The document covers properties of convex and concave polygons, including the relationship between interior and exterior angles and how to calculate the sum of interior angles using the polygon interior angles theorem. It provides examples for calculating the number of sides based on given interior angle sums, measures of angles in regular polygons, and the sum of exterior angles of convex polygons, with illustrations and exercises. Additionally, it discusses diagonals in polygons and includes practical exercises for creating tessellations.

1. Write CONVEX if the given
polygon is convex, otherwise,
write CONCAVE/NON-CONVEX
• Convex
• Concave
• Convex
• Convex • Concave • Concave
• Convex • Convex • Concave
• Convex

2. INTERIOR AND EXTERIOR
ANGLES OF A CONVEX
POLYGON
Derives inductively the relationship of exterior and interior
angles of a convex polygon (M7GE-IIIf-1)

4. a) What operation is involved whenever we put
numbers or things together?
b) After putting the pieces together, what do they
look like?
c) What can you conclude about m 1 + m 2 +
∠ ∠
m 3?
∠
d) Draw and cut a right triangle. Tear the angles and
rearrange them as what you did in the first triangle. Is
your answer in letter c the same? Explain your answer.
e) What can you say about the sum of interior angles
of a triangle?

6. INTERIOR AND
EXTERIOR ANGLES
OF POLYGONS

9. POLYGON INTERIOR ANGLES
THEOREM
• The sum of the measures of the
interior angles of a convex
polygon with n sides is (n – 2) 180°.

10. 1. Compute for the sum of interior angles of a convex decagon.
Solution: A convex decagon has 10 sides. Use the Polygon Interior Angles
Theorem and substitute 10 for n.
(n – 2) 180° = (10 – 2)(180°) (Substitute 10 for n)
= (8)(180°) (Simplify)
= 1440°
The sum of interior angles of a convex decagon is 1440°.

11. 2. Given the figure below, find the measure of C.
∠
The polygon has 5 sides, so the sum of interior
angles is 540°. (Refer to the second table)
Add the measures of the interior angles and set
the sum to be equal to 540°
m C + 75 + 150 + 90 + 90 = 540 (Add similar terms)
∠
m C + 405 = 540 (Subtract 405 from each side)
∠
m C = 135
∠

12. 3. Find the measure of an interior angle of a regular dodecagon.
Solution:
The sum of the measures of the interior angles of a dodecagon is:
(n – 2)(180°) = (12 – 2)(180°)
= (10)(180°)
= 1800°
Since the dodecagon is regular, each angle has the same measure.
Hence, we just divide the sum which is 1800° by 12 to get the measure
of one interior angle.
1800°
12 = 150°
The measure of an interior angle of a regular dodecagon is 150°.
From this example we can have a more generalized formula in getting
the measure of one interior angle of any regular polygon and that is:
(n - 2) 180°
n

13. 4. If a certain polygon’s sum of interior angle is 2700°, how many sides
does the polygon have?
If we let S be the sum of interior angles, then S = (n - 2) 180°

14. Let us use the resulting equation to answer the problem.

17. A. Compute for the sum of interior
angles of the following polygons.
1. Undecagon
2. 17-gon
3. 32-gon
4. 19-gon
5. 50-gon

18. B. Compute for the number sides of the
given polygons given the sum of their
interior angles.
6. 1800°
7. 3060°
8. 3960°
C. Compute for x.

19. A.
1. 1620°
2. 2700°
3. 5400°
4. 3060°
5. 8640°
B.
6. 12
7. 19
8. 24
C.
9. 77 °
10. 144 °

21. Art within me!
Direction: create a tessellation using at least 2 regular polygons.
Your tessellation (pattern) should cover the entire paper.
Materials: ruler, pen or pencil, coloring materials, bond paper

22. DIAGONALS
A diagonal of a polygon is a segment that joins two
nonconsecutive vertices.
Ex. 1. How many diagonals can we draw in the rectangle?

23. TRIANGLES……
Ex. 2.
How many diagonals can be drawn in the triangle?
Why?

24. What is the sum of the interior angles of a triangle?
Ex. 3. Find the missing angles.
3xº
xº 72º
A
B C
m<A =
m<B =

25. QUADRILATERALS…
In each quadrilateral draw all possible diagonals
from Vertex A.
How many triangles were formed as a result?
What do you think this means for the sum of the
interior angles of a quadrilateral?

