 Biconditional Assignment 1-19 all

11. 2.2 Biconditional and Definitions Geometry 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  q15. q  p16. p  q 17. q  p18. p  r19. r  q