Download Interior and Exterior Angles

LESSON
7.1
Name
Interior and Exterior
Angles
Class
7.1
Date
Interior and Exterior Angles
Essential Question: What can you say about the interior and exterior angles of a triangle
and other polygons?
Resource
Locker
G.6.D Verify theorems about the relationships in triangles, including…the sum of interior
angles…and apply these relationships to solve problems. Also G.5.A
Texas Math Standards
The student is expected to:
Explore 1
G.5.A
Exploring Interior Angles in Triangles
You can find a relationship between the measures of the three angles of a triangle.
An interior angle is an angle formed by two sides of a polygon with a common vertex.
So, a triangle has three interior angles.
Investigate patterns to make conjectures about geometric relationships,
including angles formed by . . . interior and exterior angles of polygons . . .
choosing from a variety of tools.
G.6.D
interior angles

Use a straightedge to draw a large triangle on a sheet
of paper and cut it out. Tear off the three corners and
rearrange the angles so their sides are adjacent and
their vertices meet at a point.

What seems to be true about placing the three interior angles of a triangle together?
Verify theorems about the relationships in triangles, including proof of
the Pythagorean Theorem, the sum of interior angles, base angles of
isosceles triangles, midsegments, and medians, and apply these
relationships to solve problems.
Mathematical Processes
G.1.G
Display, explain, and justify mathematical ideas and arguments using
precise mathematical language in written or oral communication.
They form a straight angle.
Language Objective
Work in small groups to play angle jeopardy.
ENGAGE
Essential Question: What can you say
about the interior and exterior angles of
a triangle and other polygons?
The sum of the interior angle measures of a triangle
is 180°. You can find the sum of the interior angle
measures of any n-gon, where n represents the
number of sides of the polygon, by multiplying
(n - 2)180°. In a polygon, an exterior angle forms a
linear pair with its adjacent interior angle, so the
sum of their measures is 180°. In a triangle, the
measure of an exterior angle is equal to the sum of
the measures of its two remote interior angles.
© Houghton Mifflin Harcourt Publishing Company
1.A, 1.F, 2.D.1, 2.D.2, 2.I.4

Make a conjecture about the sum of the measures of the interior angles of a triangle.
The sum of the measures of the interior angles of a triangle is 180°.
The conjecture about the sum of the interior angles of a triangle can be proven so it can be stated as a theorem. In the
proof, you will add an auxiliary line to the triangle figure. An auxiliary line is a line that is added to a figure to aid in
a proof.
The Triangle Sum Theorem
The sum of the angle measures of a triangle is 180°.

Fill in the blanks to complete the proof of the Triangle Sum Theorem.
Given: △ABC
B
4
Prove: m∠1 + m∠2 + m∠3 = 180°
A
Module 7
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7.1
HARDCOVER PAGES 291304
Resource
Locker
Quest
Essential
GE_MTXESE353886_U2M07L1.indd 351
interior angles
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© Houghto
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
View the Engage section online. Discuss the photo,
asking students to recall and describe the designs of
game boards of their favorite games. Then preview
the Lesson Performance Task.
3
Date
Class

PREVIEW: LESSON
PERFORMANCE TASK
ℓ
5
Lesson 1
351
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made throu
1
2

