Polygons

Polygons

By MARIA SEVERINO

Learning Objectives

1. To illustrate polygons in terms of:

• convexity

• angles

• sides​

Learning Competency:

Illustrates polygons: (a) convexity; (b) angles; and (c) sides M7GE-IIIe-2

Derives inductively the relationship of exterior and interior angles of a convex polygon M7GE-IIIf-1​

2. To derive inductively the relationship of interior angles of a polygon.

3. To develop one's inductive reasoning skill.​

Below are examples of polygons:​

Observe and Analyze

​

​

​

Below are not examples of polygons:​

Observe and Analyze

Multiple Select

Can you identify which represents a polygon?

Definition, Parts and Classification of a Polygon

The word “polygon” comes from two words “poly” which means “many” and “gon” which means “angles. A polygon is a closed plane figure formed by line segments.

Multiple Select

Which of these are convex polygons?

A polygon is said to be convex if the lines containing the sides of the polygon do not cross the interior of the polygon. A polygon that is not convex is called non-convex or concave.

Convex and Concave Polygons

Convex and Concave Mirrors

When all the angles and sides of a polygon are equal, the polygon is called regular. Can you name a regular three-sided polygon? How about a four-sided regular polygon? The segments forming the polygon are called sides. The points where the segments meet are called vertices. The angles formed by two adjacent sides are called the angles of the polygon. Polygons are named by writing their consecutive vertices in a clockwise or counterclockwise rotation.

​

The polygon at the right can be named as: ABCDE or BCDEA or CDEAB or DEABC. While the following are the:

​

Vertices: ​A, B, C, D, and E

Sides: ​AB, BC, CD, DE, and EA

Angles: ​∠A, ∠B, ∠C, ∠D, and ∠E

​

​

​

​∠∠∠∠∠∠

A polygon separates a plane into three sets of points: the points on the polygon, points in the interior (inside) of the polygon, and points in the exterior (outside) of the polygon

​

Here are two types of angles associated with a convex polygon: exterior angle and interior angle. An exterior angle of a convex polygon is an angle that is both supplement and adjacent to one of its interior angles.

In parallelogram SPCE​,

​∠S, ∠C, ∠E, and ∠SPC, are the interior angles,

while ∠APC is an exterior angle.

​

​

​

Consecutive vertices are vertices on the common side of the polygon. Consecutive sides are sides that have a common vertex. A segment joining two nonconsecutive vertices of a polygon is called diagonal.

​

Consecutive vertices:

​

Consecutive sides:

​

Diagonals:​

Classifications of Polygons According to number of Sides

Polygons may also be classified by the number of sides they have. Can you name each?

​

​

"Showcase your Work"

HANDS-ON ACTIVITY

"FORM ME THE FIGURE"​

​

​

Processing Questions:

a.     Were you able to derive inductively the formula in finding the sum of the interior angles of a convex polygon?

b.     How were you able to do it?

c.      What if one interior angle of the polygon is unknown, can you find its measure knowing the number of sides the polygon has?

Polygon Interior AngleTheorem

The sum of the measures of the angles of a convex polygon with n sides is (n-2)x180⁰

Examples:

a.)   Find the measures of the interior angles of a convex decagon.

A decagon has 10 sides.

Solution: (10-2) 180^o

= 8 × 180^o

= 1,440^o

The sum of the measures of the interior angles of a

decagon is ​1,440^o.

Polygon Interior AngleTheorem

Examples:

b.)   Find the measures of the interior angles of a convex polygon with 13 sides.

​

Solution: (13-2) 180^o

= 11 × 180^o

= 1,980^o

The sum of the measures of the interior angles of a polygon with 13 sides is ​1,980^o

Polygon Interior AngleTheorem

Examples:

b.)   Find the measures of the interior angles of a convex polygon with 13 sides.

​

Solution: (13-2) 180^o

​

Polygon Interior AngleTheorem

Examples:

b.)   Find the measures of the interior angles of a convex polygon with 13 sides.

​

Solution: (13-2) 180^o

= 11 × 180^o

Polygon Interior AngleTheorem

Examples:

b.)   Find the measures of the interior angles of a convex polygon with 13 sides.

​

Solution: (13-2) 180^o

= 11 × 180^o

= 1,980^o

The sum of the measures of the interior angles of a polygon with 13 sides is ​1,980^o

Polygon Interior AngleTheorem

The sum of the measures of a convex pentagon is 540^o.

 

Solution:

S + T +U + V + W = 540^o

​

Polygon Interior AngleTheorem

The sum of the measures of a convex pentagon is 540^o.

 

Solution:

S + T +U + V + W = 540^o

S + 152^o + 41^o + 96^o + 142^o = 540^o

​

Polygon Interior AngleTheorem

The sum of the measures of a convex pentagon is 540^o.

 

Solution:

S + T +U + V + W = 540^o

S + 152^o + 41^o + 96^o + 142^o = 540^o

S + 432^o = 540^o

​

Polygon Interior AngleTheorem

The sum of the measures of a convex pentagon is 540^o.

 

Solution:

S + T +U + V + W = 540^o

S + 152^o + 41^o + 96^o + 142^o = 540^o

S + 432^o = 540^o

S = 540^o - 432^o 

​

Open Ended

Find the measure of $$\angle D$$  .

Clue: What polygon is the given?

What is the sum of the interior angles of such polygon?

Polygon Interior AngleTheorem

The sum of the measures of a convex pentagon is 540^o.

 

Solution:

S + T +U + V + W = 540^o

S + 152^o + 41^o + 96^o + 142^o = 540^o

S + 431^o = 540^o

S = 540^o - 431^o 

S = 109^o

​

Corollary

The measure of each angle of an n-gon ​is (n-2)x180^o

n

Example:

a.)   Find the measures of the interior angles of a regular quadrilateral.​

Solution: (4-2)x180^o

4

= (2)x180^o

4

= 360^o

4

=​90^o

Open Ended

Dennis wants to put a fence around his regular hexagonal-shaped system of planting on his farm. What should be the measure of each interior angle of the fences?

Polygons and Relationships among its interior angles.

Choose the best answer.​

Multiple Choice

Polygons are made of line segments and curves.

• True

• False

Multiple Choice

Which of the following pictures does not illustrate a polygon?

Multiple Choice

The number of triangles in a polygon is _________ than the number of sides of the polygon.

• 3 more

• 1 less

• 2 more

• 2 less

Multiple Choice

The sum of the interior angles of a triangle is $$360\degree$$ . 

• True

• False

Multiple Choice

Which of the following are examples of consecutive vertices?

• B & D

• A & C

• D & E

• E & B

Multiple Choice

The sum of the measures of the interior angles of a convex polygon with n-sides is found by the formula ______.

• (2-n)/180^o

• (n-2)×180^o

• (2n)×180^o

• 2(180^o – n)

Multiple Choice

What is the sum of the interior angles of a regular polygon with 9 sides?

• 1,440^o

• 720^o

• 1,540^o

• 1,260^o

Multiple Choice

Find the missing angle of a convex hexagon whose other interior angles are 104^o, 29^o, 147^o, 266^o and 83^o

• 720^o

• 91^o

• 139^o

• 431^o

Multiple Choice

Leah wants to construct a regular octagon using popsicle sticks. What must be the measure of each interior angle of the octagon?

• 80^o

• 236^o

• 135^o

• 79^o

-Muhammad Iqbal

“Inductive reason, which alone makes man master of his environment, is an achievement; and when once born it must be reinforced by inhibiting the growth of other modes of knowledge.”

​

​