Download 2 Parallel and Perpendicular Lines

C H A P T E R
2
Parallel and
Perpendicular Lines
© 2010 Carnegie Learning, Inc.
2
The Giant's Causeway, on the North coast of Ireland, is made up of thousands
of basalt columns thrust up in a volcanic eruption over 60 million years ago.
The tops of the columns mainly form hexagonal (six sided) polygons, though
four, five, seven, and eight-sided polygons can also be found. You will classify
regular, irregular, concave, and convex polygons.
2.1
Transversals and Lines
Angles Formed by Transversals of
Parallel and Non-Parallel Lines | p. 79
2.2
Making Conjectures
Conjectures about Angles Formed
by Parallel Lines Cut by a
Transversal | p. 87
2.3
What’s Your Proof?
Alternate Interior Angle Theorem,
Alternate Exterior Angle Theorem,
Same-Side Interior Angle Theorem,
and Same-Side Exterior Angle
Theorem | p. 91
2.4
A Reversed Condition
Parallel Line Converse
Theorems | p. 97
2.5
Many Sides
Naming Geometric Figures | p. 105
2.6
Quads and Tris
Classifying Triangles and
Quadrilaterals | p. 113
Chapter 2
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Parallel and Perpendicular Lines
77
Introductory Problem for Chapter 2
The V Problem
You need to make a large letter “V” out of poster board to complete a school project.
You are given two pieces of rectangular poster board, each measuring 1˝ ⫻ 12˝.
1. Make your letter “V” similar to the one shown.
H
G
E
F
2
D
L
Follow these guidelines:
l Use only the two pieces of poster board and a protractor.
l Make only two cuts on each piece of poster board.
l Do not change the width of the poster board.
l The measure of ⬔GDE should be equal to 40°.
l Segments HL and FL should each measure 12 inches.
3. Name each angle in the interior of your letter “V”, then determine the measure
of each angle, if possible.
Be prepared to share your methods and solutions.
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Chapter 2
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Parallel and Perpendicular Lines
© 2010 Carnegie Learning, Inc.
2. How many angles are located in the interior of your letter “V”?
2.1
Transversals and Lines
Angles Formed by Transversals of Parallel and
Non-Parallel Lines
OBJECTIVES
KEY TERMS
In this lesson you will:
l Identify the relationship between two
lines in space.
l
l
Identify transversals.
l
Identify alternate interior and alternate
exterior angles.
l
Identify same-side interior and
same-side exterior angles.
l
l
Identify corresponding angles.
l
l
Explore relationships between the
measures of angles formed by a
transversal intersecting parallel and
non-parallel lines.
l
PROBLEM 1
© 2010 Carnegie Learning, Inc.
l
l
l
l
l
l
parallel lines
distance between a point
and a line
transversal
interior angles
exterior angles
alternate interior angles
same-side interior angles
alternate exterior angles
same-side exterior angles
corresponding angles
2
Line Relationships
In Euclidean geometry, there are four possible relationships between two lines.
Case 1: Two coplanar lines intersect at a single point.
Case 2: Two coplanar lines intersect at an infinite number of points.
Case 3: Two coplanar lines do not intersect.
Case 4: Two lines are not coplanar.
1. Identify which case is shown in each figure.
a.
b.
c.
d.
Lesson 2.1
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Transversals and Lines
79
2. Two lines intersect at a single point. Are the lines always coplanar? Explain.
3. Two lines intersect at an infinite number of points. Are the lines always
coplanar? Explain.
Take Note
Skew lines are
non-coplanar lines.
Parallel lines are coplanar lines that do not intersect.
The distance between a point and a line is the length of the segment
drawn from the point perpendicular to the line.
4. Two lines are parallel. Describe the distance between a point on one line and
the other line.
2
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5. Explain why skew lines cannot intersect.
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Parallel and Perpendicular Lines
PROBLEM 2
Angles and Angle Pairs Formed by
Intersecting Lines
The parallel symbol is ∥. The non-parallel symbol is ∦.
If 1 is parallel to 2, then the statement can be written using symbols as 1 ∥ 2,
which is read as “line 1 is parallel to line 2.” If 1 is not parallel to 2, then the
statement can be written using symbols as 1 ∦ 2, which is read as “line 1 is not
parallel to line 2.”
Sets of arrowheads are used
lines in geometric figures.
___ parallel
___ ___
___to indicate
The following figure shows AB ∥ CD and BC ∥ AD .
B
A
C
2
D
A transversal is a line that intersects two or more lines at distinct points.
A transversal is said to “cut” the lines.
© 2010 Carnegie Learning, Inc.
1. Given that 1 ∥ 2 and both lines intersect 3, name the transversal(s).
Identify the lines that each transversal intersects.
3
1
2
2. Given that 1 ∦ 2 and both lines intersect 3, name the transversal(s).
Identify the lines that each transversal intersects.
3
1
Lesson 2.1
2
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Transversals and Lines
81
When two lines are cut by a transversal, many angles are formed. Use the diagram
shown to answer Questions 3 through 10.
1 2
5 6
m
3 4
7 8
p
n
3. Identify all pairs of vertical angles.
2
4. Identify all linear pairs.
6. One pair of alternate interior angles is ⬔2 and ⬔7. Identify all other pairs of
alternate interior angles. Explain.
7. One pair of same-side interior angles is ⬔2 and ⬔3. Identify all other pairs of
same-side interior angles. Explain.
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5. Four of the eight angles are interior angles. The other four are exterior angles.
