The document discusses interior and exterior angles of polygons. It states that the sum of the interior angles of any convex n-gon is (n-2) * 180 degrees. The sum of the exterior angles of any convex n-gon is always 360 degrees. Examples are provided to demonstrate using these formulas to find missing angle measures in different polygons. The key points are that the interior angle sum is (n-2) * 180 and the exterior angle sum is always 360.

1. 3.5- Part 2
Angle Measures in
Polygons

2. Investigation: Sum of the
interior angles
1. Draw examples of 4-sided, 5-sided,
and 6-sided polygons. In each
polygon, draw all the diagonals from
1 vertex.
2. Complete the table on the next slide.
What is the pattern in the sum of the
measures of the interior angles in
any convex n-gon?

3. Polygon # of Sides # of
triangles
Sum of
angles
Triangle 3 1 180°
Quad
Pentagon
Hexagon
n-gon n

4. Polygon # of Sides # of
triangles
Sum of
angles
Triangle 3 1 180°
Quad 4 2 2·180=
360°
Pentagon 5 3 3·180=
540°
Hexagon 6 4 4·180=
720°
Dodecagon 12 10 10·180=
1800
n-gon n n-2 (n-2)180°

5. Polygon Interior Angles
Theorem
The sum of the interior
angles of a convex n-gon is
(n-2)•180°.
One angle in a regular n-gon:
n − 2( )•180
nBM #34-35

6. Exterior Angles

7. Polygon Exterior Angle
Theorem
The sum of the measures
of the exterior angles of a
convex n-gon is 360°.
BM #36

8. 1
2
3
4
5
n • interior +exterior =180n°
interior = n − 2( )180° or 180n − 360°
So, 180n − 360 + exterior = 180n
−360 + exterior = 0
exterior = 360°

9. Example 1:
• A heptagon has 4 interior
angles that measure 160° each
and two interior angles that are
right angles. What is the
measure of the other interior
angle?
BM
#34

10. Ex. 1 Solution:
• (n-2)180=interior sum
• (7-2)180=5•180=900°
• 4•160+2•90=640+180=820°
• 900-820=80°

11. Example 2:
• Find the measure of each
angle in a regular 11-gon.
BM
#35

12. Ex. 2 solution:
• (n-2)180=(11-2)180
• 9•180=1620°
• 1620÷11=147.3°

13. Example 3:
• The measure of each exterior
angle of a regular polygon
is 40°. How many sides
does the polygon have?
BM
#36

14. Ex. 3 Solution:
• 360÷40=9
• 9 sides

15. Assignment:
#22 Polygon Worksheet
#23 3.5 WS (p. 301)