Download Chapter 3: Geometry and Reasoning

3-6
Conditional
Statements
Goals
■ Identify and evaluate conditional statements.
■ Identify converses and biconditionals.
Applications
Drafting, Sports, Geography
Do you think each statement is true or false?
Explain your reasoning.
a. Denver is the capital of Colorado.
b. For all x, x2 ! x.
c. Lines m and n are parallel.
BUILD UNDERSTANDING
Many of the statements in this chapter are written
in if–then form. Statements like these are called
conditional statements, or simply conditionals. The clause following “if” is called the
hypothesis of the conditional. The clause following “then” is called the conclusion.
For example, the parallel lines postulate was presented as a conditional.
If two parallel lines are cut by a transversal ,
hypothesis
then corresponding angles are equal in measure .
conclusion
Denver, CO
A conditional is either true or false. When a conditional is true, you can justify it
in a variety of ways. For instance, you may be able to show that the conditional is
true because it follows directly from a definition. When a conditional is a
postulate, such as the parallel lines postulate, it is assumed to be true. Still other
conditionals are theorems, and these must be proved true.
To demonstrate that a conditional is false, you need to find only one example for
which the hypothesis is true but the conclusion is false. An example like this is
called a counterexample.
Example 1
Tell whether each conditional is true or false.
a. If two lines are parallel, then they are coplanar.
b. If two lines do not intersect, then they are parallel.
Solution
a. Parallel lines are defined as coplanar lines that do not intersect. So, the
conditional is true.
b. Consider skew lines k and ! shown. By the definition of skew lines, k and !
do not intersect, and so the hypothesis is true. However, also by the
definition of skew lines, k and ! are noncoplanar. Lines k and ! cannot be
parallel, and so the conclusion is false. Therefore, lines k and ! are a
counterexample, and the conditional is false.
128
Chapter 3 Geometry and Reasoning
k
!
The converse of a conditional is formed by interchanging the
hypothesis and the conclusion. The fact that a conditional is true
is no guarantee that its converse is true.
Example 2
DRAFTING People who draw plans must apply this true
statement: If two lines are parallel, then they do not intersect.
Write the converse of the statement. Is it also true?
Reading
Math
Many statements about
everyday situations can be
expressed as conditionals.
For instance, the
following is a true
conditional.
If it is raining,
then it is cloudy.
Its converse is false.
If it is cloudy,
then it is raining.
Solution
Interchange the hypothesis and the conclusion of the given
statement.
Statement:
If two lines are parallel , then they do not intersect .
hypothesis
Converse:
conclusion
If two lines do not intersect , then they are parallel .
hypothesis
conclusion
By definition, parallel lines do not intersect, and so the given statement is true.
Part b of Example 1 demonstrated that lines that do not intersect are not
necessarily parallel, and so the converse is false.
The converse of the parallel lines postulate also is assumed to be true. It is stated
as the corresponding angles postulate in the following manner.
Postulate 10
The Corresponding Angles Postulate If two lines are
cut by a transversal so that a pair of corresponding
angles are equal in measure, then the lines are parallel.
When a statement and its converse are both true, they can be combined into an
“if and only if” statement. This type of statement is called a biconditional
statement, or simply a biconditional. Every definition can be written as a
Problem Solving
biconditional.
Tip
Example 3
Write this definition as two conditionals and as a single biconditional.
A right angle is an angle whose measure is 90°.
Solution
When writing the
converse of a
conditional, you may
need to change the
wording of the
hypotheses and the
conclusion to make the
converse read clearly.
The definition leads to two true conditionals.
If an angle is a right angle, then its measure is 90°.
If the measure of an angle is 90°, then it is a right angle.
These can be combined into a single biconditional as follows.
An angle is a right angle if and only if its measure is 90°.
mathmatters3.com/extra_examples
Lesson 3-6 Conditional Statements
129
TRY THESE EXERCISES
1. TALK ABOUT IT Decide whether this conditional is true or false.
If two lines are each perpendicular to a third line, then
they are parallel to each other.
Discuss your reasoning with a classmate.
2. Write the converse of this statement.
If two angles are vertical angles, then they are equal in measure.
Are the given statement and its converse true or false?
3. Write this definition as two conditionals and as a single
biconditional.
The bisector of an angle is the ray that divides the angle
into two adjacent angles that are equal in measure.
4. NUMBER SENSE Tell whether this conditional is true or false.
If a number is less than 1, the number is a proper fraction.
Write the converse of the statement. Is the converse true or false?
5. SPORTS If a shortstop makes a bad throw to first base, the error
is charged to the shortstop. This statement is true. Write the
converse of the statement. Is it true or false?
6. GEOGRAPHY If a point is located north of the equator, it has a
northern latitude. This statement is true. Write the converse of
the statement. Is it true or false?
PRACTICE EXERCISES
• For Extra Practice, see page 672.
Sketch a counterexample that shows why each conditional is false.
7. If line t intersects lines g and h, then line t is a transversal.
8. If PQ " QR, then point Q is the midpoint of P!R!.
9. If points A, B, and C are collinear, then !!"
BA and !!"
BC are opposite rays.
10. If two angles share a common side and a common vertex, then
they are adjacent angles.
