Republic of the Philippines
DEPARTMENT OF EDUCATION
Region I
Division of Ilocos Sur
SELF-LEARNING KIT IN
MATHEMATICS
LESSON TITLE:
INTERNAL AND EXTERNAL ANGLES
OF A CONVEX POLYGON
Developed by
Writer: Christy U. Buquing,
Reviewers: Salome D. Ubarre, Jr., HT I
Apolinario R. Riotoc, T III
Marilou I. Corpuz, T II
Alegria F. Molina, PSDS
Nestor G. Villalfor, EPS - Math
Illustrator:
Administrators:
Jorge M. Reinante, SDS
Nestor C. Heraña, ASDS
Maria Salome R. Abero, CID Chief
Nestor G. Villaflor, EPS - Math
1
QUARTER #
3
SELF- LEARNING KIT #
6
FOREWORD
This Self-Learning Kit is a teacher-made module which is designed as one of
the main tools in delivering continuous learning to the students in the Division of Ilocos
Sur. This Self-Learning Kit is based on the Most Essential Learning Competency
provided by the Department of Education. This is an initiative of the Division of Ilocos
Sur to ensure Learning Continuity in the Basic Education Curriculum of the Department
of Education amidst the pandemic.
It is expected that the students will be responsible in studying the lesson for the
week and answer the given activities with honesty and ensure learning among
themselves. Each activity and assessment will be answered by the students with the
guidance of their learning resources within a specific period of time.
In the past week, the students have learned about convex polygon and its
angles which is a pre-requisite of the lesson for this week. This SLK will help the
students know more about the relationship of exterior and interior angles of a convex
polygon. Below is a short discussion on the relationship of exterior and interior angles
of a polygon and how to find the measurements of these angles followed by activities
for each day.
LEARNING COMPETENCY
At the end of the lesson, you are expected to:
1. Derives inductively the relationship of exterior and interior
angles of a convex polygon. (M7GE-IIIf-1)
1.
2
Presentation of the Lesson:
In this lesson, you will learn how to derive inductively the relationship of exterior
and interior angles of a convex polygon. Included in the discussion are the
measurement of the interior angles and exterior angles, the sum of the interior angles
as well as the exterior angles and the relationship of the angles with each other.
Short Discussion:
You have learned in your past lesson about convex polygon. A polygon is said
to be convex if the lines containing the sides of the polygon do not cross the interior of
the polygon. If any part of the diagonal contains point in the exterior of the polygon,
then the polygon is non-convex, also known as concave polygon. It has also been
taught that the Polygon Angle Sum Theorem states that “The sum of the interior angle
measures of a convex polygon with n sides is (n-2)180o”.
There are two types of angles associated with a convex polygon: exterior angle
and interior angle. Interior angles of a convex polygon are the angles on the inside of
a polygon and an exterior angle is an angle that is both supplement and adjacent to
one of its interior angles. Supplementary angles are angles that sum up to 180o. An
exterior angle is formed by one side of a polygon and the extension of a consecutive
side. Polygon Exterior Sum Theorem states that “The sum of the exterior angle
measures, one angle at each vertex, of a convex polygon is 360o”.
Illustrative Examples:
Example 1: Below is the example of an interior angle and exterior angle of a pentagon.
Exterior angle
1
6
Exterior angle
10
2 7
5
Interior angles
Exterior
angle
Exterior
angle
9 4
Exterior angle
3
8
The interior angles are ∠1, ∠2, ∠3, ∠4, ∠5 which are found inside of the convex
polygon while ∠6, ∠7, ∠8, ∠9, ∠10 are the exterior angle which are found outside the
convex polygon.
3
Example 2: This figure is a regular triangle. We know that the interior angles of a regular
polygon are congruent, the same is true with its exterior angles.
What is the sum of the exterior angles of the triangle below ?
The sum of its exterior angles is 360o.
Example 3: The example below is an irregular quadrilateral. You have learned that the
sum of the interior angles of a quadrilateral is 360o by the Polygon Angle Sum Theorem.
In this case, an exterior angle is also given, adjacent to the missing angle.
What is the measure of the missing angle of the quadrilateral?
E
α = 65˚
A
Ჾ = 115˚
B
β= 110˚
γ = 120˚
C
D
Using the Polygon Angle Sum Theorem, (n-2)180 = (4-2)180 = (2)(180) = 360
360 – (65+110+120) = 360 – 295 = 65o
4
Activity 1: Be a Detective. Look closely on the pentagon below then put a check mark
(√ ) if you have observed the following on the figure given and put a heart (
observable.
) if it’s not
_____1. There are 5 exterior angles on a convex pentagon.
_____2. The sum of the interior angles of a convex pentagon is 540o.
_____3. An interior angle and its corresponding exterior angle are complementary
(sum is 90o).