Given: △ABC
m∠3 = 180°
+ m∠2 +
Prove: m∠1
A
1
3
C
Lesson 1
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Statements
_
1. Draw line ℓ through point B parallel to AC.
4
2. m∠1 = m∠
and m∠3 = m∠
5
1
+ m∠2 + m∠
EXPLORE 1
1. Parallel Postulate
2. Alternate Interior Angles Theorem
Exploring Interior Angles in Triangles
3. Angle Addition Postulate and definition of
straight angle
3. m∠4 + m∠2 + m∠5 = 180°
4. m∠
Reasons
3
= 180°
4. Substitution Property of Equality
INTEGRATE TECHNOLOGY
Students have the option of completing the interior
angles in triangles activity either in the book or
online.
Reflect
1.
Explain how the Parallel Postulate allows you to add the auxiliary line into the triangle figure.
Since there is only one line parallel to a given line that passes through a given point, I can
draw that line into the triangle and know it is the only one possible.
2.
What does the Triangle Sum Theorem indicate about the angles of a triangle that has three angles of equal
measure? How do you know?
180
= 60, so each angle of the triangle must have a measure of 60°.
3
QUESTIONING STRATEGIES
_
Explore 2
What can you say about angles that come
together to form a straight line? Why? The
sum of the angle measures must be 180° by the
definition of a straight angle and the Angle Addition
Postulate.
Exploring Interior Angles in Polygons
To determine the sum of the interior angles for any polygon, you can use what you know
about the Triangle Sum Theorem by considering how many triangles there are in other
polygons. For example, by drawing the diagonal from a vertex of a quadrilateral, you can
form two triangles. Since each triangle has an angle sum of 180°, the quadrilateral must
have an angle sum of 180° + 180° = 360°.
quadrilateral
Is it possible for a triangle to have two obtuse
angles? Why or why not? No; the sum of
these angles would be greater than 180°.
2 triangles
A
triangle
1 triangle
4 triangles
Module 7
© Houghton Mifflin Harcourt Publishing Company
Draw the diagonals from any one vertex for each polygon. Then state the number of
triangles that are formed. The first two have already been completed.
quadrilateral
2 triangles
5 triangles
352
3 triangles
EXPLORE 2
Exploring Interior Angles in Polygons
AVOID COMMON ERRORS
6 triangles
Lesson 1
PROFESSIONAL DEVELOPMENT
GE_MTXESE353886_U2M07L1.indd 352
Integrate Mathematical Processes
2/20/14 9:04 PM
When attempting to determine the sum of the
interior angles of a polygon, some students may
divide the figure into too many triangles. For
example, a student may draw both diagonals of a
quadrilateral and conclude that the sum of the
interior angles of a polygon is 720°. Point out that
four of the angles of the triangles are not part of an
interior angle of the quadrilateral, and demonstrate
the correct division.
This lesson provides an opportunity to address Mathematical Process
TEKS G.1.G, which calls for students to “Explain and justify arguments.”
Throughout the lesson, students use hands-on investigations or geometry to
predict relationships for the interior and exterior angles of a triangle or polygon.
They prove the Triangle Sum Theorem, the Polygon Angle Sum Theorem, and the
Exterior Angle Theorem. The hands-on investigations give students a chance to
use inductive reasoning to make a conjecture. This is followed by a proof in which
students use deductive reasoning to justify their conjecture.
Interior and Exterior Angles
352
B
QUESTIONING STRATEGIES
How do you use the sum of the angles of a
triangle to find the sum of the interior angle
measures of a convex polygon? If a convex polygon
is broken up into triangles, then the sum of the
interior angles is the number of triangles times 180°.
For each polygon, identify the number of sides and triangles, and determine the angle sums.
Then complete the chart. The first two have already been done for you.
Polygon
EXPLAIN 1
Number of
Sides
Number of
Triangles
Triangle
3
1
(1)180° = 180°
Quadrilateral
4
2
(2)180° = 360°
Pentagon
5
3
( 3 ) 180° = 540°
Hexagon
6
4
( 4 ) 180° = 720°
Decagon
10
8
( 8 ) 180° = 1440°
Using Interior Angles
C
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Math Connections
Sum of Interior Angle
Measures
Do you notice a pattern between the number of sides and the number of triangles? If n represents
the number of sides for any polygon, how can you represent the number of triangles?
n-2
D
Make a conjecture for a rule that would give the sum of the interior angles for any n-gon.
Sum of interior angle measures = (n - 2)180°
You may want to review with students how to
evaluate algebraic expressions and how to use inverse
operations to solve equations.
Reflect
3.
In a regular hexagon, how could you use the sum of the interior angles to determine the measure of each
interior angle?
Since the polygon is regular, you can divide the sum by 6 to determine each interior angle
© Houghton Mifflin Harcourt Publishing Company
measure.
4.
How might you determine the number of sides for a polygon whose interior angle sum is 3240°?
Write and solve an equation for n, where (n - 2)180° = 3240°.
[# Explain 1
Using Interior Angles
You can use the angle sum to determine the unknown measure of an angle of a polygon when you know the measures
of the other angles.
The Polygon Angle Sum Theorem states that the sum of the measures of the interior angles of a convex polygon can
be found by multiplying 180° by 2 less than the number of sides. This can help you determine an unknown angle
measurement, such as for a nonagon with eight given interior angles.
Polygon Angle Sum Theorem
The sum of the measures of the interior angles of a convex polygon with n sides is (n – 2)180°.
Module 7
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Lesson 1
COLLABORATIVE LEARNING
GE_MTXESE353886_U2M07L1.indd 353
Small Group Activity
Geometry software allows students to explore the theorems in this lesson.
For the Triangle Sum Theorem and the Exterior Angle Theorem, students
should construct a triangle, measure the three angles, and use the Calculate tool
(in the Measure menu) to find the sum of the interior angle measures and also to
find the sum of the exterior angles. As students drag the vertices of the triangle to
change its shape, the individual angle measures will change, but the sum of the
measures will remain 180° for the interior angles and 360° for the exterior angles.
353
Lesson 7.1
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Example 1

