Interior and Exterior Angles of Polygons
Introduction:
1) Define Polygon
2) Name 3 reasons why a geometric figure would not be a polygon
a.
b.
c.
3) Define side of a polygon.
4) Define vertex of a polygon.
5) Define diagonal of a polygon.
6) Define regular polygon
7) List the generic names of Polygons in order, by filling in the table below.
3 sides =
4 sides =
5 sides =
6 sides =
7 sides =
8 sides =
9 sides =
10 sides =
12 sides =
n sides =
8) a. Discuss the difference between convex and concave polygons
b. Draw and label a picture of each in the space provided below
9) Match the following geometric term with its definition below
a. Parallelogram i. A figure with 4 equal sides and 4 right angles
b. Rhombus ii. A figure with 4 right angles and opposite sides
being the same length
d. Trapezoid iii. A figure with 4 equal sides and opposite angles
being the same degree measurement
e. Rectangle iv. A figure with 4 sides and only one set of those
sides are parallel
c. Square v. A figure with 4 sides and two sets of those sides are
parallel, but angles may not be right
Investigation:
Part 1
In this investigation you are going to discover an easier way to find the sum of the interior angles of a polygon, by dividing a polygon into triangles.
9) Consider the quadrilateral to the right.
Diagonal [pic] is drawn. A diagonal
is a segment connecting a vertex with a
nonadjacent vertex.
The quadrilateral is now divided into two
triangles, Triangle DEG and Triangle FEG.
Angles 1, 2, and 3 represent the interior angles
of Triangle DEG and Angles 4, 5, and 6
represent the interior angles of Triangle FEG.
m[pic]1 + m[pic]2 + m[pic]3 = _________
m[pic]4 + m[pic]5 + m[pic]6 = _________
m[pic]1 + m[pic]2 + m[pic]3 + m[pic]4 + m[pic]5 + m[pic]6 = _________
10) What is the relationship between the sum of the angles in the quadrilateral and the
sum of the angles in the two triangles?
By splitting any polygon into triangles you can find the sum of the interior
angles of the polygon.
11) Using the splitting triangle method find the sum of the interior angles of this
hexagon.
12) Draw a sketch of each polygon and use this same procedure to determine the sum
of the angles for each polygon in the table.
|Polygon |Sketch |Number of sides |Number of diagonals from 1 |Number of triangles |Interior angle sum |
| | | |vertex | | |
|Triangle |[pic] |3 |0 |1 |180o |
|Quadrilateral |[pic] |4 |1 |2 |360o |
|Pentagon | | | | | |
|Hexagon | | | | | |
|Heptagon | | | | | |
|Octagon | | | | | |
|Decagon | | | | | |
|Dodecagon | | | | | |
|n-gon | | | | | |
13) In the last row of the table you should have developed a formula for finding the
sum of the interior angles of a polygon. Use this formula to find the sum of the
interior angles of a 20-gon.
14) Write a sentence explaining how to find the sum of the interior angles of a polygon.
Part 2
Regular Polygon Nonregular Polygon
15) Compare the two polygons shown above.
How would you define a regular polygon and a nonregular polygon?
16) What is the sum of the interior angles of a hexagon?
17) What is the measure of one angle of a regular hexagon?
18) If you know the sum of the angles of a regular polygon, how can you find the
measure of one of the congruent angles?
19) Use the information from Part 1 to complete the table below:
|Regular Polygon |Interior angle sum |Measure of one angle |
|Triangle |180o | |
|Quadrilateral |360o | |
|Pentagon | | |
|Hexagon | | |
|Heptagon | | |
|Octagon | | |
|Decagon | | |
|Dodecagon | | |
|n-gon | | |
Problems:
20) What is another name for a regular triangle?
21) What is another name for a regular quadrilateral?
22) What is the interior angle sum of a 60-gon?
23) What is the measure of one interior angle of a regular 60-gon?
24) Three angles of a quadrilateral measure 98 o, 75 o, 108 o. Find the measure of the fourth angle.
25) Each interior angle of a regular polygon measures 168 o. How many sides does the polygon have?
26) If the sum of the interior angles of a polygon is 1080 o . How many sides does the polygon have?
27) If the sum of the interior angles of a polygon is 1200 o . How many sides does the polygon have? Is this polygon possible?
Webquest
HUNT FOR POLYGONS: an Internet Treasure Hunt on Polygons
(see attached page to begin)
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