Download Lesson 6-1 The Polygon Angle

Goal: To find the measures of the
interior and exterior angles of a
polygon.
15-gon
 Using
your straightedge,
draw convex polygons
with three, four, five,
and six sides. A pentagon
has been drawn at the
right.
 In
each polygon, pick a
vertex and draw the
diagonals from that one
vertex. Notice that this
divides the pentagon
into three triangles.

Copy and complete the table below. Use the fact
that the sum of the measures of the interior
angles of a triangle is 180°.
Polygon
Number of
sides
Number of
triangles
Sum of measures
of interior angles
Triangle
3
1
1 × 180° = 180°
Quadrilateral
4
2
2 × 180° = 360°
Pentagon
5
3
3 × 180° = 540°
Hexagon
6
4
4 × 180° = 720°
Heptagon
7
5
N-gon
𝑛
𝑛−2
5 × 180° = 900°
(𝑛 − 2) × 180°
Example 1: Finding a Polygon Angle Sum

What is the sum of the interior angle measures of a heptagon?
Sum =
=
=
=
𝑛 − 2 180
7 − 2 180
5 ∙ 180
900
The sum of the interior angle measures of a heptagon is 900.

Equilateral: All sides are
congruent.

Equiangular: All interior angles
are congruent.

Regular: A polygon is regular if it
is both equilateral and equiangular.
Example 2: Using the Polygon Angle-Sum Theorem

The common housefly, Musca domestica, has eyes that consist
of approximately 4000 facets. Each facet is a regular
hexagon. What is the measure of each interior angle in one
hexagonal facet?

Measure of an angle =
=
=
𝑛−2 180
𝑛
6−2 180
6
4∙180
6
= 120
 What
is 𝑚∠𝑌 in pentagon TODAY?
𝑚∠𝑇 + 𝑚∠𝑂 + 𝑚∠𝐷 + 𝑚∠𝐴 + 𝑚∠𝑌 = 5 − 2 180
110 + 90 + 120 + 150 + 𝑚∠𝑌 = 3 ∙ 180
470 + 𝑚∠𝑌 = 540
𝑚∠𝑌 = 70°
Example 4: Finding an Exterior Angle Measure
What is 𝑚∠1 in the regular octagon at the right?
𝑚∠1 =
360
𝑛
𝑚∠1 =
360
8
𝑚∠1 = 45°
P
356 – 357 #’s 7-25 all and 26-40 even