Angle 2 is an interior angle. Angle 1 is an exterior angle. Identify all other
interior and exterior angles. Explain.
8. One pair of alternate exterior angles is ⬔1 and ⬔8. Identify all other pairs of
alternate exterior angles. Explain.
9. One pair of same-side exterior angles is ⬔1 and ⬔4. Identify all other pairs of
same-side exterior angles. Explain.
10. One pair of corresponding angles is ⬔1 and ⬔3. Identify all other pairs of
corresponding angles. Explain.
2
Interior angles are angles that lie between or inside the two lines cut by
the transversal.
Exterior angles are angles that lie outside the two lines cut by the transversal.
Alternate interior angles are two angles that lie between the two lines on opposite
or alternate sides of the transversal.
© 2010 Carnegie Learning, Inc.
Same-side interior angles are two angles that lie between the two lines on the
same side of the transversal.
Alternate exterior angles are two angles that lie outside the two lines on opposite
or alternate sides of the transversal.
Same-side exterior angles are two angles that lie outside the two lines on the same
side of the transversal.
Corresponding angles are two angles that lie on the same side of the transversal in
corresponding positions.
Lesson 2.1
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Transversals and Lines
83
PROBLEM 3
The Measures of Angles
1. Draw two non-parallel lines cut by a transversal, number each angle, and then
use a protractor to measure each angle.
2
3. Describe the relationships between the measures of each pair of angles when
the lines are not parallel and parallel.
a. Alternate interior angles
b. Alternate exterior angles
c. Corresponding angles
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Parallel and Perpendicular Lines
© 2010 Carnegie Learning, Inc.
2. Draw two parallel lines cut by a transversal, number each angle, and then use a
protractor to measure each angle.
d. Same-side interior angles
e. Same-side exterior angles
4. Summarize your conclusions in the chart. The relationships are either congruent
or not congruent, supplementary or not supplementary.
Type of Angles
When Two Parallel
Lines are Cut by a
Transversal
When Two Non-Parallel
Lines are Cut by a
Transversal
Alternate interior angles
2
Alternate exterior angles
Corresponding angles
Same-side interior angles
Same-side exterior angles
5. Did you use inductive or deductive reasoning to summarize your conclusion?
© 2010 Carnegie Learning, Inc.
6. Compare the measures of the angles everyone used and your chart to the
charts of the rest of your class. What do you notice?
7. Can we state the relationships in the chart as conjectures or theorems?
How many instances would be enough evidence to be considered a proof of
the relationships? Explain.
Be prepared to share your methods and solutions.
Lesson 2.1
|
Transversals and Lines
85
© 2010 Carnegie Learning, Inc.
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2.2
Making Conjectures
Conjectures about Angles Formed by Parallel
Lines Cut by a Transversal
OBJECTIVES
KEY TERMS
In this lesson you will:
l Use the Corresponding Angle Postulate.
l Make conjectures about pairs of angles
formed by the intersection of a transversal
and parallel lines.
PROBLEM 1
l
l
Corresponding Angle Postulate
conjecture
2
The Corresponding Angle Postulate
The Corresponding Angle Postulate states: “If two parallel lines are intersected by
a transversal, then corresponding angles are congruent.”
3
y
4
7 8
1 2
© 2010 Carnegie Learning, Inc.
5 6
w
x
Use the diagram shown to answer Questions 1 through 5.
1. Name the congruent pairs of corresponding angles in the figure.
A conjecture is a hypothesis that something is true; the hypothesis can later be
proved or disproved. Conjectures are generalizations made through
inductive reasoning.
Lesson 2.2
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Making Conjectures
87
2. Make a conjecture about alternate interior angles. Explain your reasoning.
3. Make a conjecture about alternate exterior angles. Explain your reasoning.
2
4. Make a conjecture about same-side interior angles. Explain your reasoning.
6. Did you use inductive or deductive reasoning to make each conjecture.
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© 2010 Carnegie Learning, Inc.
5. Make a conjecture about same-side exterior angles. Explain your reasoning.
PROBLEM 2
Determine the Measure of Unknown Angles
Use the Corresponding Angle Postulate to determine the measures of the
unknown angles.
1. If a ∥ d and m⬔1 ⫽ 38°, determine the measure of all unknown angles.
1
5
2
6
a
3
7
2
4
8
d
c
© 2010 Carnegie Learning, Inc.
2. If a ∥ d and m⬔1 ⫽ 67°, determine the measure of all unknown angles.
c
3 4
7 8
1 2
5 6
d
a
Lesson 2.2
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Making Conjectures
89
PROBLEM 3
Summary
In this lesson, you made several conjectures about pairs of angles formed by a
transversal intersecting two parallel lines. These conjectures can be stated as
theorems if you prove them to be true.
List each conjecture.
1. Alternate Interior Angle Conjecture:
2. Alternate Exterior Angle Conjecture:
2
3. Same-Side Interior Angle Conjecture:
Be prepared to share your methods and solutions.
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© 2010 Carnegie Learning, Inc.
4. Same-Side Exterior Angle Conjecture:
2.3
What’s Your Proof?
Alternate Interior Angle Theorem, Alternate Exterior
Angle Theorem, Same-Side Interior Angle Theorem, and
Same-Side Exterior Angle Theorem
OBJECTIVES
KEY TERMS
In this lesson you will:
l Prove the Alternate Interior Angle Theorem.
l Prove the Alternate Exterior Angle Theorem.
l Prove the Same-Side Interior Angle Theorem.
l Prove the Same-Side Exterior Angle Theorem.
l
l
l
l
Alternate Interior Angle Theorem
Alternate Exterior Angle Theorem
Same-Side Interior Angle
Theorem
Same-Side Exterior Angle
Theorem
2
Previously, you made conjectures about angle pairs formed by two parallel lines cut by
a transversal. If you can prove that a conjecture is true, then it becomes a theorem.