Write the converse of each statement. Then tell whether the given
statement and its converse are true or false.
11. If points J, K, and L are coplanar, then they are collinear.
12. If point Y is the midpoint of X!Z!, then XY # YZ " XZ.
13. If the sum of the measures of two angles is 90°, then the angles are
complementary.
14. If two angles are supplementary, then the sum of their measures is greater
than 90°.
15. If two lines are perpendicular, then they do not intersect.
16. If m!QRS " m!SRT, then !!"
RS bisects !QRT.
130
Chapter 3 Geometry and Reasoning
Write each definition as two conditionals and as a single biconditional.
17. The midpoint of a segment is the point that divides it into two segments of
equal length.
18. Perpendicular lines are two lines that intersect to form right angles.
19. A transversal is a line that intersects two or more coplanar lines in different
points.
20. Vertical angles are two angles whose sides form two pairs of opposite rays.
GEOMETRIC CONSTRUCTION The corresponding angles postulate
provides a method for constructing parallel lines.
t
X
m
In the figure at the right, you see the finished construction of a line
parallel to line ! through point P. Trace line ! and point P onto a sheet of
paper and repeat the construction. Then complete the statements below
that outline the steps of the construction.
P
Y
A
!
21. Step 1: Using a straightedge, draw any line ___?__ through point P
intersecting line !. Label the intersection point ___?__.
Q
B
22. Step 2: With the compass point at ___?__, draw an arc intersecting lines t and
!. Label the intersection points ___?__ and ___?__.
23. Step 3: Using the same radius as in Step 2, place the compass point at
point ___?__ and draw an arc intersecting line t. Label the intersection
point ___?__.
24. Step 4: Place the compass point at point ___?__ and the pencil at
point ___?__. Using this radius, draw an arc that intersects line !.
25. Step 5: Using the same radius as in Step 4, place the compass point
at point ___?__ and draw an arc that intersects the arc you drew in
Step 3. Label the intersection point ___?__.
26. Step 6: Draw line ___?__ through points P and Y. m!___?__ " m!___?__ ,
and so ___?__ ! ___?__.
EXTENDED PRACTICE EXERCISES
WRITING MATH Explain why each of the following is not a good definition.
27. Vertical angles are two angles whose sides form opposite rays.
28. A line segment is part of a line.
29. Complementary angles are adjacent angles whose exterior sides form a right angle.
30. Skew lines are noncoplanar lines that do not intersect.
MIXED REVIEW EXERCISES
Find each length. (Lesson 3-1)
31. In the figure below, AC " 130. Find BC.
3x ! 4
32. In the figure below, LM " 94. Find LN.
4x
A
B
C
33. In the figure below, JL " 88. Find KL.
3y " 8
J
L
mathmatters3.com/self_check_quiz
3p
M
N
34. In the figure below, QS " 41. Find QR.
9y
K
7p #14
2z ! 14
L
Q
3z " 18
R
S
Lesson 3-6 Conditional Statements
131
Name _________________________________________________________
RETEACHING
Date ____________________________
3-6
CONDITIONAL STATEMENTS
Conditional statements are written in an if-then (hypothesis/conclusion) form.
The converse of a conditional statement is formed by interchanging the hypothesis
and the conclusion.
A counterexample proves a conditional or converse is false.
E x a m p l e
— —
— —
—
Write the converse of this statement: If RS || TU, then RS , TU, and RT are coplanar.
Then decide whether the statement and its converse are true or false.
If false, give a counterexample.
Solution
!S
!, T
!U
! and R
!T
! are coplanar, then !
RS
! || T
!U
!.
Converse: If R
Original statement is true, since parallel lines are coplanar by definition and for any
two points in a plane, the line joining them lies in the plane (Postulate 3).
Converse is false. Counterexample: Each pair of coplanar lines could intersect to
form a triangle.
! EXERCISES
Write the converse of each statement. Then tell whether the given statement and its
converse are true or false. If false, give a counterexample.
1. If points A, B, C and D lie on both plane L and plane M, then points A, B, C, and D
are collinear.
2. If m"IQJ ! m"HQJ " 180°, then "IQJ and "HQJ are obtuse angles.
3. If three lines have one point in common, then they are coplanar.
4. If two lines are skew, then they are not coplanar.
© Glencoe/McGraw-Hill
88
MathMatters 3
Name _________________________________________________________
EXTRA PRACTICE
Date ____________________________
3-6
CONDITIONAL STATEMENTS
! EXERCISES
Sketch a counterexample that shows why each conditional is
false. Use your own paper.
1. If point A is the midpoint of !
CD
!, then C
!A
!—
| !
DA
!.
➝
➝
2. If XY and XZ are opposite rays, then point X is the midpoint of !
YZ
!.
➝
3. If "RST and "TSV are congruent, then ST bisects "RSV.
Write the converse of each statement. Then tell whether the given
statement and its converse are true or false.
4. If points A, B, and C are collinear, then B is the midpoint of !
AC
!.
5. If point X is the vertex of "1 and "2, then "1 and "2 are adjacent angles.
6. If two angles are complementary, then both of the angles are acute.
7. If two lines intersect, then they are parallel.
8. If an angle is obtuse, then its supplement is acute.
Write each definition as two conditionals and as a single biconditional.
