1. Warm-Up #28 Monday 5/2
Solve for x Find AB

2. Warm-Up #29 Tuesday, 5/3
1. Prove why they are similar and then solve for x.
2. Solve for x and y

3. Interior and Exterior Angles of Polygons
Homework
Interior and Exterior Angles of Polygons

4. Naming Polygons closed figure
A polygon is a _____________ in a plane formed by segments, called sides.
closed figure
angles
A polygon is named by the number of its _____ or ______.
sides
tri
A triangle is a polygon with three sides. The prefix ___ means three.

5. Prefix Number of Sides Name of Polygon Naming Polygons
Prefixes are also used to name other polygons.
Prefix
Number of
Sides
Name of
Polygon
tri- 3
triangle
quadri- 4
quadrilateral
penta- 5
pentagon
hexa- 6
hexagon
hepta- 7
heptagon
octa- 8
octagon
nona- 9
nonagon
deca- 10
decagon

6. Terms Naming Polygons A vertex is the point of intersection of
two sides.
Consecutive vertices are
the two endpoints of any
side. U T S Q R P
A segment whose
endpoints are
nonconsecutive
vertices is a
diagonal.
Sides that share a vertex
are called consecutive
sides.

7. An equilateral polygon has all _____ congruent. sides
An equiangular polygon has all ______ congruent.
angles
A regular polygon is both ___________ and ___________.
equilateral
equiangular
equilateral
but not
equiangular
equiangular
but not
equilateral
regular,
both equilateral
and equiangular
Investigation: As the number of sides of a series of regular polygons increases, what do you
notice about the shape of the polygons?

8. Diagonals and Angle Measure Number of Diagonals from One Vertex
Make a table like the one below.
1) Draw a convex quadrilateral.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2
2(180) = 360

9. Diagonals and Angle Measure Number of Diagonals from One Vertex
1) Draw a convex pentagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2
2(180) = 360
pentagon 5 2 3
3(180) = 540

10. Diagonals and Angle Measure Number of Diagonals from One Vertex
1) Draw a convex hexagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2
2(180) = 360
pentagon 5 2 3
3(180) = 540
hexagon 6 3 4
4(180) = 720

11. Diagonals and Angle Measure Number of Diagonals from One Vertex
1) Draw a convex heptagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2
2(180) = 360
pentagon 5 2 3
3(180) = 540
hexagon 6 3 4
4(180) = 720
heptagon 7 4 5
5(180) = 900

12. Diagonals and Angle Measure Number of Diagonals from One Vertex
1) Any convex polygon.
2) All possible diagonals from one vertex.
3) How many triangles?
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2
2(180) = 360
pentagon 5 2 3
3(180) = 540
hexagon 6 3 4
4(180) = 720
heptagon 7 4 5
5(180) = 900
n-gon n
n - 3
n - 2
(n – 2)180
Theorem
If a convex polygon has n sides, then the sum of the measure of its interior angles is (n – 2)180.

13. Theorem of interior angles
If a convex polygon has n sides, then the sum of the measure of its interior angles is (n – 2)180.

14. Ex. 1: Finding measures of Interior Angles of Polygons
Find the value of x in the diagram shown:
142
88
136
105
136 x

15. SOLUTION:
88
136
142
105 x
The sum of the measures of the interior angles of any hexagon is (6 – 2) ● 180 = 4 ● 180 = 720.
Add the measure of each of the interior angles of the hexagon.

16. SOLUTION: 136 + 136 + 88 + 142 + 105 +x = 720. 607 + x = 720
The sum is 720
Simplify.
Subtract 607 from each side.
The measure of the sixth interior angle of the hexagon is 113.

17. EXTERIOR ANGLE THEOREMS

18. EXTERIOR ANGLE THEOREMS

20. Ex. 2: Finding the Number of Sides of a Polygon
The measure of each interior angle is 140. How many sides does the polygon have?

21. Solution: = 140 Corollary to Thm. 11.1 (n – 2) ●180= 140n
Multiply each side by n.
180n – 360 = 140n
Distributive Property
Addition/subtraction props.
40n = 360
n = 9
Divide each side by 40.

22. Ex. 3: Finding the Measure of an Exterior Angle

23. Ex. 3: Finding the Measure of an Exterior Angle

24. Ex. 3: Finding the Measure of an Exterior Angle

25. Using Angle Measures in Real Life Ex
Using Angle Measures in Real Life Ex. 4: Finding Angle measures of a polygon

26. Using Angle Measures in Real Life Ex
Using Angle Measures in Real Life Ex. 5: Using Angle Measures of a Regular Polygon

27. Using Angle Measures in Real Life Ex
Using Angle Measures in Real Life Ex. 5: Using Angle Measures of a Regular Polygon

28. Using Angle Measures in Real Life Ex
Using Angle Measures in Real Life Ex. 5: Using Angle Measures of a Regular Polygon
Sports Equipment: If you were designing the home plate marker for some new type of ball game, would it be possible to make a home plate marker that is a regular polygon with each interior angle having a measure of:
135°?
145°?

29. Using Angle Measures in Real Life Ex
Using Angle Measures in Real Life Ex. : Finding Angle measures of a polygon