PROBLEM 1
Alternate Interior Angle Conjecture or Theorem?
© 2010 Carnegie Learning, Inc.
The Alternate Interior Angle Conjecture states: “If two parallel lines are intersected by
a transversal, then alternate interior angles are congruent.”
w
1 2
3 4
x
5 6
7 8
z
1. Use the diagram to write the given statement for the Alternate Interior
Angle Conjecture.
Given:
Prove: ⬔3 ⬔6
Lesson 2.3
|
What’s Your Proof?
91
2. Complete the flow chart proof of the Alternate Interior Angle Conjecture by
writing the reason for each statement in the boxes provided.
Given
Corresponding Angle Postulate
2
Transitive Property
Vertical angles are congruent
Congratulations! You have just proven the Alternate Interior Angle Conjecture.
It is now known as the Alternate Interior Angle Theorem. You can now use this
theorem as a valid reason in proofs.
3. Create a two-column proof of the Alternate Interior Angle Theorem.
Prove that ⬔4 ⬔5.
Reasons
© 2010 Carnegie Learning, Inc.
Statements
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PROBLEM 2
Alternate Exterior Angle Conjecture or Theorem?
The Alternate Exterior Angle Conjecture states: “If two parallel lines are intersected
by a transversal, then alternate exterior angles are congruent.”
1. Draw and label a diagram illustrating this conjecture.
2
© 2010 Carnegie Learning, Inc.
2. Use the diagram to write the given and prove statements for the Alternate
Exterior Angle Conjecture.
Given:
Prove:
3. Prove the Alternate Exterior Angle Conjecture.
Congratulations! You have just proven the Alternate Exterior Angle Conjecture.
It is now known as the Alternate Exterior Angle Theorem. You can now use this
theorem as a valid reason in proofs.
Lesson 2.3
|
What’s Your Proof?
93
PROBLEM 3
Same-Side Interior Angle Conjecture
or Theorem?
The Same-Side Interior Angle Conjecture states: “If two parallel lines are intersected by a
transversal, then interior angles on the same side of the transversal are supplementary.”
1. Draw and label a diagram illustrating this conjecture.
2
2. Use the diagram to write the given and prove statements for the Same-Side
Interior Angle Conjecture.
Given:
Prove:
© 2010 Carnegie Learning, Inc.
3. Prove the Same-Side Interior Angle Conjecture.
Congratulations! You have just proven the Same-Side Interior Angle Conjecture.
It is now known as the Same-Side Interior Angle Theorem. You can now use this
theorem as a valid reason in proofs.
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PROBLEM 4
Same-Side Exterior Angle Conjecture
or Theorem?
The Same-Side Exterior Angle Conjecture states: “If two parallel lines are intersected
by a transversal, then exterior angles on the same side of the transversal are
supplementary.”
1. Draw and label a diagram illustrating this conjecture.
2
2. Use the diagram to write the given and prove statements for the Same-Side
Exterior Angle Conjecture.
Given:
Prove:
© 2010 Carnegie Learning, Inc.
3. Prove Same-Side Exterior Angle Conjecture.
Lesson 2.3
|
What’s Your Proof?
95
Congratulations! You have just proven the Same-Side Exterior Angle Conjecture. It
is now known as the Same-Side Exterior Angle Theorem. You can now use this
theorem as a valid reason in proofs.
PROBLEM 5
Summary
1. Did you use inductive or deductive reasoning to prove each theorem?
If two parallel lines are intersected by a transversal, then:
a. corresponding angles are congruent.
b. alternate interior angles are congruent.
c. alternate exterior angles are congruent.
d. same-side interior angles are supplementary.
e. same-side exterior angles are supplementary.
2
Corresponding Angle Postulate: If two parallel lines are intersected by a
transversal, then corresponding angles are congruent.
Alternate Interior Angle Theorem: If two parallel lines are intersected by a
transversal, then alternate interior angles are congruent.
Same-Side Interior Angle Theorem: If two parallel lines are intersected
by a transversal, then interior angles on the same side of the transversal
are supplementary.
Same-Side Exterior Angle Theorem: If two parallel lines are intersected
by a transversal, then exterior angles on the same side of the transversal
are supplementary.
Be prepared to share your methods and solutions.
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Alternate Exterior Angle Theorem: If two parallel lines are intersected by a
transversal, then alternate exterior angles are congruent.
2.4
A Reversed Condition
Parallel Line Converse Theorems
OBJECTIVE
KEY TERMS
In this lesson you will:
l
l
Write and prove parallel line
converse conjectures.
l
l
PROBLEM 1
converse
Corresponding Angle
Converse Postulate
Alternate Interior Angle
Converse Theorem
l
l
l
Alternate Exterior Angle
Converse Theorem
Same-Side Interior Angle
Converse Theorem
Same-Side Exterior Angle
Converse Theorem
2
Converses
The converse of a conditional statement written in the form “If p, then q” is the
statement written in the form “If q, then p.” The converse is a new statement that
results when the hypothesis and conclusion of the conditional statement
are interchanged.
© 2010 Carnegie Learning, Inc.