9. Coplanar points are points that lie in the same plane.
10. A segment is a part of a line that begins at one endpoint and ends at another.
© Glencoe/McGraw-Hill
89
MathMatters 3
Review and Practice Your Skills
PRACTICE
LESSON 3-5
Draw the next figure in each pattern. Then describe the tenth figure in the
pattern.
1.
2.
O
O
O
3.
4.
5.
6.
7.
PRACTICE
O
8.
LESSON 3-6
Sketch a counterexample to show why each conditional is false.
9. If !ABC and !DEF are supplements, then m!ABC ! m!DEF.
10. If three points are coplanar, then they are collinear.
11. If two lines are skew, then they intersect.
Write the converse of each statement. Then tell whether the given statement
and its converse are true or false.
12. If two lines intersect, then they are perpendicular.
13. If C is the midpoint of A!B!, then AB " 2(AC ).
14. If two angles are vertical angles, then their supplements are equal.
Write each definition as two conditionals and as a single biconditional.
15. Perpendicular lines are lines that intersect to form right angles.
16. Skew lines are noncoplanar lines.
132
Chapter 3 Geometry and Reasoning
Name _________________________________________________________
ENRICHMENT
Date ____________________________
3-6
CATEGORICAL PROPOSITIONS
A categorical proposition is a statement about an entire category
or class of things. There are four different standard forms of
categorical propositions.
All S is P.
No S is P.
Some S is P.
Some S is not P.
All dogs are friendly.
No dogs are friendly.
Some dogs are friendly.
Some dogs are not friendly.
Venn diagrams can be used to illustrate categorical propositions.
E x a m p l e
1
E x a m p l e
2
Diagram “All S is P.”
Solution
Diagram “Some S is not P.”
Solution
The shading shows that this part of
the diagram has no members.
The X shows that this part of the
diagram has at least one member.
! EXERCISES
Draw a Venn diagram for each categorical proposition.
1. No S is P.
2. Some S is P.
Write the converse of each of the four standard forms. Then draw a Venn diagram for
each one.
3.
4.
5.
6.
7. Which of the standard forms are logically equivalent to their converses?
© Glencoe/McGraw-Hill
90
MathMatters 3
3-6
Lesson Planning
NCTM Standards/Strands
■
■
■
■
■
■
■
■
Number & Operations
Algebra
Geometry
Problem Solving
Reasoning & Proof
Communication
Connections
Representation
■ Identify and evaluate conditional statements.
■ Identify converses and biconditionals.
Applications
Drafting, Sports, Geography
True; this is a
a. Denver is the capital of Colorado. fact.
b. For all x, x2 ! x. False; !"12"" < "12".
c. Lines m and n are parallel.
2
Cannot tell; there is no information given to
identify line m and n.
BUILD UNDERSTANDING
hypothesis
counterexample
Many of the statements in this chapter are written
in if–then form. Statements like these are called
conditional statements, or simply conditionals. The clause following “if” is called the
hypothesis of the conditional. The clause following “then” is called the conclusion.
For example, the parallel lines postulate was presented as a conditional.
Materials Needed
paper/pencil
Lesson Resources
then corresponding angles are equal in measure .
conclusion
Example 1
5-MINUTE WARM-UP
Tell whether each conditional is true or false.
a. If two lines are parallel, then they are coplanar.
b. If two lines do not intersect, then they are parallel.
Is each statement true or false?
1. If a figure is a square, then it
has four sides. true
2. If a figure has four sides, then
it is a square. false
Solution
a. Parallel lines are defined as coplanar lines that do not intersect. So, the
conditional is true.
b. Consider skew lines k and ! shown. By the definition of skew lines, k and !
do not intersect, and so the hypothesis is true. However, also by the
definition of skew lines, k and ! are noncoplanar. Lines k and ! cannot be
parallel, and so the conclusion is false. Therefore, lines k and ! are a
counterexample, and the conditional is false.
Introduction to Lesson 3-6
Have students work in small groups
to answer the questions. Point out
that while the first two statements
are either true or false, the third
cannot be determined without
knowing more about lines m and n.
3. If an angle is bisected by a ray,
then the two adjacent angles
formed are equal in measure.
If an angle is divided by a ray
into two adjacent angles that
are equal in measure, then the
ray bisects the angle. An angle
is bisected by a ray if and only
if the two adjacent angles
formed are equal in measure.
hypothesis
To demonstrate that a conditional is false, you need to find only one example for
which the hypothesis is true but the conclusion is false. An example like this is
called a counterexample.
Getting Started
ADDITIONAL ANSWERS
If two parallel lines are cut by a transversal ,
Denver, CO
A conditional is either true or false. When a conditional is true, you can justify it
in a variety of ways. For instance, you may be able to show that the conditional is
true because it follows directly from a definition. When a conditional is a
postulate, such as the parallel lines postulate, it is assumed to be true. Still other
conditionals are theorems, and these must be proved true.
Warm-Up Transparency 8
Reteaching 3-6
Extra Practice 3-6
Enrichment 3-6
128
Goals
Do you think each statement is true or false?
