Polygons & Quadrilaterals

All sides and all angles congruent.

Where two sides of a polygon meet is called a vertex!

ABCDE

A diagonal is formed by connecting any two non-adjacent vertices.

Triangle 180 °

Quadrilateral 360 °

Pentagon 540 °

Hexagon 720 °

Heptagon or Septagon 900 °

Octagon 1080 °

Nonagon 1260 °

Decagon 1440 °

180 ° (𝑛−2 )

The sum of the measures of the exterior angles of ANY POLYGON is always

360 degrees!

Interior and Exterior angles.

Interior Angle Exterior Angle

An interior angle and its corresponding exterior angle form a straight line (straight angle), therefore they add up to 180 degrees.

𝑆=180 (𝑛−2 )

𝑆=180 (6−2 )

𝑆=180 (4 )

𝑆=720

We are going to use exterior angles to help us on this one.

𝟏𝟎𝟖°

To find an exterior angle, just subtract the interior from 180.

180−108=¿72

𝟕𝟐°

Since the sum of the exterior angles of any polygon is 360 degrees and we know one of the exterior angles, we can just divide 360 by 72. We can do this because we are dealing with a REGULAR POLYGON.

36072

=5 Polygon has 5 sides.

The sum of the measures of the interior angles of a quadrilateral is 360 degrees.

2 𝑥+2 𝑥+𝑥+𝑥=3606 𝑥=3606 𝑥6

=3606

𝑥=60

∠𝐴=𝑥=60∠𝐵=2𝑥=2 (60 )=120∠𝐶=2𝑥=2 (60 )=120∠𝐷=𝑥=60

Sum of exterior angles = 360

3608

=45

Each exterior angle has 45 degrees in it.

𝟒𝟓180−45=135𝟏𝟑𝟓

Each interior angle has 135 degrees in it.

a.

a.

a.

b.

b.

b. b.

c.

c.

c.

c.

360 ° 360 ° 360 ° 360 °

4.

4.

4.

6.

6.

6.

8.

8.

8.

10.

10.

10.

12a.

12b.

12c.

12c.

13a.

13a.

13a.

13a.

13c.

13c. 140

13c.

13c.

14a. 14a.

14a.

14a. +2 +2

14a.

14c. 14c.

14c.

14c. +2 +2

14c.

14f. 14f.

14f.

14f. +2 +2

14f.

14h. 14h.

14h.

14h. +2 +2

14h.

Remember, an interior angle and its corresponding exterior angle are supplementary (add up to 180).

2 𝑥+𝑥=1803 𝑥=1803𝑥3

=1803

𝑥=60

𝐸𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝐴𝑛𝑔𝑙𝑒=2 𝑥=2 (60 )=120

360120

=3 𝑠𝑖𝑑𝑒𝑠

3 𝑥+4 𝑥+5 𝑥+6 𝑥=36018 𝑥=36018 𝑥18

=36018

𝑥=20

6 𝑥=6 (20 )=120

Homework

•Page 6#1-5, 7

a.

a.

a.

a.

b.

b.

b.

b.

c.

c.

c.

c.

d.

d.

d.

d.

a. a.

a.

a. +2 +2

a.

b. b.

b.

b. +2 +2

b.

c. c.

c.

c. +2 +2

c.

d. d.

d.

d. +2 +2

d.

a.

b.

c. c.

c. c.

c. c.

a.

a.

b.

b.

c.

c.

d.

d.

a. Ex

a.

b. Ex

b.

c. Ex

c.

d. Ex

d.

𝑺𝒖𝒎𝒐𝒇 𝒆𝒙𝒕𝒆𝒓𝒊𝒐𝒓=𝟑𝟔𝟎𝑺𝒖𝒎𝒐𝒇 𝒊𝒏𝒕𝒆𝒓𝒊𝒐𝒓=𝟒 (𝒆𝒙𝒕𝒆𝒓𝒊𝒐𝒓 )

𝑺𝒖𝒎𝒐𝒇 𝒊𝒏𝒕𝒆𝒓𝒊𝒐𝒓=𝟒 (𝟑𝟔𝟎 )

𝑺𝒖𝒎𝒐𝒇 𝒊𝒏𝒕𝒆𝒓𝒊𝒐𝒓=𝟏𝟒𝟒𝟎