Geometry (H) Lesson 2.3
2.3 Notes: Biconditionals
Several years ago a rancher in Yucca Valley, California, got into
trouble for having too many horses. He said that he had only 3, but the
local officials said that he had 29. The disagreement centered on the
definition of a horse.
According to the breeding industry, a horse is “a male animal at
least five years old and 58 inches high at the base of the neck.”
“Horsefeathers,” said the officials. “If it looks like a horse, trots like a horse and smells like a
horse, it is a horse.”
Disagreements about the meanings of words frequently come up in courts of law. To
prevent arguments in debates, key terms are often defined first. Variations in a word’s meaning
seldom cause much of a problem in everyday communication, but they are serious when we are
trying to reason precisely. For this reason, definitions play an important role in mathematics.
When we define a word in geometry, the word and its definition are understood to have
exactly the same meaning. For example, when we define a “triangle” as “a polygon that has three
sides,” we can say not only that, “if a figure is a triangle, then it is a polygon that has three sides,”
but also that, “If a figure is a polygon that has three sides, then it is a triangle.”
-Geometry: Seeing, Doing, Understanding, Harold R. Jacobs
Lesson Objectives: In this lesson, we will be learning about another type of statement called a
biconditional statement as well as what makes a good definition in mathematics. You should be
able to do the following:
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Know what a true biconditional statement is
Write a biconditional statement given a conditional statement and its converse
Identify two separate conditional statements in a biconditional statement
Determine whether a definition fits the criteria for a good definition in mathematics
Biconditional Statements
1. Write the converse of the following conditional statement:
Conditional: If an angle measures 90 degrees, then it is a right angle.
Converse:
2. What is the truth value of each conditional above?
Biconditional Statements
A biconditional is
ESSENTIAL UNDERSTANDING
Geometry (H) Lesson 2.3
Example 1: Write the two conditional statements above as a single biconditional statement.
Biconditional:
3. Consider the conditional statement: If two angles have equal measure, then the angles are
congruent.
Use the conditional statement along with its converse to write a biconditional statement.
Biconditional:
Biconditional Statements and Symbols
Abbreviation for “if and only if”:
Example 2: What are two conditional statements that form the following biconditional statement?
Two numbers are reciprocals if and only if their product is 1.
Conditional:
Converse:
4. Convert the following statement into a biconditional statement.
Three collinear points lie on the same line.
Biconditional:
Conditional:
Converse:
Geometry (H) Lesson 2.3
We have learned that undefined terms such as point, line and plane are building blocks of
geometry. We understand the meaning of the terms intuitively. Now we will look at what makes a
good definition.
Good Definitions
A good definition is a statement that can help you identify or classify an object. A good definition
has several important components.
Criteria for a Good Definition
1.
2.
3.
Example 3: Determine whether the following is a good definition for a straight angle:
A straight angle is an angle that measures 180 degrees.
One way to show a statement is NOT a good definition is to find a counterexample.
5. Determine whether the following definitions are good definitions using the criteria above.
a. A fish is an animal that swims.___________________________________________
b. A triangle is a figure with three sharp corners._______________________________
c. Giraffes are animals with very long necks.__________________________________
d. A penny is a coin worth one cent._________________________________________
e. A circle is the intersection of a plane and a sphere.___________________________
Geometry (H) Lesson 2.3
6. Find the flaws in the definition below of an obtuse angle. Then fix the definition so that it is a
good definition using our criteria.
An obtuse angle is a really big angle that measures more than 90 degrees and has a
greater measure than acute angles.
Assignment 2B
Biconditional Statements (2 – 3)
At the end of this assignment, you should be able to do the following:
o Combine a conditional statement and its coverse to form a biconditional statement
o State the truth value of a biconditional statement
o Recognize whether a definition is a good defintion
Part I: Practice
1. Each conditional statement below is true. Write its converse. If the converse is also true,
combine the statements as a biconditional.
a. If a number is divisible by 20, then it is even.
b. If x = 12, then 2x – 5 = 19.
c. If x = 3, then x = 3.
d. If two segments have the same length, then they are congruent.
Geometry (H) Lesson 2.3
2. Write the two statements that combine to form each conditional.
a. A line bisects a segment if and only if the line intersects the segment at its midpoint.
b. An angle is a right angle if and only if its measure is 90 degrees.
c. Yesterday was Monday if and only if today is Tuesday.
3. For each statement below, test each statement to see if it is reversible. If so, write it as a true
biconditional. If not, write not reversible.
a. A cat is an animal with whiskers.
b. Complementary angles are two angles with measures that sum to 90 degrees.
c. Two angles that form a linear pair are adjacent.
d. A rectangle is a four-sided figure with at least one right angle.
4. Is each statement below a good definition? If not, explain.
a. A segment is part of a line.
b. The red wolf is an endangered animal.
c. Perpendicular lines are two lines that intersect to form right angles.
Geometry (H) Lesson 2.3
Part II: Application and Problem Solving
5. Write your own true biconditional statement.
6. Your friend defines an obtuse angle as an angle with measure greater than 90 degrees. Is this a
good definition? Explain.
7. Which conditional and its converse form a true biconditional?
8. Write each as a biconditional.
a. Points in Quadrant III have two negative coordinates.
b. A hexagon is a six-sided polygon.
9. Reading Math Let statements p, q, r, and s be as follows:
Substitute for p, q, r, and s, and write each statementt the way you would read it.
a. p  q
b. p  r
c. p  s
d. p  s
Geometry (H) Lesson 2.3
10. Use the figures below to write a good definition of a line in spherical geometry.