The Corresponding Angle Postulate states: “If two parallel lines are intersected by a
transversal, then the corresponding angles are congruent.”
The Corresponding Angle Converse Postulate states: “If two lines intersected by a
transversal form congruent corresponding angles, then the lines are parallel.”
The Corresponding Angle Converse Postulate is used to prove new conjectures formed
by writing the converses of the parallel lines theorems.
Lesson 2.4
|
A Reversed Condition
97
For each parallel line theorem:
• Identify the hypothesis p and conclusion q.
• Write the converse of the theorem as a conjecture.
1. Alternate Interior Angle Theorem: If two parallel lines are intersected by a
transversal, then the alternate interior angles are congruent.
Hypothesis p:
Conclusion q:
Alternate Interior Angle Converse Conjecture:
2. Alternate Exterior Angle Theorem: If two parallel lines are intersected by a
transversal, then the alternate exterior angles are congruent.
Hypothesis p:
2
Conclusion q:
Alternate Exterior Angle Converse Conjecture:
3. Same-Side Interior Angle Theorem: If two parallel lines are intersected by a
transversal, then the same-side interior angles are supplementary.
Hypothesis p:
Conclusion q:
4. Same-Side Exterior Angle Theorem: If two parallel lines are intersected by a
transversal, then the same-side exterior angles are supplementary.
Hypothesis p:
Conclusion q:
Same-Side Exterior Angle Converse Conjecture:
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© 2010 Carnegie Learning, Inc.
Same-Side Interior Angle Converse Conjecture:
PROBLEM 2 Construction
Parallel Lines
1. Given line r and transversal s, use the Corresponding Angle Converse Postulate
to construct a line parallel to line r. Write the steps.
2
r
s
© 2010 Carnegie Learning, Inc.
2. Which line is a transversal?
3. Which lines are parallel?
Lesson 2.4
|
A Reversed Condition
99
PROBLEM 3
Proving the Parallel Line Converse Conjectures
The Alternate Interior Angle Converse Conjecture states: “If two lines intersected by a
transversal form congruent alternate interior angles, then the lines are parallel.”
w
1 2
3 4
x
2
5 6
7 8
z
1. Use the diagram to write the given and prove statements for the Alternate
Interior Angle Converse Conjecture.
Given:
Prove:
© 2010 Carnegie Learning, Inc.
2. Prove the Alternate Interior Angle Converse Conjecture.
Congratulations! You have just proven the Alternate Interior Angle Converse
Conjecture. It is now known as the Alternate Interior Angle Converse Theorem.
You can now use this theorem as a valid reason in proofs.
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The Alternate Exterior Angle Converse Conjecture states: “If two lines intersected by
a transversal form congruent alternate exterior angles, then the lines are parallel.”
w
1 2
3 4
x
5 6
7 8
z
3. Use the diagram to write the given and prove statements for the Alternate
Exterior Angle Converse Conjecture.
Given:
Prove:
2
© 2010 Carnegie Learning, Inc.
4. Prove the Alternate Exterior Angle Converse Conjecture.
Congratulations! You have just proven the Alternate Exterior Angle Converse
Conjecture. It is now known as the Alternate Exterior Angle Converse Theorem.
You can now use this theorem as a valid reason in proofs.
Lesson 2.4
|
A Reversed Condition
101
The Same-Side Interior Angle Converse Conjecture states: “If two lines intersected
by a transversal form supplementary same-side interior angles, then the lines
are parallel.”
w
1 2
3 4
x
2
5 6
7 8
z
5. Use the diagram to write the given and prove statements for the Same-Side
Interior Angle Converse Conjecture.
Given:
Prove:
© 2010 Carnegie Learning, Inc.
6. Prove the Same-Side Interior Angle Converse Conjecture.
Congratulations! You have just proven the Same-Side Interior Angle Converse
Conjecture. It is now known as the Same-Side Interior Angle Converse Theorem.
You can now use this theorem as a valid reason in proofs.
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The Same-Side Exterior Angle Converse Conjecture states: “If two lines intersected
by a transversal form supplementary same-side exterior angles, then the lines
are parallel.”
w
1 2
3 4
x
5 6
7 8
2
z
7. Use the diagram to write the given and prove statements for the Same-Side
Exterior Angle Converse Conjecture.
Given:
Prove:
© 2010 Carnegie Learning, Inc.
8. Prove the Same-Side Exterior Angle Converse Conjecture.
Congratulations! You have just proven the Same-Side Exterior Angle Converse
Conjecture. It is now known as the Same-Side Exterior Angle Converse Theorem.
You can now use this theorem as a valid reason in proofs.
Lesson 2.4
|
A Reversed Condition
103
PROBLEM 4
Summary
Corresponding Angle Converse Postulate: If two lines intersected by a transversal
form congruent corresponding angles, then the lines are parallel.
Each converse conjecture you have proven is a new theorem.
Alternate Interior Angle Converse Theorem: If two lines intersected by a
transversal form congruent alternate interior angles, then the lines are parallel.
Alternate Exterior Angle Converse Theorem: If two lines intersected by a
transversal form congruent alternate exterior angles, then the lines are parallel.
Same-Side Interior Angle Converse Theorem: If two lines intersected by a
transversal form supplementary same-side interior angles, then the lines are parallel.
2
Same-Side Exterior Angle Converse Theorem: If two lines intersected by a
transversal form supplementary same-side exterior angles, then the lines are parallel.
© 2010 Carnegie Learning, Inc.
Be prepared to share your methods and solutions.
