Chapter 8 QUADRILATERALS
Lesson 8.1 Notes: Finding angle measures in polygons
Recall: What is an exterior angle? Draw them!
Draw each exterior angle (only one direction) and find each exterior angle measure.
Next, find the sum of the exterior angles of each polygon.
50°
140°
30°
140°
80°
100°
80°
140°
SUM = ________
SUM = ________
140°
SUM = ________
Memorize this!
Polygon EXTERIOR ANGLES theorem
The sum of the measures of the exterior angles of an n-gon is _________________
__________!
Recall: A regular polygon is ______________________ and ______________________ .
To find the measure of ONE EXTERIOR angle of a regular n-gon
ONE Exterior angle
=
or
(recall that “n” is the number of sides)
number of sides =
Memorize these also!
The exterior angle is
the star of the show!
Examples:
A) What is the measure of each exterior angle in a regular octagon?
B) One exterior angle of a regular polygon is 18°. How many sides does this polygon have?
C)
What is the value of x for this regular polygon?
x°
For any polygon, the exterior and interior angles create a _________________________.
This means their sum is always ___________.
Examples:
A)
Steps:
1. Find the EXTERIOR angle x first = _________
y°
x°
2. x and y make a linear pair! The measure of interior angle y = _______
B) Find the measure of an exterior angle AND the measure of an interior angle of a regular nonagon.
Exterior angle = __________
Interior angle = __________
Diagonals of a Polygon:
A
Draw all diagonals
Draw all diagonals from vertex A
Do you know the sum of the measures of the interior angles of any of the polygons below?
A
A
SUM = ________
A
A
SUM = ________
Polygon Interior Angles Theorem:
SUM = ________
SUM = ________
Note: Polygon does NOT have to be regular!
The sum of the interior angle measures of a convex n-gon is:
Another formula
to Memorize!
In words:
If you forget this formula … a shortcut is to draw the diagonals and count the number of triangles created!
Examples:
A) Find the sum of the interior angles of a regular decagon.
B) Find the sum of the interior angles of a 14-gon.
C) The sum of the interior angles of a polygon is 1080°. How many sides does the polygon have?
If you like the exterior angle
formula … you do not need to
use or memorize this formula!
Measure of One Interior Angle of a Regular n-gon
One Interior angle =
180(𝑛 − 2)
𝑛
Examples:
A) What is the measure of one interior angle of a regular dodecagon?
Method 1 (formula method)
Method 2 (exterior angle method)
B) One interior angle of a regular polygon is 175°. Find the number of sides of the polygon.
Method 1 (formula method)
Method 2 (exterior angle method)
Combination examples:
A) Is it possible for a regular polygon to have an interior angle with a measure of 70°?
Why or why not? (This question is on nearly every SOL test!)
B) What type of polygon is this?
3x°
x°
38°
What is the SUM of the measures of the interior angles?
100°
2x°
Write an equation and solve for x:
C) Find the following measures for a regular 15-gon.
Sum of the interior angles: __________
Sum of the exterior angles: __________
Measure of each interior angle: __________
Measure of each exterior angle: __________
D) The measure of one interior angle of a regular polygon is 160°, find the following:
Hint: find the number of sides first!
Number of sides: __________
Sum of the exterior angles: __________
Sum of the interior angles : __________
Measure of each exterior angle: __________