Explain your reasoning.
Vocabulary
conditional
conclusion
biconditional
Conditional
Statements
128
k
!
Chapter 3 Geometry and Reasoning
t
7.
8.
A
P
g
B
5
h
5
Q
R
9.
B
A
C
10. There are two possible counterexamples. In the following figure,
!AXB and !AXC share a common side
and a common vertex, but they also
have interior points in common. Therefore, they are not adjacent angles.
Chapter 3 Geometry and Reasoning
X
C
In the figure below, !MON and !NOP
share a common side and a common
vertex, but they
do not lie in the
M
N
same plane.
P
O
Therefore, they
are not adjacent.
The converse of a conditional is formed by interchanging the
hypothesis and the conclusion. The fact that a conditional is true
is no guarantee that its converse is true.
Example 2
DRAFTING People who draw plans must apply this true
statement: If two lines are parallel, then they do not intersect.
Write the converse of the statement. Is it also true?
Reading
Math
Chalkboard Examples
Many statements about
everyday situations can be
expressed as conditionals.
For instance, the
following is a true
conditional.
If it is raining,
then it is cloudy.
Write the converse of this statement: If a number is divisible by six,
then it is divisible by three. Then
decide whether the statement and
its converse are true or false. If a
number is divisible by three, then it
is divisible by six. The statement is
true; the converse is false.
Interchange the hypothesis and the conclusion of the given
statement.
Statement:
If two lines are parallel , then they do not intersect .
Converse:
If two lines do not intersect , then they are parallel .
hypothesis
conclusion
hypothesis
conclusion
Supplementary Example 3
By definition, parallel lines do not intersect, and so the given statement is true.
Part b of Example 1 demonstrated that lines that do not intersect are not
necessarily parallel, and so the converse is false.
Write this definition as two conditionals and a single biconditional: A
triangle is a polygon with three
sides. If a polygon is a triangle,
then it has three sides. If a polygon
has three sides, then it is a triangle.
A polygon is a triangle if and only if
it has three sides.
The converse of the parallel lines postulate also is assumed to be true. It is stated
as the corresponding angles postulate in the following manner.
Postulate 10
Tell whether this conditional is true
or false: If two lines are coplanar,
then they are not skew lines. true
Supplementary Example 2
Its converse is false.
If it is cloudy,
then it is raining.
Solution
Supplementary Example 1
The Corresponding Angles Postulate If two lines are
cut by a transversal so that a pair of corresponding
angles are equal in measure, then the lines are parallel.
When a statement and its converse are both true, they can be combined into an
“if and only if” statement. This type of statement is called a biconditional
statement, or simply a biconditional. Every definition can be written as a
Problem Solving
biconditional.
Tip
Example 3
Write this definition as two conditionals and as a single biconditional.
A right angle is an angle whose measure is 90°.
Solution
When writing the
converse of a
conditional, you may
need to change the
wording of the
hypotheses and the
conclusion to make the
converse read clearly.
Name _________________________________________________________
RETEACHING
Date ____________________________
3-6
CONDITIONAL STATEMENTS
The definition leads to two true conditionals.
Conditional statements are written in an if-then (hypothesis/conclusion) form.
The converse of a conditional statement is formed by interchanging the hypothesis
and the conclusion.
A counterexample proves a conditional or converse is false.
If an angle is a right angle, then its measure is 90°.
If the measure of an angle is 90°, then it is a right angle.
E x a m p l e
— —
— —
—
Write the converse of this statement: If RS || TU, then RS , TU, and RT are coplanar.
Then decide whether the statement and its converse are true or false.
If false, give a counterexample.
These can be combined into a single biconditional as follows.
Solution
An angle is a right angle if and only if its measure is 90°.
mathmatters3.com/extra_examples
Reteaching Worksheet 3-6
Lesson 3-6 Conditional Statements
129
#, #
TU
# and #
RT
# are coplanar, then #
RS
# || #
TU
#.
Converse: If #
RS
Original statement is true, since parallel lines are coplanar by definition and for any
two points in a plane, the line joining them lies in the plane (Postulate 3).
Converse is false. Counterexample: Each pair of coplanar lines could intersect to
form a triangle.
" EXERCISES
Write the converse of each statement. Then tell whether the given statement and its
converse are true or false. If false, give a counterexample.
11. Converse; If points J, K, and L are
collinear, then they are coplanar. The
given statement is false. Its converse is
true.
12. Converse: If XY ! YZ " XZ, then point
Y is the midpoint of XZ
!. The given
statement is true. Its converse is false.
13. Converse: If two angles are complementary, then the sum of their measures is 90º. Both the given statement
and its converse are true.
14. Converse: If the sum of the measures
of two angles is greater than 90º, then
the angles are supplementary. The
given statement is true. Its converse is
false.
15. Converse: If two lines do not intersect,
then they are perpendicular. Both the
given statement and its converse are
false.
16. Converse: If RS
!" bisects !QRT, then
m!QRS " m!SRT. The given statement is false. Its converse is true.
1. If points A, B, C and D lie on both plane L and plane M, then points A, B, C, and D
are collinear.