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2.5
Many Sides
Naming Geometric Figures
OBJECTIVES
KEY TERMS
In this lesson you will:
l Classify geometric figures as polygons,
triangles, quadrilaterals, pentagons,
hexagons, heptagons, octagons,
nonagons, and decagons.
l Define diagonals of geometric figures.
l Define consecutive sides and angles.
l Define opposite sides and angles.
l Classify polygons as concave, convex,
regular, and irregular.
l
l
l
l
l
l
l
l
l
l
© 2010 Carnegie Learning, Inc.
PROBLEM 1
triangle
quadrilateral
adjacent or
consecutive sides
consecutive or
adjacent interior
angles
opposite sides
opposite angles
diagonal
convex figure
concave figure
reflex angle
l
l
l
l
l
l
l
l
l
polygon
regular polygon
irregular polygon
pentagon
hexagon
heptagon
octagon
nonagon
decagon
2
Triangles
You have already worked with basic geometric figures such as points, lines,
rays, planes, line segments, and angles. These basic figures can be used to build
more complex geometric figures. A triangle is the simplest closed three-sided
geometric figure.
1. Draw three different triangles.
Lesson 2.5
|
Many Sides
105
2. How many line segments, angles, and vertices are needed to form a triangle?
A triangle is named using three capital letters representing the vertices. Triangle
RAD can be written using symbols as 䉭RAD and is read as
“triangle RAD.” The triangle shown could be named 䉭RAD, 䉭ADR, or 䉭DRA.
A
D
R
The root word “tri” means “three,” so triangle literally means “three angles.”
A triangle is a closed figure because it has a well-defined interior and exterior.
3. Label the vertices of the triangles in Question 1, and then use symbols to name
each triangle.
2
4. Shade the interior of 䉭RAD shown in Question 2.
5. Name the three sides and the three angles of 䉭RAD.
Sides:
Angles:
Side 2
Side 1
Side 3
7. Compare the triangle that you constructed with the triangles that your
classmates constructed. What do you observe? Why?
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6. Construct a triangle with the three sides shown.
PROBLEM 2
Quadrilaterals
The root word “quad” means “four” and the root word “lateral” means “side.”
So, a quadrilateral is a closed geometric figure that has four sides. A quadrilateral is
named using four capital letters representing the vertices, listed in order clockwise
or counterclockwise.
1. Draw quadrilateral ABCD.
2
2. How many angles, sides, and vertices are needed to form a quadrilateral?
Use quadrilateral ABCD to answer Questions 3 through 7.
Adjacent sides or consecutive sides of a quadrilateral are two sides that share a
common endpoint.
© 2010 Carnegie Learning, Inc.
3. Name two pairs of consecutive sides.
Consecutive angles or adjacent interior angles of a quadrilateral are two angles
that share a common side.
4. Name two pairs of consecutive angles.
Opposite sides of a quadrilateral are two sides that do not share a common endpoint.
5. Name two pairs of opposite sides.
Opposite angles of a quadrilateral are two angles that do not share a
common side.
6. Name two pairs of opposite angles.
Lesson 2.5
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Many Sides
107
A diagonal of a closed geometric figure is a line segment whose endpoints are two
vertices that do not share a common side.
7. Draw the diagonals of Quadrilateral ABCD, and then name each diagonal.
8. Construct a quadrilateral with the four sides shown. Label and name the
quadrilateral.
Side 3
Side 1
2
Side 4
Side 2
Quadrilaterals ABCD and EFGH are shown with congruent sides indicated.
H
D
A
C
B
E
G
F
10. Draw and name the diagonals in each figure.
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© 2010 Carnegie Learning, Inc.
9. Compare the quadrilateral that you constructed with the quadrilaterals that your
classmates constructed. What do you observe? Why?
11. What is the difference between the diagonals of the quadrilaterals?
A convex figure is a closed geometric figure where line segments connecting any two
points in the interior of the figure are completely in the interior of the figure. Convex figures
can also be described as figures with all interior angles measuring less than to 180°.
A concave figure is a closed geometric figure that is not convex. Concave figures can
also be described as figures that have at least one interior reflex angle. A reflex angle is
any angle with a measure greater than 180° and less than 360°.
12. Classify each quadrilateral as convex or concave. If the quadrilateral is
concave, draw a line segment that connects two points in the interior such that
the line segment is not completely in the interior of the figure.
2
© 2010 Carnegie Learning, Inc.
PROBLEM 3
Polygons
Triangles and quadrilaterals are examples of geometric figures with many sides. The
root word “poly” means “many” and the root word “gon” means “side.” A polygon
is a closed figure that is formed by joining three or more line segments at their
endpoints. A regular polygon is a polygon with all sides congruent and all angles
congruent. An irregular polygon is a polygon that is not regular.
The root word “penta” means “five.” A pentagon is a five-sided polygon.
1. Sketch a convex pentagon, a concave pentagon, and a regular pentagon.
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2. Construct a pentagon with the five sides shown.
3. Compare the pentagon that you constructed with those that your classmates
constructed. What do you observe? Why?
2
4. Draw and label pentagon ABCDE.
6. Name two pairs of consecutive sides and two pairs of consecutive angles in
pentagon ABCDE.
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5. Draw and name the diagonals of pentagon ABCDE.
7. Does a pentagon have opposite sides? Opposite angles? Explain.
The root word “hexa” means “six.” A hexagon is a six-sided polygon.
8. Sketch a convex hexagon, a concave hexagon, and a regular hexagon.
The root word “hepta” means “seven.” A heptagon is a seven-sided polygon.