_____4. The interior angle and its corresponding exterior angle belong to the same
plane.
_____5. Two consecutive exterior angles always have a sum of 180o.
_____6. The sum of the exterior angles of the polygon is 360o.
_____7. Interior angle and its corresponding exterior angle of the convex pentagon
are supplementary (sum is 180o).
_____8. The interior and its corresponding exterior angle of the pentagon are not
adjacent to each other.
_____9. When the exterior angle is drawn, it forms a straight line along with its
interior angle.
_____10. The exterior angles of the polygon measures 65 o,70o,80o,60o and 85o.
_____11. The interior angles of the polygon measures 65o,70o,80o,60o and 85o.
_____12. The sum of the exterior angles of the pentagon is 540o.
_____13. The interior angle measuring 100o has its corresponding exterior angle
measuring 85o.
_____14. The interior angle which measures 120o has its exterior angle of 60o.
_____15. The number of interior angles of a polygon is equal to the number of its
corresponding exterior angles.
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Activity 2: Legendary…Savage…Wiped Out!
Be a legendary hero by destroying all the turrets all the way to the enemy’s base
for you to win the game. To do this, answer each question in every turret until you come
to the enemy’s base to destroy it and win the match. (5 seconds ‘til the enemy reaches
the battlefield, smash them)
_______1. The sum of the exterior angles of a convex polygon always measures
360o. True or False
________2. Interior angles and its exterior angles are always supplementary which
means their sum is ___.
________3. An interior angle and its corresponding exterior angle are adjacent to
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each other. True or False.
________4. Tell whether the following are possible measurements of the exterior
angles of a convex triangle, 60o, 100o, 85o. Yes or No. Why?
___________________
________5. If the interior angle of a convex octagon measures 85o, what is the
measurement of its corresponding exterior angle?
________6. Given that the measure of an exterior angle of a convex quadrilateral is
100o, what is the measure of its corresponding interior angle?
________7. Suppose the measure of the exterior angles of a triangle are 45o, 65o,
and ___. Find the corresponding exterior angle of the missing interior
angle.
________8. If the measurement of one of the interior angles of a regular decagon is
144, what is the measurement of one of its exterior angles?
________9. What is the measurement of one of the exterior angles of a regular
15-gon?
________10. Suppose the measurement of the interior angles of a convex
quadrilateral are 60o, 75o, 80o and ____. Find the missing interior angle
and the corresponding exterior angles for each interior angles.
VICTORY!!!!!!!!
Activity 3: Mix and Match.
From the mix of interior angles below, find its match by choosing which among
the others is its corresponding exterior angle. Write your answers on a separate sheet.
7
A. 132o
B. 150o
55o
E. 140o
45o
G. 130o
25o
30o
40o
8
Activity 4: Adventure Time!
Imagine you are Super Mario and your goal is to save Princess Peach. In order
to go where the princess is, you have to go through many obstacles and to overcome
each, you have to answer the questions in each station.
Solution:
START
STATION 1
TASK: Find the
value of x.
Solution:
STATION 2
TASK:Find the measure of
the interior angle and value
of x.
Solution:
STATION 3
TASK:Find the
measure of the
exterior angle
of this regular
hexagon
STATION 4
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Solution:
TASK: Find the
measurements
of the interior
angles of the
two triangles
STATION 5
Solution:
TASK:Find the value
of x then find the
measure of each
exterior angles.
FINISH
Generalization:
Interior angles are angles inside the convex polygon while exterior angles are
angles outside the polygon. Interior angles and exterior angles are supplementary and
adjacent to each other. The sum of all the exterior angles of a convex polygon is always
360o as stated by the Polygon Exterior Sum Theorem.
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Application/Assessment:
1. Tell in these pictures the difference and/or relationship of interior ang exterior
angles of a convex polygon.
1.
2.
3.
4.
5.
______________________________
______________________________
______________________________
______________________________
______________________________
2. Suppose the two objects (hourglass and shutter) are of regular sizes(all sides
are equal), what are the measurements of their interior angles and exterior
angles?
SOLUTION FOR SHUTTER
SOLUTION FOR
HOURGLASS
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ANSWER KEY
Activity 1: Be a Detective.
__√__1. There are 5 exterior angles on a convex pentagon.
__√__2. The sum of the interior angles of a convex pentagon is 540o.
_____3. An interior angle and its corresponding exterior angle are complementary
(sum is 90o).
__√__4. The interior angle and its corresponding exterior angle belong to the same
plane.
_____5. Two consecutive exterior angles always have a sum of 180o.
__√__6. The sum of the exterior angles of the polygon is 360 o.
__√__7. Interior angle and its corresponding exterior angle of the convex pentagon are
supplementary (sum is 180o).
