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3-5 The Polygon Angle-Sum Theorem
Lesson Objectives: Classify polygons.
Find the sums of the measures of the
interior and exterior angles of polygons.
Lesson 3-5 The Polygon Angle-Sum Theorem
A polygon is a closed figure in a plane made up of segments called sides
that intersect only at their endpoints called vertices, and no adjacent sides are collinear.
Vocabulary
a polygon Not a polygon Not a polygon
not a closed figure two sides intersect between endpoints
Lesson 3-5 The Polygon Angle-Sum Theorem
Classifying Polygons: A convex polygon does not have a diagonal with points outside the
polygon. A concave polygon has at least one diagonal with points outside the
polygon.
Vocabulary
Connection to Science The study of optics teaches
that a convex lens causes rays
of light to come together and
that a concave lens causes
rays of light to spread apart.
Convex lenses are used in
microscopes and telescopes.
Eyeglasses may be either
convex or concave.
a convex polygon
a concave polygon
A diagonal of a polygon is a segment that connects two non-consecutive vertices.
Lesson 3-5 The Polygon Angle-Sum Theorem
Classifying Polygons: An equilateral polygon has all sides congruent. An equiangular polygon has all interior angles congruent. A regular polygon is both equilateral and equiangular.
Vocabulary
n, # of sides Name of polygon
3 triangle
4 quadrilateral
5 pentagon
6 hexagon
7 heptagon
8 octagon
9 nonagon
10 decagon
12 dodecagon
n n-gon
You can classify a polygon by the number of sides it has.
Lesson 3-5 The Polygon Angle-Sum Theorem
Theorem 3-14: Polygon Angle-Sum Theorem
Theorem 3-15: Polygon Exterior Angle Sum Theorem
Key Concepts
The sum of the measures of the angles of an n-gon is (n-2) 180.
m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360°.
For the pentagon,
The sum of the measures of the exterior angles of a polygon at each vertex, is 360.
Lesson 3-5 The Polygon Angle-Sum Theorem
Classifying Polygons
Finding a Polygon Angle-Sum
Classify the polygon at the right by its sides. Identify it as convex or concave.
12
outside
concave
Find the sum of the measures of the angles of a decagon.
10 10
Polygon Angle-Sum
10 10
8
1440°
Lesson 3-5 The Polygon Angle-Sum Theorem
Examples
4
360
Polygon Angle-Sum (n-2)180
90 100 (42)180
190
190 170
m∠X m∠X
2m∠X
2
170
170
85
Lesson 3-5 The Polygon Angle-Sum Theorem
Example
congruent
supplement
supplements
congruent
Hexagon ; convex octagon ; concave
Lesson 3-5 The Polygon Angle-Sum Theorem
Sum of s = (n 2)180
Sum of s = (13 2)180 Sum of s = 1980
Sum of s = (n 2)180
720 = (n 2)180
4 = n 2
6 = n
Divide each side by 180.
Add 2 to each side.
Sum of s = (n 2)180
Sum of s = (5 2)180
Sum of s = 540
Measure of each = 5
540= 108
Lesson 3-5 The Polygon Angle-Sum Theorem
Theorem 3-15: Polygon Exterior Angle Sum Theorem
The sum of the measures of the exterior angles of a polygon
at each vertex, is 360.
Recall…
1
1
Measure of each exterior 1 = 6
360= 60
No, 2 is not formed by extending one side of the polygon.
m1 + m2 = 90
60 + m2 = 90
m2 = 30
Lesson 3-5 The Polygon Angle-Sum Theorem
Go to PRACTICE 3–5 page 301
Start working on it and complete the
worksheet as homework for the day.
