1. PUT HW ON THE CORNER OF YOUR DESK!
GT Geometry Drill 10/21/14
1. Identify the hypothesis of the conditional
statement “Two angles are complementary if
the sum of their measures is 90 degrees.”
2. Provide a counterexample to show that
each statement is false.
If the sum of two integers is even, then the
integers are even
3. Write the converse of each and then
decided if it is true or false.
Angle is a right angle if its measure is 90
degrees

2. Is the conclusion a result of inductive or
deductive reasoning?
There is a myth that you can balance an egg on
its end only on the spring equinox. A person
was able to balance an egg on July 8,
September 21, and December 19. Therefore this
myth is false.
Since the conclusion is based on a pattern of
observations, it is a result of inductive reasoning.

3. Is the conclusion a result of inductive or
deductive reasoning?
There is a myth that the Great Wall of China is
the only man-made object visible from the
Moon. The Great Wall is barely visible in
photographs taken from 180 miles above Earth.
The Moon is about 237,000 miles from Earth.
Therefore, the myth cannot be true.
The conclusion is based on logical reasoning from
scientific research. It is a result of deductive
reasoning.

4. There is a myth that an eelskin wallet will
demagnetize credit cards because the skin of
the electric eels used to make the wallet holds
an electric charge. However, eelskin products
are not made from electric eels. Therefore, the
myth cannot be true. Is this conclusion a result
of inductive or deductive reasoning?
The conclusion is based on logical reasoning from
scientific research. It is a result of deductive
reasoning.

5. 2-4
Biconditional Statements
and Definitions
Review properties of equality and use
them to write algebraic proofs.
Identify properties of equality and
congruence.
Holt McDougal Geometry
Objectives
Write and analyze biconditional
statements.

6. 2-4
Biconditional Statements
and Definitions
deductive reasoning
Holt McDougal Geometry
Vocabulary
biconditional statement
definition
polygon
triangle
quadrilateral

7. 2-4
Biconditional Statements
and Definitions
Deductive reasoning is the process of using
logic to draw conclusions from given facts,
definitions, and properties.
Holt McDougal Geometry

8. 2-4
Biconditional Statements
and Definitions
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.”
Holt McDougal Geometry

9. 2-4
Biconditional Statements
and Definitions
p q means p q and q p
Writing Math
The biconditional “p if and only if q” can also be
written as “p iff q” or p  q.
Holt McDougal Geometry

10. 2-4
Biconditional Statements
and Definitions
Example 1A: Identifying the Conditionals within a
Biconditional Statement
Write the conditional statement and converse
within the biconditional.
An angle is obtuse if and only if its measure is
greater than 90° and less than 180°.
Let p and q represent the following.
p: An angle is obtuse.
q: An angle’s measure is greater than 90° and
less than 180°.
Holt McDougal Geometry

11. 2-4
Biconditional Statements
and Definitions
Let p and q represent the following.
p: An angle is obtuse.
q: An angle’s measure is greater than 90° and
less than 180°.
Holt McDougal Geometry
Example 1A Continued
The two parts of the biconditional p  q are p  q
and q  p.
Conditional: If an  is obtuse, then its measure is
greater than 90° and less than 180°.
Converse: If an angle's measure is greater
than 90° and less than 180°, then it is obtuse.

12. 2-4
Biconditional Statements
and Definitions
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.
Holt McDougal Geometry

13. 2-4
Biconditional Statements
and Definitions
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.
Holt McDougal Geometry

14. 2-4
Biconditional Statements
and Definitions
In the glossary, a polygon is defined as a closed
plane figure formed by three or more line
segments.
Holt McDougal Geometry

15. 2-4
Biconditional Statements
and Definitions
A triangle is defined as a three-sided polygon,
and a quadrilateral is a four-sided polygon.
Holt McDougal Geometry

16. 2-4
Biconditional Statements
and Definitions
Example 4: Writing Definitions as Biconditional
Write each definition as a biconditional.
A figure is a pentagon if and only if it is a 5-sided
polygon.
Holt McDougal Geometry
Statements
A. A pentagon is a five-sided polygon.
B. A right angle measures 90°.
An angle is a right angle if and only if it measures 90°.

17. 2-4
Biconditional Statements
and Definitions
Check It Out! Example 4
Write each definition as a biconditional.
4a. A quadrilateral is a four-sided polygon.
A figure is a quadrilateral if and only if it is a
4-sided polygon.
4b. The measure of a straight angle is 180°.
An  is a straight  if and only if its measure is
180°.
Holt McDougal Geometry

18. 2-4
Biconditional Statements
and Definitions
A proof is an argument that uses logic, definitions,
properties, and previously proven statements to show
that a conclusion is true.
An important part of writing a proof is giving
justifications to show that every step is valid.
Holt McDougal Geometry

19. 2-4
Biconditional Statements
and Definitions
Holt McDougal Geometry

20. 2-4
Biconditional Statements
and Definitions
Remember!
The Distributive Property states that
a(b + c) = ab + ac.
Holt McDougal Geometry

21. 2-4
Biconditional Statements
and Definitions
Example 1: Solving an Equation in Algebra
Solve the equation 4m – 8 = –12. Write a
justification for each step.
4m – 8 = –12 Given equation
+8 +8 Addition Property of Equality
4m = –4 Simplify.
m = –1 Simplify.
Holt McDougal Geometry
Division Property of Equality

22. 2-4
Biconditional Statements
and Definitions
Check It Out! Example 1
Solve the equation . Write a justification for
each step.
t = –14 Simplify.
Holt McDougal Geometry
Given equation
Multiplication Property of Equality.

