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6-1 Practice 1: Presenting Polygons Guided Notes
NOTES: There are 3 requirements for a figure to be a polygon. Re-write each statement below in
your own words to help you determine if a shape is a polygon.
A polygon is a plane figure …
1) Formed by three or more line segments called sides.
2) Where each side intersects exactly two other sides.
3) Only the endpoints of the sides are points of intersection.
(ex 1) Circle the figures below that are polygons!
(ex 2) Draw your own example of a polygon:
Each common endpoint of two consecutive sides is called a vertex of the polygon.
(ex 3) Draw a star to label a vertex in this duck sign.
If a line containing a side of a polygon does not contain any interior
points of the polygon, the polygon is called a convex polygon.
convex polygon
If a line containing a side of a polygon does contain at least one interior
point of the polygon, the polygon is
not convex or concave.
nonconvex (or
concave) polygon
A polygon is regular if it is both equilateral and equiangular.
Write in your own words:
A polygon is irregular if
(ex 4) Tell whether each polygon is regular or irregular. Tell whether it is concave or convex.
A diagonal of a polygon is a line segment whose endpoints are two non-consecutive vertices of the
polygon.
B
C
AC is a diagonal
USE A STRAIGHTEDGE TO
DRAW ANOTHER DIAGONAL
IN QUADRILATERAL ABCD.
A
D
USE YOUR PRIOR KNOWLEDGE OR GEOMETRY TEXTBOOK TO FINISH THE
TABLE 
Polygon Names
Number
of Sides
Triangle
3
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
n
ngon
THEOREM: The sum of the measures of the interior angles of a quadrilateral is 360.
THIS IS IMPORTANT, REMEMBER THIS!
For this quadrilateral, draw a diagonal dividing this shape into two triangles. Please note that each
triangle has 2 sides that are sides from the quadrilateral!
1) How many degrees are there in each triangle above?
2) What is the sum of the measures of the interior angles for a quadrilateral? Explain your
answer!
3) Are there are always this many degrees in a quadrilateral? Try this shape:
4) For this pentagon, how many triangles can we draw in the interior from the same point?
5) What is the sum of the measures of the interior angles for a pentagon? Explain your answer!
Let’s try this for a hexagon:
6) For this hexagon, how many triangles can we draw in the interior from the same point?
7) What is the sum of the measures of the interior angles for a hexagon? Explain!
Let’s try this for a decagon:
8) For this decagon, how many triangles can we draw in the interior (where each triangle has 2
sides from the sides of the decagon)?
9) What is the sum of the measures of the interior angles for a pentagon? Explain!
**Key Question: Is there a rule or formula that we can write which allows us to find the sum of the
interior angles for ANY polygon! Brainstorm below:
THEOREM: The sum of the measures of the interior angles of a convex n-gon is 180(n ─ 2).
THIS IS IMPORTANT, REMEMBER THIS!
B
C
(ex 5) Find the sum of the interior angle measures of a convex
heptagon.
A
180
180
180
D
E
(ex 6) Find the measure of each interior angle in a convex heptagon
A pentagon has 5 sides. Two
diagonals may be drawn from a
single vertex, which forms 3
triangles. (180)(3) = 540  .
Thus, the sum of the measures of
the inter ior angles of a pentagon is
180(n - 2) = 180(5 - 2) = 180(3) = 540.
COROLLARY: The measure of each interior angle of a regular n-gon is
THIS IS IMPORTANT, REMEMBER THIS!
(ex 7) Find the measure of each interior angle of a regular 16-gon.
(ex 8)
180(n  2)
n
.
THEOREM: The sum of the measures of the exterior angles of a convex polygon, one angle at
each vertex, is 360º. (The example below demonstrates this theorem using a hexagon.)
THIS IS IMPORTANT, REMEMBER THIS!
A
1
B
2
6
F
C
3
5
E
4
m 1 + m  2 + m  3 + m  4 + m  5 + m  6 = 360
D
COROLLARY: The measure of each exterior angle of a regular n-gon is
THIS IS IMPORTANT, REMEMBER THIS!
(ex 9) Find the measure of each exterior angle of a regular 20-gon.
(ex 10)
360
n
.
Activity/Homework
Save this for when I get back tomorrow! Try not to miss me too much 
1.
Use the information in
the diagram to solve for x and y.
.
Use the diagram on the right to answer questions 2 – 5
2. Name the polygon by the number of sides it has.
3. Polygon ABCDEFG is one name for the polygon. State two
other names.
4. Name all of the diagonals that have vertex E as an endpoint.
5. Name the nonconsecutive angles to  A.
6. Solve for x in the accompanying diagram to the right.
7. Classify the shape of the polygon by its number of sides. Then
state whether the polygon is convex or concave.
8.
Find the value of x.
9.
The measure of each interior angle of a regular polygon is 144º. How many sides does
the polygon have?
10. Find the value of x.