_____8. The interior and its corresponding exterior angle of the pentagon are not
adjacent to each other.
__√__9. When the exterior angle is drawn, it forms a straight line along with its interior
angle.
__√__10. The exterior angles of the polygon measures 65 o,70o,80o,60o and 85o.
_____11. The interior angles of the polygon measures 65o,70o,80o,60o and 85o.
_____12. The sum of the exterior angles of the pentagon is 540o.
_____13. The interior angle measuring 100o has its corresponding exterior angle
measuring 85o.
__√__14. The interior angle which measures 120o has its exterior angle of 60o.
__√__15. The number of interior angles is equal to the number of its corresponding
exterior angles.
Activity 2: Legendary…Savage…
___True_1. The sum of the exterior angles of a convex polygon always measures 360o.
____180_2. Interior angles and its exterior angles are always supplementary which
means their sum is ___.
___True_3. An interior angle and its corresponding exterior angle are adjacent to each
other.
_No. The sum is not 180o__4. Tell whether the following are possible measurements
of the exterior angles of a convex triangle, 60 o, 100o, 85o.
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___95o_5. If the interior angle of a convex octagon measures 85o, what is the
measurement of its corresponding exterior angle?
___80o_6. Given that the measure of an exterior angle of a convex quadrilateral is
100o, what is the measure of its corresponding interior angle?
_110o_7. Suppose the measure of the exterior angles of a triangle are 45 o, 65o, and
___. Find the corresponding exterior angle of the missing interior angle.
_36o__8. If the measurement of one of the interior angles of a regular decagon is 144,
what is the measurement of one of its exterior angles?
_24o__9. What is the measurement of one of the exterior angles of a regular 15-gon?
120o, 105o, 100o and 35o respectively Suppose the measurement of the interior
angles of a convex quadrilateral are 60o, 75o, 80o and 145o . Find the missing interior
angle and the corresponding exterior angles for each interior angles.
Activity 4: Adventure Time.
Solution:
START
STATION
1
TASK: Find the
value of x.
95+x =
180
x= 180 95
x = 85
Solution:
60+(x-10) = 180
x - 10= 180 – 60
x = 120 + 10
x = 130
Int. Ang.: x-10
130-10 = 120
STATION 2
TASK:Find the measure of
the interior angle and value
of x.
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Solution:
STATION 3
TASK:Find the
measure of the
exterior angle
of this regular
hexagon
360 ÷ 6 = 60o
Or find first the
interior angle.
[(n-2)180] ÷ 6
=[(6-2)180] ÷6
=[(4)(180)] ÷ 6
=120
180-120 = 60o
STATION 4
Solution:
180 – 120 = 60o
180 – 120 = 60o
60 + 60 = 120
(n-2)180
(3-2)180
(1)180 = 180
180- 120 = 60o
TASK: Find the
measures of the
interior angles
of the two
triangles
STATION 5
Solution:
TASK:Find the value
of x then find the
measure of each
exterior angles.
9x+2x+7x+4x+5x+6x
+3x = 360o
36x=360
(36x)/36 = (360)/36
x = 10
9(10) = 90
2(10) = 20
7(10) = 70
4(10) = 40
5(10) = 50
6(10) = 60
3(10) = 30
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FINISH
Application/Assessment:
1. Tell in this picture the difference and/or relationship of interior and exterior
angles of a convex polygon.
1. Interior angles are on the inside while exterior
angles are on the outside of the convex polygon
2. Interior and exterior angles are supplementary
3. Interior and exterior angles are adjacent to each
other
4. The sum of all exterior angles of a convex
polygon is 360o
5. Every interior angle has its corresponding
exterior angle.
2. Suppose the two objects (hourglass and shutter) are of regular sizes (all sides
are equal), what are the measurements of their interior angles and exterior
angles?
SOLUTION FOR SHUTTER
SOLUTION FOR HOURGLASS
For both triangles:
For the interior angles:
[(n-2)180] ÷ n = [(3-2)180]÷3=
[(1)180]÷3 = 60
[(n-2)180] ÷ n = [(8-2)180] ÷ 8= [(6)180]
÷8 = 135
The measurement of each interior angle
is 60o since the triangle is regular.
The measurement of each interior angle
is 135o since it is a regular octagon.
360 ÷ 3 = 120o
360 ÷ 8 = 45o
The measurement of each exterior
angle is 120o since the triangle is
regular.
The measurement of each exterior angle
is 45o since it is a regular octagon.
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ACTIVITY SHEET
Quarter 3-Module 6
Instruction: Cut this page then answer activity 3(Mix and Match) only on page 7 to
8. Do not forget to write your name and section below.
Activity 3: Mix and Match.
A. 132o
B. 150o
55o
E. 140o
45o
G. 130o
25o
30o
40o
Cut this line.
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