If points A, B, C, and D are collinear, then points A, B, C, and D
lie in both plane L and plane M.; true; false; Counterexample:
↔
AD lies on plane L and plane L is parallel to plane M.
2. If m!IQJ # m!HQJ ! 180°, then !IQJ and !HQJ are obtuse angles.
If !IQJ and !HQJ are obtuse angles, then m!IQJ # m!HQJ
! 180°.; false; Counterexample: If m!IQJ $ 30° and m!HQJ
$ 170°, then m!IQJ # m!HQJ $ 200° 180°.; true
3. If three lines have one point in common, then they are coplanar.
If three lines are coplanar, then they have one point in common.
false; Counterexample: 2 lines can be coplanar and a third line
can intersect that plane at only one point.; false; 3 parallel lines
4. If two lines are skew, then they are not coplanar.
If two lines are not coplanar, then they are skew.; true; true
Lesson 3-6 Conditional Statements
129
TRY THESE EXERCISES
Lesson Wrap-up
1. TALK ABOUT IT Decide whether this conditional is true or false.
False; it is possible that the two lines are noncoplanar.
If two lines are each perpendicular to a third line, then
they are parallel to each other.
QUICK ASSESSMENT
Discuss your reasoning with a classmate.
Ask the following questions to
determine if students understand
the content presented in this lesson.
1. When is a conditional statement
true? when it is always true;
when it can be shown that if the
hypothesis is true, then the conclusion must also be true
2. Give an example of a mathematical statement that is true
although its converse is false.
Answers may vary.
3. Explain how a definition and a
biconditional statement are
related. Any definition can be
written as two true conditional
statements (the statement and
its converse), which can be written as a biconditional.
2. Write the converse of this statement.
If two angles are equal in measure, then they are vertical angles.
If two angles are vertical angles, then they are equal in measure.
Are the given statement and its converse true or false?
Given statement is true. Converse is false.
3. Write this definition as two conditionals and as a single
biconditional. See additional answers.
The bisector of an angle is the ray that divides the angle
into two adjacent angles that are equal in measure.
4. NUMBER SENSE Tell whether this conditional is true or false.
False. Negative integers are less than 1.
If a number is less than 1, the number is a proper fraction.
Write the converse of the statement. Is the converse true or false?
Converse: If the number is a proper fraction, then it is less than 1. True.
5. SPORTS If a shortstop makes a bad throw to first base, the error
is charged to the shortstop. This statement is true. Write the
converse of the statement. Is it true or false? If an error is charged to
the shortstop, then the shortstop made a bad throw to first base. False.
6. GEOGRAPHY If a point is located north of the equator, it has a
northern latitude. This statement is true. Write the converse of
the statement. Is it true or false?
If a point has a northern latitude, then the point is located north
of the equator. True.
PRACTICE EXERCISES
ASSIGNMENT GUIDE
Basic: 1–26, 31, 32
Enriched: 1–32
Sketch a counterexample that shows why each conditional is false.
See additional answers.
7. If line t intersects lines g and h, then line t is a transversal.
8. If PQ $ QR, then point Q is the midpoint of P#R#.
9. If points A, B, and C are collinear, then !!"
BA and !!"
BC are opposite rays.
ADDITIONAL ANSWERS
17. Conditionals: If a point is the
midpoint of a segment, then it
divides the segment into two
segments of equal length; if a
point divides a segment into
two segments of equal length,
then it is the midpoint of the
segment. Biconditional: A point
is the midpoint of a segment if
and only if it divides the segment into two segments of
equal length.
18. Conditionals: If two lines are
perpendicular, then they intersect to form right angles; 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.
19. Conditionals: If a line is a transversal, then it intersects two or
more coplanar lines in different
points; if a line intersects two or
more coplanar lines in different
points, then it is a transversal.
Biconditional: A line is a transversal if and only if it intersects
130 Chapter 3 Geometry and Reasoning
• For Extra Practice, see page 672.
10. If two angles share a common side and a common vertex, then
they are adjacent angles.
Write the converse of each statement. Then tell whether the given
statement and its converse are true or false. See additional answers.
11. If points J, K, and L are coplanar, then they are collinear.
12. If point Y is the midpoint of X#Z#, then XY # YZ $ XZ.
13. If the sum of the measures of two angles is 90°, then the angles are
complementary.
14. If two angles are supplementary, then the sum of their measures is greater
than 90°.
15. If two lines are perpendicular, then they do not intersect.
16. If m!QRS $ m!SRT, then !!"
RS bisects !QRT.
130
Chapter 3 Geometry and Reasoning
two or more coplanar lines in different points.
20. Conditionals: If two angles are vertical angles, then their sides form two
pairs of opposite rays; if the sides of
two angles form two pairs of opposite rays, then they are vertical
angles. Biconditional: Two angles are
vertical angles if and only if their
sides form two pairs of opposite rays.
27. Write the given definition as two conditional statements: If two angles are
vertical angles, then their sides form
opposite rays is true. However, If the
sides of two angles form opposite
rays, then the angles are vertical
angles is false. Here is a counterexample in which the sides of !1 and !2
form a pair of opposite rays, but the
angles are not vertical angles. For this
reason, it is necessary
to define vertical
1 2
angles as two angles
whose sides form two
pairs of opposite rays.