Determine the unknown angle measures.
QUESTIONING STRATEGIES
For the nonagon shown, find the unknown angle measure x°.
How do you use the sum of the interior angle
measures of a polygon to find the measure of
an unknown interior angle? Use the Polygon Sum
Theorem to find the total measure of the interior
angles, then solve an algebraic equation to find the
unknown angle.
First, use the Polygon Angle Sum Theorem to find the sum of the interior angles:
n=9
(n - 2)180° = (9 - 2)180° = (7)180° = 1260°
Then solve for the unknown angle measure, x°:
98°
125 + 130 + 172 + 98 + 200 + 102 + 140 + 135 + x = 1260
102°
125°
130°
x = 158
140°
The unknown angle measure is 158°.

x°
135°
200°
172°
Determine the unknown interior angle measure of a convex octagon in which the measures
of the seven other angles have a sum of 940°.
n= 8
Sum =
(
)
8 - 2 180° =
940 + x =
(
6
) 180° =
1080°
1080
x = 140
The unknown angle measure is 140° .
Reflect
5.
any regular polygon.
Your Turn
6.
Determine the unknown angle measures in this pentagon.
n=5
Sum = (5 - 2)180° = (3)180° = 540°
270 + 2x = 540
x°
x°
© Houghton Mifflin Harcourt Publishing Company
How might you use the Polygon Angle Sum Theorem to write a rule for determining the measure of each
interior angle of any regular convex polygon with n sides?
(n - 2)180°
gives the measure of an interior angle for
You can divide the angle sum by n. __
n
2x = 270
x = 135
Each unknown angle measure is 135°.
Module 7
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Lesson 1
DIFFERENTIATE INSTRUCTION
GE_MTXESE353886_U2M07L1 354
Manipulatives
2/22/14 11:17 PM
Have students fold and crease the four corners of a sheet of paper. Next, ask them
to open the folds to reveal a creased polygon shape. Have students classify the
polygon (octagon). Ask them to find the sum of the interior and exterior angle
measures (1080°; 360°). Then have students measure the interior and exterior
angles to verify their sums.
Interior and Exterior Angles
354
Determine the measure of the fourth interior
angle of a quadrilateral if you know the other
three measures are 89°, 80°, and 104°.
7.
EXPLAIN 2
8.
n=4
Proving the Exterior Angle Theorem
Sum = (6 - 2)180° = (4)180° = 720°
Sum = (4 - 2)180° = 2(180°) = 360°
b + 2b + 69 + 108 + 135 + 204 = 720
89 + 80 + 104 + x = 360
QUESTIONING STRATEGIES
Determine the unknown angle measures in a
hexagon whose six angles measure 69°, 108°,
135°, 204°, b°, and 2b°.
n=6
3b + 516 = 720
3b = 204
x = 87
b = 68
The unknown angle measure is 87°.
How does finding the measure of an exterior
angle differ from finding the measure of an
interior angle? The measure of an exterior angle is
the supplement of its adjacent interior angle
because the angles form linear pairs with the
interior angles. The measure of an interior angle is
not found by using linear pairs.
2b = 136
The two unknown angle measures are 68°
and 136°.
Explain 2
Proving the Exterior Angle Theorem
An exterior angle is an angle formed by one side of a polygon and the extension of an adjacent side. Exterior angles
form linear pairs with the interior angles.
A remote interior angle is an interior angle that is not adjacent to the exterior angle.
Why is the Exterior Angle Theorem
sometimes called a corollary of the Triangle
Sum Theorem? because the Exterior Angle
Theorem follows from the Triangle Sum Theorem
Remote
interior
angles
Example 2
Exterior angle
Follow the steps to investigate the relationship between each exterior angle
of a triangle and its remote interior angles.
Step 1 Use a straightedge to draw a triangle with angles 1, 2, and 3. Line up your
straightedge along the side opposite angle 2. Extend the side from the vertex at