26. PENTAGONS….
In the following pentagon draw all possible diagonals
from Vertex A.
How many triangles were formed as a result?
What do you think this means for the sum of the
interior angles of a quadrilateral?

27. POLYGONS….
Now take a second and draw in all diagonals from
one vertex of each polygon on your worksheet. Is
there a relationship between the sides and the
number of triangles?
Hexagon
Nonagon
Octagon
Decagon

28. SUM OF THE INTERIOR ANGLES
OF A CONVEX POLYGON.
SI = (n – 2) • 180
Ex. 4. Find the sum of the interior angles of a convex
heptagon.
Ex. 5. Find the sum of the interior angles of a convex 15-gon.

29. Ex. 6. Solve for x.
Ex. 7. Solve for y.
Not drawn to scale
120º
48º
xº
2xº
5yº
9yº 92º
139º
71º

30. REGULAR POLYGONS
A REGULAR polygon is Equilateral and
Equiangular (all sides and all angles ).
To Find the measure of each Interior Angle of a regular convex
polygon.
 
( 2) 180
n
n
 

31. Ex. 8. Find the measure of each angle in a
regular convex octagon.
Ex. 9. The measure of each interior angle
of a regular polygon is 165º. How many
sides does the polygon have?

32. EXTERIOR ANGLES
The Exterior Angle of any polygon forms
a linear pair with an Interior angle of a
polygon.
Ex. <1 is an exterior angle. <1 and <2 form a linear pair.
<1
<2 m<1 + m<2 = 180

33. SUM OF THE EXTERIOR ANGLES
OF A CONVEX POLYGON.
360º
ALL Exterior Angles of EVERY
polygon add up to 360º

34. POLYGON EXTERIOR
ANGLES THEOREM
•The sum of the measures of
the exterior angles of a
convex polygon, one angle at
each vertex, is 360°.
•m 1 + m 2 + · · · + m n =
∠ ∠ ∠
360°

35. ILLUSTRATIVE EXAMPLES:
1. Given the figure below, find the value of x.
Solution:
x + 80 + 120 = 360 Polygon Exterior Angles Theorem
x + 200 = 360 Combining like terms
x + 200 – 200 = 360 – 200 Subtraction Property of Equality
x = 160
Therefore, the value of x is 160

36. ILLUSTRATIVE EXAMPLES:
2. Given the figure below, find the value of x.
x + 60 + 95 + 83 + 52 = 360 Polygon Exterior Angles Theorem
x + 290 = 360 Combining like terms
x + 290 – 290 = 360 – 290 Subtraction Property of Equality
x = 70
Therefore, the value of x is 70.

37. ILLUSTRATIVE EXAMPLES:
3. A convex heptagon has exterior angles with measures 30°, 33°, 45°, 54°,67°, and 79°.
What is the measure of an exterior angle at the seventh vertex?
Solution:
Let x = be the seventh exterior angle
x + 30+ 33+45+ 54+67+ 79 = 360
x + 308 = 360
x + 308 – 308 = 360 – 308
x = 52

38. ILLUSTRATIVE EXAMPLES:
4. Find the measure of one of the exterior angles of a regular Dodecagon.
Solution:
Since the sum of interior angles is always 360° and the
given polygon is a regular polygon, we simply divide
360° by the number of sides. A regular dodecagon
has 12 sides, so:
360°=360° = 30°
n 12
Hence, one exterior angle of a regular dodecagon
measures 30°.

39. Directions: refer to the given figures and find the value of x.
5. The stop sign shown is in the shape of a regular octagon.
Find the measure of one of its exterior angles.

40. FILL IN THE BLANKS TO MAKE
EACH STATEMENT TRUE.
1. The sum of the measures of the angles of a convex
polygon with n sides is _____________.
2. The sum of the measures of the exterior angles of any
convex polygon is ___________.
3. The measure of each angle of a regular polygon with n
sides is ___________.
4. The sum of the measures of each exterior angles of a
regular pentagon is ____________.
5. The sum of the measure of the interior angles of a
convex ______ is 1800°.

41. What is the sum of the exterior angles of a convex triangle?
What is the sum of the exterior angles of a convex 300-gon?

42. REGULAR POLYGONS
To Find the measure of each Exterior
Angle of a regular convex polygon.
360
n


43. Find the measure of each exterior angle of a regular heptagon.
The measure of each exterior angle of a regular polygon is 40º.
How many sides does it have?