Write each definition as two conditionals and as a single biconditional.
See additional answers.
17. The midpoint of a segment is the point that divides it into two segments of
equal length.
Extra Practice Worksheet 3-6
18. Perpendicular lines are two lines that intersect to form right angles.
19. A transversal is a line that intersects two or more coplanar lines in different
points.
Name _________________________________________________________
20. Vertical angles are two angles whose sides form two pairs of opposite rays.
GEOMETRIC CONSTRUCTION The corresponding angles postulate
provides a method for constructing parallel lines.
CONDITIONAL STATEMENTS
t
" EXERCISES
X
m
In the figure at the right, you see the finished construction of a line
parallel to line ! through point P. Trace line ! and point P onto a sheet of
paper and repeat the construction. Then complete the statements below
that outline the steps of the construction.
P
Y
A
!
21. Step 1: Using a straightedge, draw any line ___?__ through point P
intersecting line !. Label the intersection point ___?__. t, Q
Q
Date ____________________________
3-6
EXTRA PRACTICE
Sketch a counterexample that shows why each conditional is
false. Use your own paper. Check students’ drawings.
1. If point A is the midpoint of C
#D
#, then #
CA
#—
| D
#A
#.
➝
➝
2. If XY and XZ are opposite rays, then point X is the midpoint of Y
#Z
#.
➝
3. If !RST and !TSV are congruent, then ST bisects !RSV.
Write the converse of each statement. Then tell whether the given
statement and its converse are true or false.
B
If B is the
midpoint of #
AC
#, then points A, B, and C are collinear; false; true
If point X is the vertex of !1 and !2, then !1 and !2 are adjacent angles. If !1
and !2 are adjacent angles, then they have a common vertex;
false; true
If two angles are complementary, then both of the angles are acute. If two
angles are acute, then the angles are complementary.; true; false
If two lines intersect, then they are parallel. If two lines are parallel, then
the line intersect.; false; false
If an angle is obtuse, then its supplement is acute. If the supplement of an
angle is acute, then the angle is obtuse.; true; true
4. If points A, B, and C are collinear, then B is the midpoint of #
AC
#.
22. Step 2: With the compass point at ___?__, draw an arc intersecting lines t and
!. Label the intersection points ___?__ and ___?__. Q, A, B
5.
23. Step 3: Using the same radius as in Step 2, place the compass point at
point ___?__ and draw an arc intersecting line t. Label the intersection
point ___?__. P, X
6.
7.
8.
24. Step 4: Place the compass point at point ___?__ and the pencil at
point ___?__. Using this radius, draw an arc that intersects line !. A, B
Write each definition as two conditionals and as a single biconditional.
If points are coplanar,
then they lie in the same plane. If points lie in the same plane,
then the points are coplanar. Points are coplanar if and only if
they lie in the same plane
9. Coplanar points are points that lie in the same plane.
25. Step 5: Using the same radius as in Step 4, place the compass point
at point ___?__ and draw an arc that intersects the arc you drew in
Step 3. Label the intersection point ___?__. X, Y
10. A segment is a part of a line that begins at one endpoint and ends at another.
26. Step 6: Draw line ___?__ through points P and Y. m!___?__ $ m!___?__ ,
and so ___?__ ! ___?__. m, XPY, AQB, m, !
If a figure is a segment, then it is a part of a line that
begins at one endpoint and ends at another. If a figure is a part of
a line that begins at one endpoint and ends at another, then the
figure is a segment. A figure is a segment if and only if it is part
of a line that begins at one endpoint and ends at another.
EXTENDED PRACTICE EXERCISES
WRITING MATH Explain why each of the following is not a good definition.
See additional answers.
27. Vertical angles are two angles whose sides form opposite rays.
28. A line segment is part of a line.
29. Complementary angles are adjacent angles whose exterior sides form a right angle.
Enrichment Worksheet 3-6
30. Skew lines are noncoplanar lines that do not intersect.
MIXED REVIEW EXERCISES
Name _________________________________________________________
Find each length. (Lesson 3-1)
ENRICHMENT
31. In the figure below, AC $ 130. Find BC. 72 32. In the figure below, LM $ 94. Find LN. 70
3x ! 4
4x
A
B
C
33. In the figure below, JL $ 88. Find KL. 72
3y # 8
J
L
mathmatters3.com/self_check_quiz
3p
M
N
CATEGORICAL PROPOSITIONS
A categorical proposition is a statement about an entire category
or class of things. There are four different standard forms of
categorical propositions.
All S is P.
No S is P.
Some S is P.
Some S is not P.
34. In the figure below, QS $ 41. Find QR. 32
9y
K
7p #14
2z ! 14
L
Q
3z # 18
R
Date ____________________________
3-6
All dogs are friendly.
No dogs are friendly.
Some dogs are friendly.
Some dogs are not friendly.
Venn diagrams can be used to illustrate categorical propositions.
E x a m p l e
S
Lesson 3-6 Conditional Statements
131
1
E x a m p l e
2
Diagram “All S is P.”
Diagram “Some S is not P.”
Solution
Solution
The shading shows that this part of
the diagram has no members.
The X shows that this part of the
diagram has at least one member.