angle 3. You have just constructed an exterior angle. The exterior angle is drawn
supplementary to its adjacent interior angle.
∠4 is an exterior angle.
© Houghton Mifflin Harcourt Publishing Company
2
It forms a linear pair with interior angle ∠3.
1
3
4
Its remote interior angles are ∠1 and ∠2.
Step 2 Use the diagram. You know the sum of the measures of the interior angles of a
triangle.
m∠1 + m∠2 + m∠3 = 180 °
Since an exterior angle is supplementary to its adjacent interior angle, you also know:
m∠3 + m∠4 = 180 °
Make a conjecture: What can you say about the measure of the exterior angle and the measures of its
remote interior angles?
Conjecture: The measure of the exterior angle is the same as the sum of the measures of
its two remote interior angles.
Module 7
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Lesson 1
LANAGUAGE SUPPORT
GE_MTXESE353886_U2M07L1 355
Communicate Math
Have students write clues about interior and exterior angles in polygons,
for example: “The sum of the interior angles of this three-sided polygon is
180 degrees” or “The sum of the exterior angles of this three-sided polygon is
360 degrees.” Have each student write two clue cards about different polygons.
They then read their clues to the rest of the group, and the group must decide
which polygon fits the clue.
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Lesson 7.1
2/22/14 11:19 PM
The conjecture you made in Step 2 can be formally stated as a theorem.
AVOID COMMON ERRORS
Exterior Angle Theorem
Some students may confuse the theorems in this
lesson and incorrectly assume that the sum of the
interior angles of a polygon is 360°. Remind students
of the Triangle Sum Theorem. Have them draw an
equilateral triangle and show that its interior angle
measures add to 180° and its exterior angle measures
add to 360°.
The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior
angles.
Step 3 Complete the proof of the Exterior Angle Theorem.
2
∠4 is an exterior angle.
It forms a linear pair with interior angle ∠3.
1
3
Its remote interior angles are ∠1 and ∠2.
4
By the Triangle Sum Theorem , m∠1 + m∠2 + m∠3 = 180°.
Also, m∠3 + m∠4 = 180° because they are supplementary and make a straight angle.
By the Substitution Property of Equality, then, m∠1 + m∠2 + m∠3 = m∠ 3 + m∠ 4 .
Subtracting m∠3 from each side of this equation leaves
m∠1 + m∠2 = m∠4 .
This means that the measure of an exterior angle of a triangle is equal to the sum of the measures of the
remote interior angles.
Reflect
9.
Discussion Determine the measure of each exterior angle. Add them together. What can you say about
their sum? Explain.
40°
The exterior angles will measure 140°, 120°, and 100°. Their sum is 360°. Each exterior angle
is equal to the sum of the measures of the two remote interior angles, and the sum of all 3
exterior angles includes each interior angle twice.
10. According to the definition of an exterior angle, one of the sides of the triangle must be extended in order
to see it. How many ways can this be done for any vertex? How many exterior angles is it possible to draw
for a triangle? for a hexagon?
Two exterior angles can be drawn from any vertex by extending either side, so a triangle
© Houghton Mifflin Harcourt Publishing Company
60°
can have 6 exterior angles. You could draw 12 different exterior angles for a hexagon.
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Interior and Exterior Angles
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Explain 3
EXPLAIN 3
Using Exterior Angles
You can apply the Exterior Angle Theorem to solve problems with unknown angle measures by writing and solving
equations.
Using Exterior Angles
Example 3

