Polygons & Quadrilaterals

All sides and all angles congruent.

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

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

𝑆=180 𝑛−2 𝑆=𝑠𝑢𝑚 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒𝑠 𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠 𝑆=180 6−2 𝑆=180 4 𝑆=720

We are going to use exterior angles to help us on this one. To find an exterior angle, just subtract the interior from 180. 180−108= 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. 𝟏𝟎𝟖° 𝟕𝟐° 360 72 =5 Polygon has 5 sides.

2𝑥+2𝑥+𝑥+𝑥=360 ∠𝐴=𝑥=60 ∠𝐵=2𝑥=2 60 =120 6𝑥=360 ∠𝐶=2𝑥=2 60 =120 The sum of the measures of the interior angles of a quadrilateral is 360 degrees. 2𝑥+2𝑥+𝑥+𝑥=360 ∠𝐴=𝑥=60 ∠𝐵=2𝑥=2 60 =120 6𝑥=360 ∠𝐶=2𝑥=2 60 =120 6𝑥 6 = 360 6 ∠𝐷=𝑥=60 𝑥=60

Sum of exterior angles = 360 360 8 =45 Each exterior angle has 45 degrees in it. 𝟏𝟑𝟓 𝟒𝟓 180−45=135 Each interior angle has 135 degrees in it.

a. 𝑆=180 𝑛−2 b. 𝑆=180 𝑛−2 a. 𝑆=180 3−2 b. 𝑆=180 7−2 a. 𝑆=180 b. 𝑆=180 5 b. 𝑆=900 c. 𝑆=180 𝑛−2 c. 𝑆=180 9−2 c. 𝑆=180 7 c. 𝑆=1260

360° 360° 360° 360° 4. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟒 =𝟗𝟎 8. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟗 =𝟒𝟎 4. 𝐈𝐧𝐭=𝟏𝟖𝟎−𝟗𝟎 8. 𝐈𝐧𝐭=𝟏𝟖𝟎−𝟒𝟎 4. 𝐈𝐧𝐭=𝟗𝟎 8. 𝐈𝐧𝐭=𝟏𝟒𝟎 6. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟔 =𝟔𝟎 10. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟐𝟎 =𝟏𝟖 6. 𝐈𝐧𝐭=𝟏𝟖𝟎−𝟔𝟎 10. 𝐈𝐧𝐭=𝟏𝟖𝟎−𝟏𝟖 6. 𝐈𝐧𝐭=𝟏𝟐𝟎 10. 𝐈𝐧𝐭=𝟏𝟔𝟐

12a. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟑𝟎 =𝟏𝟐 𝒔𝒊𝒅𝒆𝒔 12c. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟔𝟎 =𝟔 𝒔𝒊𝒅𝒆𝒔 12b. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟒𝟓 =𝟖 𝒔𝒊𝒅𝒆𝒔 12c. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟏𝟐𝟎 =𝟑 𝒔𝒊𝒅𝒆𝒔 13a. 𝐄𝐱𝐭=𝟏𝟖𝟎−𝒊𝒏𝒕𝒆𝒓𝒊𝒐𝒓 13c. 𝐄𝐱𝐭=𝟏𝟖𝟎−𝒊𝒏𝒕𝒆𝒓𝒊𝒐𝒓 13a. 𝐄𝐱𝐭=𝟏𝟖𝟎−𝟗𝟎 13c. 𝐄𝐱𝐭=𝟏𝟖𝟎−140 13a. 𝐄𝐱𝐭=𝟗𝟎 13c. 𝐄𝐱𝐭=𝟒𝟎 13a. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟗𝟎 =𝟒 𝒔𝒊𝒅𝒆𝒔 13c. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟒𝟎 =𝟗 𝒔𝒊𝒅𝒆𝒔

14a. 𝑆=180 𝑛−2 14c. 𝑆=180 𝑛−2 14a. 180=180 𝑛−2 14c. 540=180 𝑛−2 14a. 180 180 = 180 180 𝑛−2 14c. 540 180 = 180 180 𝑛−2 14a. 1=𝑛−2 14c. 3=𝑛−2 +2 +2 +2 +2 14a. 3=𝑛 14c. 5=𝑛

14f. 𝑆=180 𝑛−2 14h. 𝑆=180 𝑛−2 14f. 2700=180 𝑛−2 14h. 3600=180 𝑛−2 14f. 2700 180 = 180 180 𝑛−2 14h. 3600 180 = 180 180 𝑛−2 14f. 15=𝑛−2 14h. 20=𝑛−2 +2 +2 +2 +2 14f. 17=𝑛 14h. 22=𝑛

2𝑥+𝑥=180 𝐸𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝐴𝑛𝑔𝑙𝑒=2𝑥=2 60 =120 3𝑥=180 360 120 =3 𝑠𝑖𝑑𝑒𝑠 Remember, an interior angle and its corresponding exterior angle are supplementary (add up to 180). 2𝑥+𝑥=180 𝐸𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝐴𝑛𝑔𝑙𝑒=2𝑥=2 60 =120 3𝑥=180 360 120 =3 𝑠𝑖𝑑𝑒𝑠 3𝑥 3 = 180 3 𝑥=60