" EXERCISES
28. Write the given definition as two conditionals: If a figure is a line segment,
then it is part of a line is true. However, If a figure is part of a line, then
it is a line segment is false. A ray also
is part of a line. It is necessary to
specify that a line segment is part
of a line that begins at one endpoint
and ends at another.
29. Write the given definition as two conditionals: If two angles are adjacent
angles whose exterior sides form a
right angle, then they are complementary is true. However, If two
angles are complementary, then they
are adjacent angles whose exterior
sides form a right angle is false.
Complementary angles are not necessarily adjacent.
30. The given definition is a true statement, but it contains too much information to be a good definition. It is
not necessary to include the phrase
that do not intersect because the
term noncoplanar lines already
indicates that the two lines do
not intersect.
Draw a Venn diagram for each categorical proposition.
1. No S is P.
S
2. Some S is P.
P
P
S
X
Write the converse of each of the four standard forms. Then draw a Venn diagram for
each one.
All P is S.
3.
S
5.
No P is S.
4.
P
Some P is S.
P
S
6.
Some P is not S.
P
S
X
7. Which of the standard forms are logically equivalent to their converses?
No S is P. ↔ No P is S. and Some S is P. ↔ Some P is S.
Lesson 3-6 Conditional Statements
131
Skills Practice
Review and Practice Your Skills
Vocabulary Review
PRACTICE
Lesson 3-5
inductive reasoning conjecture
sequence
Lesson 3-6
conditional
conclusion
biconditional
LESSON 3-5
Draw the next figure in each pattern. Then describe the tenth figure in the
pattern. See additional answers for drawings.
1.
2.
O
O
O
hypothesis
counterexample
O
same as second figure
19 units horizontal; 10 units vertical
ASSIGNMENT GUIDE
3.
4.
All students: 1–24
10 units long
ADDITIONAL ANSWERS
5.
1.
6.
same orientation as second figure, but
with ten lines in the interior
2.
7.
3.
O
1
3
512 units each ""9 as long as original line.
180 °
"
512 individual angles, each !"
512 "
8.
same as second figure
5.
13–sided polygon with all diagonals
PRACTICE
LESSON 3-6
Sketch a counterexample to show why each conditional is false.
6.
See additional answers for sample answers.
9. If !ABC and !DEF are supplements, then m!ABC ! m!DEF.
10. If three points are coplanar, then they are collinear.
11. If two lines are skew, then they intersect.
7.
Write the converse of each statement. Then tell whether the given statement
and its converse are true or false.
If two lines are perpendicular, then they
12. If two lines intersect, then they are perpendicular. intersect. false, true
13. If C is the midpoint of A#B#, then AB $ 2(AC ). If AB $ 2(AC), then C is the midpoint of A#B#. true, false
14. If two angles are vertical angles, then their supplements are equal.
8.
If the supplements of two angles are equal, then they are vertical angles. true, false
Write each definition as two conditionals and as a single biconditional.
9.
C
15. Perpendicular lines are lines that intersect to form right angles. See additional answers.
16. Skew lines are noncoplanar lines. If lines are skew, they are noncoplanar. If lines are noncoplanar,
D
30%
A
10.
#
X
132
150%
B E
then they are skew. Two lines are skew if and only if they are noncoplanar.
Chapter 3 Geometry and Reasoning
F
Y
4.
15. If two lines are perpendicular, then they intersect to form right angles. If two
lines intersect to form right angles, then they are perpendicular. Two lines
are perpendicular if and only if they intersect to form right angles.
Z
11.
"
!
132 Chapter 3 Geometry and Reasoning
© Glencoe/McGraw-Hill
A9
8
9
10
5.
6.
7.
© Glencoe/McGraw-Hill
7
4.
D10 # 1430
D 9 # 429
D 8 # 132
D 7 # 42
D 6 # 14
D5 # 5
6
D4 # 2
5
3.
D3 # 1
4
Number of
Dissections
3
Number
of Sides
87
MathMatters 3
D7 ! D3D6 ! D4D5 ! D5D4 ! D6D3 ! D7
D8 ! D3D7 ! D4D6 ! D5D5 ! D6D4 ! D7D3 ! D8
D9 ! D3D8 ! D4D7 ! D5D6 ! D6D5 ! D7D4 ! D8D3 ! D9
D6 ! D3D5 ! D4D4 ! D5D3 ! D6
D5 ! D3D4 ! D4D3 ! D5
D4 ! D3D3 ! D4
D3 ! D3
14 ways
2 ways
Formula for Dn
Study the patterns in this chart. Then fill in the missing values.
2. Show all the different ways to dissect a convex
six-sided polygon into 4 triangles. How many are there?
1. Show all the different ways to dissect a convex
four-sided polygon into 2 triangles. How many are there?