2
9. Sketch a convex heptagon, a concave heptagon, and a regular heptagon.
The root word “octa” means “eight.” An octagon is an eight-sided polygon.
© 2010 Carnegie Learning, Inc.
10. Sketch a convex octagon, a concave octagon, and a regular octagon.
The root word “nona” means “nine.” A nonagon is a nine-sided polygon.
11. Sketch a convex nonagon, a concave nonagon, and a regular nonagon.
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The root word “deca” means “ten.” A decagon is a ten-sided polygon.
12. Sketch a convex decagon, a concave decagon, and a regular decagon.
13. Classify each polygon as regular or irregular, and then classify each polygon as
convex or concave. Explain your reasoning.
a.
b.
c.
d.
© 2010 Carnegie Learning, Inc.
2
Be prepared to share your methods and solutions.
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2.6
Quads and Tris
Classifying Triangles and Quadrilaterals
OBJECTIVES
In this lesson you will:
l Classify triangles.
l Classify quadrilaterals.
KEY TERMS
l
l
l
l
l
l
l
PROBLEM 1
equilateral triangle
isosceles triangle
scalene triangle
equiangular triangle
acute triangle
right triangle
obtuse triangle
l
l
l
l
l
l
l
square
rectangle
rhombus
parallelogram
kite
trapezoid
counterexample
2
Triangles
Triangles are classified based on the characteristics of their angles, or the
characteristics of their sides.
© 2010 Carnegie Learning, Inc.
The root word “equi” means “equal” and the root word “lateral” means “sides.”
An equilateral triangle is a triangle with all sides congruent.
1. Construct an equilateral triangle using the side shown.
2. Compare the triangle that you constructed with the triangles that your
classmates constructed. What do you observe? Why?
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An isosceles triangle is a triangle with at least two congruent sides.
3. Construct an isosceles triangle using one of the congruent sides shown.
Indicate the congruent sides.
4. Compare the triangle that you constructed with the triangles that your
classmates constructed. What do you observe? Why?
2
A scalene triangle is a triangle with no congruent sides.
The root word “equi” means “equal” and the root word “angular” means “angle.”
An equiangular triangle is a triangle with all angles congruent.
6. Use your protractor to measure each angle of the triangle you constructed in
Question 1. What do you observe?
7. How are equilateral and equiangular triangles related? Explain.
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5. Draw three different scalene triangles.
An acute triangle is a triangle that has three angles that each measure less than 90°.
8. Draw three different acute triangles.
A right triangle is a triangle that has exactly one right angle.
9. Construct three different right triangles.
2
© 2010 Carnegie Learning, Inc.
10. Compare the right triangles that you constructed with the right triangles your
classmates constructed. What do you observe? Why?
An obtuse triangle is a triangle that has exactly one angle measuring greater than 90°.
11. Draw three different obtuse triangles.
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115
PROBLEM 2
Quadrilaterals
Quadrilaterals are classified based on the characteristics of their angles and the
characteristics of their sides.
A square is a quadrilateral with all sides congruent and all angles congruent.
1. Construct a square using the side shown.
2
2. Compare the squares that you constructed with the squares that your
classmates constructed. What do you observe? Why?
A rectangle is a quadrilateral with opposite sides congruent and all
angles congruent.
© 2010 Carnegie Learning, Inc.
3. Construct a rectangle using the two non-congruent sides shown.
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4. Compare the rectangle that you constructed with the rectangles that your
classmates constructed. What do you observe? Why?
A rhombus is a quadrilateral with all sides congruent. The plural of rhombus
is rhombi.
5. Construct a rhombus using the side shown.
2
6. Compare the rhombus that you constructed with the rhombi that your
classmates constructed. What do you observe? Why?
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
© 2010 Carnegie Learning, Inc.
7. Construct a parallelogram using the two non-congruent sides shown.
8. Compare the parallelogram that you constructed with the parallelograms your
classmates constructed. What do you observe? Why?
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A kite is a quadrilateral with two pairs of consecutive congruent sides with opposite
sides that are not congruent.
9. Construct a kite using the non-congruent sides shown.
10. Compare the kite that you constructed with the kites that your classmates
constructed. What do you observe? Why?
2
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
12. Compare the trapezoid that you constructed with the trapezoids that your
classmates constructed. What do you observe? Why?
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11. Construct a trapezoid using the starter line.
PROBLEM 3
True or False
To prove a conjecture, you must prove that it is true for all cases. But to disprove
a conjecture, you only need to show that it is not true for at least one case. A
counterexample is an example that demonstrates that a conjecture is not true.
Decide whether each statement about triangles or quadrilaterals is true or false.
Explain your answer and draw an example or counterexample, if possible.
1. All equilateral triangles are equiangular triangles.
2
© 2010 Carnegie Learning, Inc.
2. An isosceles triangle can be an obtuse, acute, or right triangle.
3. A scalene triangle can be an obtuse, acute, or right triangle.
4. A right triangle can be an obtuse triangle.
5. All squares are rectangles.
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119
6. All rectangles are squares.
7. All squares are rhombi.
8. All rhombi are squares.
2
9. All squares are parallelograms.
10. All rectangles are parallelograms.
12. All trapezoids are parallelograms.
13. Did you use inductive or deductive reasoning to determine if each statement
was true or false?
Be prepared to share your methods and solutions.
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11. All rhombi are parallelograms.
