Warm UpWrite a conditional statement from each of the following.

1. The intersection of two lines is a point.

2. An odd number is one more than a multiple of 2.

3. Write the converse of the conditional “If Pedro lives in Chicago, then he lives in Illinois.” Find its truth value.

If two lines intersect, then they intersect in a point.

If a number is odd, then it is one more than a multiple of 2.

If Pedro lives in Illinois, then he lives in Chicago; False.

13. H: Animal is tabby C: it is a cat

14. H: two lines intersect C: four angles are formed

15. H: 8 oz of cereal cost $2.99 C: 16 oz of cereal cost $5.98

16. If a patient is ill, then you should monitor the patient’s heart rate.

17. If the batter makes 3 strikes, then the batter is out.

18. If segments are congruent, then they have equal measures.

19. T

20. F, 2 planes intersect at 1 LINE

21. T

22. Conv: In an event is unlikely to occur, then the probability of the event is 0.1. F

Inv: If the probability of an event is not 0.1, then the event is likely to occur. F

Contr: If an event is likely to occur, then the probability of an event is not 0.1. T

23. Conv: If the air temp is 32°F or less, then freezing rain is falling. F

Inv: If freezing rain is not falling, then the air temp is greater than 32°F. F

Cont: If the air temp is greater than 32°F, then freezing rain is not falling. T

24. T 25. T 26. T 27. F 28. T 29. F

34. If an animal is a dolphin, then it is a mammal.

35. If a person is a Texan, then the person is an American.

36. If x < -4, then x < -1.

38. X = 5

40.

1 2

42. If a mineral is calcite, then it has a hardness of 3. T

44. If a mineral is not apatite, then it has a hardness less than 5. F

46. If a mineral has a hardness of 3, then it is not apatite. T

When you combine a conditional statement and its converse, you create a biconditional statement.

A biconditional statement is a statement that can be written in the form “p if and only if q.” This means “if p, then q” and “if q, then p.”

p q means p q and q p

The biconditional “p if and only if q” can also be written as “p iff q” or p q.

Writing Math

Example 1a

Let x and y represent the following.

x: An angle is acute.

y: An angle has a measure that is greater than 0 and less than 90.

An angle is acute iff its measure is greater than 0° and less than 90°.

Write the conditional statement and converse within the biconditional.

Conditional: If an angle is acute, then its measure is greater than 0° and less than 90°.

The two parts of the biconditional x y are x y and y x.

Converse: If an angle’s measure is greater than 0° and less than 90°, then the angle is acute.

Example 1b

Cho is a member if and only if he has paid the $5 dues.

Write the conditional statement and converse within the biconditional.

Conditional: If Cho is a member, then he has paid the $5 dues.

Let x and y represent the following.

x: Cho is a member.

y: Cho has paid his $5 dues.

The two parts of the biconditional x y are x y and y x.

Converse: If Cho has paid the $5 dues, then he is a member.

Example 2a

If the date is July 4th, then it is Independence Day.

For the conditional, write the converse and a biconditional statement.

Converse: If it is Independence Day, then the date is July 4th.

Biconditional: It is July 4th if and only if it is Independence Day.

Example 3a

An angle is a right angle iff its measure is 90°.

Determine if the biconditional is true. If false, give a counterexample.

Conditional: If an angle is a right angle, then its measure is 90°.

The conditional is true.

Converse: If the measure of an angle is 90°, then it is a right angle.

The converse is true.

Since the conditional and its converse are true, the biconditional is true.

For a biconditional statement to be true, both the conditional statement and its converse must be true. If either the conditional or the converse is false, then the biconditional statement is false.

Example 3b

y = –5 y2 = 25

Determine if the biconditional is true. If false, give a counterexample.

Conditional: If y = –5, then y2 = 25.

The conditional is true.

Converse: If y2 = 25, then y = –5.

The converse is false.

The converse is false when y = 5. Thus, the biconditional is false.

In geometry, biconditional statements are used to write definitions.

A definition is a statement that describes a mathematical object and can be written as a true biconditional.

In the glossary, a polygon is defined as a closed plane figure formed by three or more line segments.

Think of definitions as being reversible. Postulates, however are not necessarily true when reversed.

Helpful Hint

Example 4

4a. A quadrilateral is a four-sided polygon.

4b. The measure of a straight angle is 180°.

Write each definition as a biconditional.

A figure is a quadrilateral if and only if it is a 4-sided polygon.

An is a straight if and only if its measure is 180°.