! EXERCISES
© Glencoe/McGraw-Hill
88
If two lines are not coplanar, then they are skew.; true; true
4. If two lines are skew, then they are not coplanar.
MathMatters 3
If three lines are coplanar, then they have one point in common.
false; Counterexample: 2 lines can be coplanar and a third line
can intersect that plane at only one point.; false; 3 parallel lines
3. If three lines have one point in common, then they are coplanar.
If !IQJ and !HQJ are obtuse angles, then m!IQJ ! m"HQJ
" 180°.; false; Counterexample: If m!IQJ # 30° and m!HQJ
# 170°, then m!IQJ ! m!HQJ # 200° 180°.; true
2. If m"IQJ ! m"HQJ " 180°, then "IQJ and "HQJ are obtuse angles.
If points A, B, C, and D are collinear, then points A, B, C, and D
lie in both plane L and plane M.; true; false; Counterexample:
↔
AD lies on plane L and plane L is parallel to plane M.
1. If points A, B, C and D lie on both plane L and plane M, then points A, B, C, and D
are collinear.
Write the converse of each statement. Then tell whether the given statement and its
converse are true or false. If false, give a counterexample.
! EXERCISES
Solution
!S
!, !
TU
! and R
!T
! are coplanar, then !
RS
! || T
!U
!.
Converse: If R
Original statement is true, since parallel lines are coplanar by definition and for any
two points in a plane, the line joining them lies in the plane (Postulate 3).
Converse is false. Counterexample: Each pair of coplanar lines could intersect to
form a triangle.
E x a m p l e
— —
— —
—
Write the converse of this statement: If RS || TU, then RS , TU, and RT are coplanar.
Then decide whether the statement and its converse are true or false.
If false, give a counterexample.
Conditional statements are written in an if-then (hypothesis/conclusion) form.
The converse of a conditional statement is formed by interchanging the hypothesis
and the conclusion.
A counterexample proves a conditional or converse is false.
3-6
Date ____________________________
CONDITIONAL STATEMENTS
RETEACHING
Name _________________________________________________________
There are five ways to dissect a convex five-sided polygon into three triangles.
3-5
Date ____________________________
PATTERNS WITH DISSECTIONS
ENRICHMENT
Name _________________________________________________________
Answers (Lesson 3-5 and 3-6)
MathMatters 3
© Glencoe/McGraw-Hill
A10
© Glencoe/McGraw-Hill
89
MathMatters 3
If a figure is a segment, then it is a part of a line that
begins at one endpoint and ends at another. If a figure is a part of
a line that begins at one endpoint and ends at another, then the
figure is a segment. A figure is a segment if and only if it is part
of a line that begins at one endpoint and ends at another.
10. A segment is a part of a line that begins at one endpoint and ends at another.
9.
Coplanar points are points that lie in the same plane. If points are coplanar,
then they lie in the same plane. If points lie in the same plane,
then the points are coplanar. Points are coplanar if and only if
they lie in the same plane
Write each definition as two conditionals and as a single biconditional.
8.
7.
6.
5.
4. If points A, B, and C are collinear, then B is the midpoint of !
AC
!.
If B is the
midpoint of A
!C
!, then points A, B, and C are collinear; false; true
If point X is the vertex of "1 and "2, then "1 and "2 are adjacent angles. If "1
and "2 are adjacent angles, then they have a common vertex;
false; true
If two angles are complementary, then both of the angles are acute. If two
angles are acute, then the angles are complementary.; true; false
If two lines intersect, then they are parallel. If two lines are parallel, then
the line intersect.; false; false
If an angle is obtuse, then its supplement is acute. If the supplement of an
angle is acute, then the angle is obtuse.; true; true
Write the converse of each statement. Then tell whether the given
statement and its converse are true or false.
Sketch a counterexample that shows why each conditional is
false. Use your own paper. Check students’ drawings.
1. If point A is the midpoint of !
CD
!, then C
!A
!—
| !
DA
!.
➝
➝
2. If XY and XZ are opposite rays, then point X is the midpoint of !
YZ
!.
➝
3. If "RST and "TSV are congruent, then ST bisects "RSV.
All dogs are friendly.
No dogs are friendly.
Some dogs are friendly.
Some dogs are not friendly.
P
S
X
2. Some S is P.
S
Some P is S.
P
All P is S.
6.
4.
P
S
X
P
Some P is not S.
S
No P is S.
© Glencoe/McGraw-Hill
90
No S is P. ↔ No P is S. and Some S is P. ↔ Some P is S.
7. Which of the standard forms are logically equivalent to their converses?
5.
3.
MathMatters 3
Write the converse of each of the four standard forms. Then draw a Venn diagram for
each one.
S
1. No S is P.
P
The X shows that this part of the
diagram has at least one member.
Draw a Venn diagram for each categorical proposition.
! EXERCISES
The shading shows that this part of
the diagram has no members.
Solution
2
Diagram “Some S is not P.”
E x a m p l e
Solution
1
Diagram “All S is P.”
E x a m p l e
Venn diagrams can be used to illustrate categorical propositions.
All S is P.
No S is P.
Some S is P.
Some S is not P.
A categorical proposition is a statement about an entire category
or class of things. There are four different standard forms of
categorical propositions.
3-6
Date ____________________________
! EXERCISES
ENRICHMENT
Name _________________________________________________________
CATEGORICAL PROPOSITIONS
3-6
Date ____________________________
CONDITIONAL STATEMENTS
EXTRA PRACTICE
Name _________________________________________________________
Answers (Lesson 3-6)
MathMatters 3