3𝑥+4𝑥+5𝑥+6𝑥=360 18𝑥=360 18𝑥 18 = 360 18 𝑥=20 6𝑥=6 20 =120

Homework Page 6 #1-5, 7

a. 𝑆=180 𝑛−2 b. 𝑆=180 𝑛−2 a. 𝑆=180 4−2 b. 𝑆=180 6−2 a. 𝑆=180 2 b. 𝑆=180 4 a. 𝑆=360 b. 𝑆=720 c. 𝑆=180 𝑛−2 d. 𝑆=180 𝑛−2 c. 𝑆=180 9−2 d. 𝑆=180 13−2 c. 𝑆=180 7 d. 𝑆=180 11 c. 𝑆=1260 d. 𝑆=1980

a. 𝑆=180 𝑛−2 b. 𝑆=180 𝑛−2 a. 1800=180 𝑛−2 b. 2700=180 𝑛−2 a. 1800 180 = 180 180 𝑛−2 b. 2700 180 = 180 180 𝑛−2 a. 10=𝑛−2 b. 15=𝑛−2 +2 +2 +2 +2 a. 12=𝑛 b. 17=𝑛

c. 𝑆=180 𝑛−2 d. 𝑆=180 𝑛−2 c. 540=180 𝑛−2 d. 2160=180 𝑛−2 c. 540 180 = 180 180 𝑛−2 d. 2160 180 = 180 180 𝑛−2 c. 3=𝑛−2 d. 12=𝑛−2 +2 +2 +2 +2 c. 5=𝑛 d. 14=𝑛

b. 𝑆=180 𝑛−2 𝑆=180 5−2 𝑆=180 3 a. 𝑆=180 𝑛−2 𝑆=540 𝑆=180 4−2 𝑅𝐴=540−116−138−94−88 𝑆=180 2 𝑅𝐴=104 𝑆=360 𝑅𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝐴𝑛𝑔𝑙𝑒=360−42−75−118 𝑅𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝐴𝑛𝑔𝑙𝑒=125

c. 𝑆=180 𝑛−2 c. 𝑆=180 6−2 c. 𝑆=180 4 c. 𝑆=720 c. 𝑅𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝐴𝑛𝑔𝑙𝑒=720−95−154−80−145−76 c. 𝑅𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝐴𝑛𝑔𝑙𝑒=170

a. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟓 =𝟕𝟐 d. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟏𝟓 =𝟐𝟒 a. 𝐈𝐧𝐭=𝟏𝟖𝟎−𝟕𝟐=𝟏𝟎𝟖 d. 𝐈𝐧𝐭=𝟏𝟖𝟎−𝟐𝟒=𝟏𝟓𝟔 b. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟐𝟒 =𝟏𝟓 b. 𝐈𝐧𝐭=𝟏𝟖𝟎−𝟏𝟓=𝟏𝟔𝟓 c. 𝐄𝐱𝐭= 𝟑𝟔𝟎 𝟖 =𝟒𝟓 c. 𝐈𝐧𝐭=𝟏𝟖𝟎−𝟒𝟓=𝟏𝟑𝟓

a. Ex𝐭=𝟏𝟖𝟎−𝟏𝟔𝟐=𝟏𝟖 c. Ex𝐭=𝟏𝟖𝟎−𝟏𝟒𝟎=𝟒𝟎 a. 𝟑𝟔𝟎 𝟏𝟖 =𝟐𝟎 𝒔𝒊𝒅𝒆𝒔 c. 𝟑𝟔𝟎 𝟒𝟎 =𝟗 𝒔𝒊𝒅𝒆𝒔 b. Ex𝐭=𝟏𝟖𝟎−𝟏𝟒𝟒=𝟑𝟔 d. Ex𝐭=𝟏𝟖𝟎−𝟏𝟔𝟖=𝟏𝟐 b. 𝟑𝟔𝟎 𝟑𝟔 =𝟏𝟎 𝒔𝒊𝒅𝒆𝒔 d. 𝟑𝟔𝟎 𝟏𝟐 =𝟑𝟎 𝒔𝒊𝒅𝒆𝒔

𝑺𝒖𝒎 𝒐𝒇 𝒆𝒙𝒕𝒆𝒓𝒊𝒐𝒓=𝟑𝟔𝟎 𝑺𝒖𝒎 𝒐𝒇 𝒊𝒏𝒕𝒆𝒓𝒊𝒐𝒓=𝟒 𝒆𝒙𝒕𝒆𝒓𝒊𝒐𝒓 𝑺𝒖𝒎 𝒐𝒇 𝒊𝒏𝒕𝒆𝒓𝒊𝒐𝒓=𝟒 𝟑𝟔𝟎 𝑺𝒖𝒎 𝒐𝒇 𝒊𝒏𝒕𝒆𝒓𝒊𝒐𝒓=𝟏𝟒𝟒𝟎