Determine the measure of the specified angle.
Find m∠B.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Patterns
A
2z°
Encourage students to make a table listing the sums
of the exterior angles of regular triangles,
quadrilaterals, pentagons, and hexagons. Ask them
what they notice about the sum of the exterior angles.
(The sum is always 360°.) Ask them to find the
pattern in the measure of each individual exterior
angle for these regular polygons. (They each have the
same measure.)
C
B
Write and solve an equation relating the exterior and remote interior angles.
145 = 2z + 5z - 2
145 = 7z - 2
z = 21
Now use this value for the unknown to evaluate the expression for the required angle.
m∠B = (5z - 2)° = (5(21) - 2)° = (105 - 2)° = 103°
CONNECT VOCABULARY

Find m∠PRS.
Q
© Houghton Mifflin Harcourt Publishing Company
Help students understand the meanings of interior,
exterior, and remote by writing the definitions on note
cards. An interior angle is inside the figure, an exterior
angle is outside the figure, and a remote interior angle
is interior and away from the exterior angle. Relate
the idea of a remote interior angle to a television
remote control that sends a signal across the room
and away from you.
(5z - 2)°
145°
D
R
(3x - 8)°
(x + 2)°
P
S
Write an equation relating the exterior and remote interior angles.
Solve for the unknown.
3x - 8 = (x + 2) + 90
3x - 8 = x + 92
2x = 100
x = 50
Use the value for the unknown to evaluate the expression for the required angle.
m∠PRS = (3x - 8)° = (3(50) - 8)° = 142°
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Your Turn
QUESTIONING STRATEGIES
Determine the measure of the specified angle.
11. Determine m∠N in △MNP.
What kind of angle is formed by extending
one of the sides of a triangle? What is its
relationship to the adjacent interior angle? an
exterior angle; the angles are supplementary.
12. If the exterior angle drawn measures 150°, and
the measure of ∠D is twice that of ∠E, find the
measure of the two remote interior angles.
N
(3x + 7)°
D
150°
E
(5x + 50)°
63°
M
P
F
G
ELABORATE
x + 2x = 150
Q
3x = 150
5x + 50 = (3x + 7) + 63
QUESTIONING STRATEGIES
x = 50
5x + 50 = 3x + 70
How do you use the sum of the interior angle
measures of a regular polygon to find the
measure of each interior angle? Divide the sum of
the interior angles by the number of sides.
m∠E = x° = 50°
2x = 20
m∠D = 2x° = 100°
x = 10
m∠N = (3x + 7)° = (3(10) + 7)° = 37°
What happens to the measure of each exterior
angle as the number of sides of a regular
polygon increases? Why? The measures get smaller
and smaller because the sum must remain 360°.
Elaborate
13. In your own words, state the Polygon Angle Sum Theorem. How does it help you find
unknown angle measures in polygons?
Possible answer: The sum of the measures of the interior angles of a convex polygon
equals (n - 2)180°. You can use it to find an unknown measure of an interior angle of a
polygon when you know the measures of the other angles.
SUMMARIZE THE LESSON
Have students fill out a chart to summarize the
theorems in this lesson. Sample:
© Houghton Mifflin Harcourt Publishing Company
14. When will an exterior angle be acute? Can a triangle have more than one acute exterior angle?
Describe the triangle that tests this.
An exterior angle will be acute when paired with an obtuse adjacent interior angle;
therefore, the triangle must be obtuse. Since a triangle must have two or three acute
interior angles, at least two exterior angles must be obtuse.
15. Essential Question Check-In Summarize the rules you have discovered about the interior
and exterior angles of triangles and polygons.
The sum of the measures of the interior angles of a triangle is 180°. The sum of the
Triangle Sum Theorem
m∠1 + m∠2 + m∠3 = 180°
2
where n represents the number of sides of the polygon. The measure of an exterior angle
Polygon Sum Theorem
(n - 2) 180° = (6 - 2) 180° = 720°
of a triangle is equal to the sum of the measures of the two remote interior angles.
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1
measures of the interior angles for any polygon can be found by the rule (n - 2)180°,
Lesson 1
22/01/15 8:53 AM
Exterior Angle Theorem
m∠4 = m∠1 + m∠2
2
1
3
4
Interior and Exterior Angles
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