23. 2-4
Biconditional Statements
and Definitions
Like algebra, geometry also uses numbers, variables,
and operations. For example, segment lengths and
angle measures are numbers. So you can use these
same properties of equality to write algebraic proofs in
geometry.
Helpful Hint
Holt McDougal Geometry
A B
AB represents the length AB, so you can think of
AB as a variable representing a number.

24. 2-4
Biconditional Statements
and Definitions
Example 3: Solving an Equation in Geometry
Write a justification for each step.
NO = NM + MO
4x – 4 = 2x + (3x – 9)
Holt McDougal Geometry
Segment Addition Post.
Substitution Property of
Equality
4x – 4 = 5x – 9 Simplify.
–4 = x – 9
5 = x
Subtraction Property of
Equality
Addition Property of
Equality

25. 2-4
Biconditional Statements
and Definitions
Check It Out! Example 3
Write a justification for each step.
mABC = mABD + mDBC  Add. Post.
8x° = (3x + 5)° + (6x – 16)° Subst. Prop. of Equality
8x = 9x – 11 Simplify.
–x = –11 Subtr. Prop. of Equality.
x = 11
Holt McDougal Geometry
Mult. Prop. of Equality.

26. 2-4
Biconditional Statements
and Definitions
You learned in Chapter 1 that segments with
equal lengths are congruent and that angles with
equal measures are congruent. So the Reflexive,
Symmetric, and Transitive Properties of Equality
have corresponding properties of congruence.
Holt McDougal Geometry

27. 2-4
Biconditional Statements
and Definitions
Holt McDougal Geometry

28. 2-4
Biconditional Statements
and Definitions
Remember!
Numbers are equal (=) and figures are congruent
().
Holt McDougal Geometry

29. 2-4
Biconditional Statements
and Definitions
Example 4: Identifying Property of Equality and
Identify the property that justifies each statement.
A. QRS  QRS
B. m1 = m2 so m2 = m1
C. AB  CD and CD  EF, so AB  EF.
D. 32° = 32°
Holt McDougal Geometry
Congruence
Symm. Prop. of
=
Trans. Prop of

Reflex. Prop. of
.
Reflex. Prop. of
=

30. 2-4
Biconditional Statements
and Definitions
Check It Out! Example 4
Identify the property that justifies each statement.
4a. DE = GH, so GH = DE.
4b. 94° = 94°
4c. 0 = a, and a = x. So 0 = x.
4d. A  Y, so Y  A
Holt McDougal Geometry
Sym. Prop. of =
Reflex. Prop. of =
Trans. Prop. of =
Sym. Prop. of 

31. 2-4
Biconditional Statements
and Definitions
Holt McDougal Geometry
Lesson Quiz: Part I
Solve each equation. Write a justification for each
step.
1.
Given
z – 5 = –12 Mult. Prop. of =
z = –7 Add. Prop. of =

32. 2-4
Biconditional Statements
and Definitions
Holt McDougal Geometry
Lesson Quiz: Part II
Solve each equation. Write a justification for each
step.
2. 6r – 3 = –2(r + 1)
Given
6r – 3 = –2r – 2
8r – 3 = –2
Distrib. Prop.
Add. Prop. of =
6r – 3 = –2(r + 1)
8r = 1 Add. Prop. of =
Div. Prop. of =

33. 2-4
Biconditional Statements
and Definitions
Holt McDougal Geometry
Lesson Quiz: Part III
Identify the property that justifies each
statement.
3. x = y and y = z, so x = z.
4. DEF  DEF
5. AB  CD, so CD  AB.
Trans. Prop. of =
Reflex. Prop. of 
Sym. Prop. of 

34. 2-4
Biconditional Statements
and Definitions
Converse: If an  measures 90°, then the  is right.
Biconditional: An  is right iff its measure is 90°.
Holt McDougal Geometry
Lesson Quiz
1. For the conditional “If an angle is right, then
its measure is 90°,” write the converse and a
biconditional statement.
2. Determine if the biconditional “Two angles are
complementary if and only if they are both
acute” is true. If false, give a counterexample.
False; possible answer: 30° and 40°
3. Write the definition “An acute triangle is a
triangle with three acute angles” as a
biconditional.
A triangle is acute iff it has 3 acute s.