Chapter 2 Checklist
KEY TERMS
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
parallel lines (2.1)
distance between a point
l
and a line (2.1)
l
transversal (2.1)
l
interior angles (2.1)
l
exterior angles (2.1)
alternate interior angles (2.1) l
same-side interior angles (2.1) l
alternate exterior angles (2.1) l
same-side exterior angles (2.1) l
corresponding angles (2.1) l
l
conjecture (2.2)
l
converse (2.4)
l
triangle (2.5)
l
quadrilateral (2.5)
l
adjacent or consecutive
sides (2.5)
consecutive or adjacent
interior angles (2.5)
opposite sides (2.5)
opposite angles (2.5)
diagonal (2.5)
convex figure (2.5)
concave figure (2.5)
reflex angle (2.5)
polygon (2.5)
regular polygon (2.5)
irregular polygon (2.5)
pentagon (2.5)
hexagon (2.5)
heptagon (2.5)
octagon (2.5)
nonagon (2.5)
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
decagon (2.5)
equilateral triangle (2.6)
isosceles triangle (2.6)
scalene triangle (2.6)
equiangular triangle (2.6)
acute triangle (2.6)
right triangle (2.6)
obtuse triangle (2.6)
square (2.6)
rectangle (2.6)
rhombus (2.6)
parallelogram (2.6)
kite (2.6)
trapezoid (2.6)
counterexample (2.6)
2
POSTULATES
© 2010 Carnegie Learning, Inc.
l
Corresponding Angle
Postulate (2.2)
l
Corresponding Angle
Converse Postulate (2.4)
l
Same-Side Exterior Angle
Theorem (2.3)
Alternate Interior Angle
Converse Theorem (2.4)
Alternate Exterior Angle
Converse Theorem (2.4)
l
isosceles triangle (2.6)
right triangle (2.6)
square (2.6)
rectangle (2.6)
l
THEOREMS
l
l
l
Alternate Interior Angle
Theorem (2.3)
Alternate Exterior Angle
Theorem (2.3)
Same-Side Interior Angle
Theorem (2.3)
l
l
l
Same-Side Interior Angle
Converse Theorem (2.4)
Same-Side Exterior Angle
Converse Theorem (2.4)
CONSTRUCTIONS
l
l
l
l
l
parallel lines (2.4)
triangle (2.5)
quadrilateral (2.5)
pentagon (2.5)
equilateral triangle (2.6)
l
l
l
l
l
l
l
rhombus (2.6)
parallelogram (2.6)
kite (2.6)
trapezoid (2.6)
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2.1
Identifying Angle Pairs Formed by Intersecting Lines
When two lines are cut by a transversal, many angles are formed.
• Alternate interior angles are two angles that lie between the two lines on
opposite (or alternate) sides of the transversal.
• Same-side interior angles are two angles that lie between the two lines on the
same side of the transversal.
• Alternate exterior angles are two angles that lie outside the two lines on
opposite (or alternate) sides of the transversal.
• Same-side exterior angles are two angles that lie outside the two lines on the
same side of the transversal.
• Corresponding angles are two angles that lie on the same side of the
transversal in corresponding positions.
2
Examples:
t
1
2
4 3
a
5
8
6
7
b
The pairs of alternate interior angles are angles 3 and 5 and angles 4 and 6.
The pairs of alternate exterior angles are angles 1 and 7 and angles 2 and 8.
The pairs of same-side exterior angles are angles 1 and 8 and angles 2 and 7.
The pairs of corresponding angles are angles 1 and 5, angles 2 and 6, angles 3
and 7, and angles 4 and 8.
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The pairs of same-side interior angles are angles 3 and 6 and angles 4 and 5.
2.2
Using the Corresponding Angle Postulate
The Corresponding Angle Postulate states: “If two parallel lines are intersected by a
transversal, then corresponding angles are congruent.”
Examples:
r
130°
50°
s
1
2
t
The angle that measures 50° and ⬔1 are corresponding angles.
So, m⬔1 ⫽ 50º.
2
The angle that measures 130° and ⬔2 are corresponding angles.
So, m⬔2 ⫽ 130º.
2.3
Using the Alternate Interior Angle Theorem
The Alternate Interior Angle Theorem states: “If two parallel lines are intersected by a
transversal, then alternate interior angles are congruent.”
Examples:
117°
© 2010 Carnegie Learning, Inc.
63°
1
2
m
n
t
The angle that measures 63° and ⬔1 are alternate interior angles.
So, m⬔1 ⫽ 63º.
The angle that measures 117° and ⬔2 are alternate interior angles.
So, m⬔2 ⫽ 117º.
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2.3
Using the Alternate Exterior Angle Theorem
The Alternate Exterior Angle Theorem states: “If two parallel lines are intersected by
a transversal, then alternate exterior angles are congruent.”
Examples:
t
d
59°
121°
c
1
2
2
The angle that measures 121° and ⬔1 are alternate exterior angles. So, m⬔1 ⫽ 121º.
The angle that measures 59° and ⬔2 are alternate exterior angles. So, m⬔2 ⫽ 59º.
2.3
Using the Same-Side Interior Angle Theorem
The Same-Side Interior Angle Theorem states: “If two parallel lines are intersected by
a transversal, then same-side interior angles are supplementary.”
Examples:
1 2
p
q
t
The angle that measures 81° and ⬔1 are same-side interior angles.
So, m⬔1 ⫽ 180º ⫺ 81º ⫽ 99º.
The angle that measures 99° and ⬔2 are same-side interior angles.
So, m⬔2 ⫽ 180º ⫺ 99º ⫽ 81º.
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81° 99°
2.3
Using the Same-Side Exterior Angle Theorem
The Same-Side Exterior Angle Theorem states: “If two parallel lines are intersected
by a transversal, then same-side exterior angles are supplementary.”
Examples:
t
105°
75°
g
1 2
h
2
The angle that measures 105° and ⬔1 are same-side exterior angles.
So, m⬔1 ⫽ 180º ⫺ 105º ⫽ 75º.
The angle that measures 75° and ⬔2 are same-side exterior angles.
So, m⬔2 ⫽ 180º ⫺ 75º ⫽ 105º.
2.4
Using the Corresponding Angle Converse Postulate
The Corresponding Angle Converse Postulate states: “If two lines intersected by a
transversal form congruent corresponding angles, then the lines are parallel.”
Example:
j
© 2010 Carnegie Learning, Inc.
80°
k
80°
t
Corresponding angles have the same measure. So, j k.
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2.4
Using the Alternate Interior Angle Converse Theorem
The Alternate Interior Angle Converse Theorem states: “If two lines intersected by a
transversal form congruent alternate interior angles, then the lines are parallel.”
Example:
t
58°
58°
m
Alternate interior angles have the same measure. So, m.
2
2.4
Using the Alternate Exterior Angle Converse Theorem
The Alternate Exterior Angle Converse Theorem states: “If two lines intersected by a
transversal form congruent alternate exterior angles, then the lines are parallel.”
Example:
x
y
125°
t
125°
2.4
Using the Same-Side Interior Angle Converse Theorem
The Same-Side Interior Angle Converse Theorem states: “If two lines intersected
by a transversal form supplementary same-side interior angles, then the lines
are parallel.”
Example:
t
37°
143°
w
v
Same-side interior angles are supplementary: 37° ⫹ 143° ⫽ 180°. So, v w.
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Alternate exterior angles have the same measure. So, x y.
2.4
Using the Same-Side Exterior Angle Converse Theorem
The Same-Side Exterior Angle Converse Theorem states: “If two lines intersected
by a transversal form supplementary same-side exterior angles, then the lines are
parallel.”
Example:
b
c
131°
49°
t
Same-side exterior angles are supplementary: 131° ⫹ 49° ⫽ 180°. So, b c.
2.5
2
Classifying Polygons
A triangle is a three-sided polygon.
A quadrilateral is a four-sided polygon.
A pentagon is a five-sided polygon.
A hexagon is a six-sided polygon.
A heptagon is a seven-sided polygon.
An octagon is an eight-sided polygon.
A nonagon is a nine-sided polygon.
A decagon is a ten-sided polygon.
© 2010 Carnegie Learning, Inc.
Examples:
The polygon has
five sides. So, it is
a pentagon.
The polygon has
three sides. So, it
is a triangle.
The polygon has
nine sides. So, it is
a nonagon.
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2.5
Determining Whether Polygons are Convex or Concave
A convex polygon is a polygon in which line segments connecting any two points in
the interior of the polygon are completely in the interior of the polygon. Each interior
angle in a convex polygon measures less than 180°.
A concave polygon is a polygon that is not convex. Concave polygons have at least
one interior reflex angle. A reflex angle is any angle with a measure greater than 180°.
Examples:
The pentagon in the previous example is convex because it contains no
reflex angles.
The triangle in the previous example is convex because it contains no reflex angles.
The nonagon in the previous example is concave because it contains three
reflex angles.
2
2.5
Classifying Triangles by Sides
An equilateral triangle is a triangle with all sides congruent.
An isosceles triangle is a triangle with at least two congruent sides.
A scalene triangle is a triangle with no congruent sides.
The triangle has all
sides congruent. So,
the triangle is both
equilateral and isosceles.
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The triangle has two
congruent sides.
So, the triangle
is isosceles.
The triangle has no
congruent sides.
So, the triangle
is scalene.
© 2010 Carnegie Learning, Inc.
Examples:
2.5
Classifying Triangles by Angles
An equiangular triangle is a triangle with all angles congruent.
An acute triangle is a triangle that has three angles that each measure less than 90°.
A right triangle is a triangle that has exactly one right angle.
An obtuse triangle is a triangle that has exactly one angle measuring greater
than 90°.
Examples:
80°
60°
The triangle has all angles
congruent. So, the triangle
is equiangular.
40°
2
Each angle of the triangle
measures less than 90°.
So, the triangle is acute.
25°
© 2010 Carnegie Learning, Inc.
110°
The triangle has one right
angle. So, the triangle is a
right triangle.
45°
The triangle has one angle
measuring greater than 90°.
So, the triangle is obtuse.
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2.5
Classifying Quadrilaterals
A square is a quadrilateral with all sides congruent and all angles congruent.
A rectangle is a quadrilateral with opposite sides congruent and all
angles congruent.
A rhombus is a quadrilateral with all sides congruent.
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
A kite is a quadrilateral with two pairs of consecutive congruent sides with opposite
sides that are not congruent.
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Examples:
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The quadrilateral is a square. It is also a
rectangle. It is also a rhombus. It is also
a parallelogram.
The quadrilateral is a rectangle. It
is also a parallelogram.
The quadrilateral is a rhombus.
It is also a parallelogram.
The quadrilateral is
a parallelogram.
The quadrilateral is a kite.
The quadrilateral is a trapezoid.
Parallel and Perpendicular Lines
© 2010 Carnegie Learning, Inc.
2