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Interior Angles
The figure below is formed with three sets of parallel lines. Remembering the
theorems about parallel lines and their angles, put a 1 in all the angles you can find
that are congruent to 1 . Put a 2 in all the angles that are congruent to  2 . Do
the same for 3 .
Below are two theorems you learned in Grade 8 as you investigated all the
properties of parallel lines. Write the converse of each theorem.
If two parallel lines are cut by a transversal, then corresponding angles are
congruent.
Converse:__________________________________________________________
Is the above converse a true statement? Explain why or why not.
If two parallel lines are cut by a transversal, then alternate interior angles
are congruent.
Converse:__________________________________________________________
Is the above converse a true statement? Explain why or why not.
2
1
3
Now study the drawing on the previous page. Do you notice anything about the
angles around a point? How many degrees make a straight angle? How many
degrees around the center of a circle? What conjecture can you draw about the 3
angles in a triangle, 1,  2 and  3?
Using a protractor measure the angles in the triangle and quadrilateral below.
F
D
A
E
C
B
G
What is the sum of the interior angles in Δ ABC?
Can you draw a triangle that does not comply with your conjecture?
Do you believe the sum of the interior angles of a triangle is always the same
value? Remember that you are using a protractor to measure. Could you and your
partner measure the same angle and get a different answer? If you didn’t get the
same answer, what is the source of the difference? Justify your thinking about the
sum of the interior angles in all triangles.
If two angles of one triangle are congruent to two angles of another triangle, then
the third angles of the triangles are ____________________________. Justify
your thinking.
What is the sum of the interior angles in Quadrilateral DEGF on the previous page?
Now select one vertex in the quadrilateral and draw a
diagonal from that vertex. (as seen at the right)
D
F
5
6
1
m  1 + m  2 + m  3 = _________
m  4 + m  5 + m  6 = _________
E
2
3
4
G
What can you conjecture about the sum of the angles in Quadrilateral DEFG?
m  1 + m  2 + m  3 + m  4 + m  5 + m  6 = _________
Could you use the idea used in the quadrilateral above to find the sum of the angles
in a pentagon? A hexagon? Any polygon?
In your own words describe how to find the sum of the interior angles of a polygon?
Now use this procedure to complete the table below.
Polygon
Sketch
Number
of sides
Number
of
diagonals
from 1
vertex
Number
of
triangles
Interior
angle
sum
Triangle
(3-gon)
Quadrilateral
(4-gon)
Pentagon
(5-gon)
Hexagon
(6-gon)
Heptagon
(7-gon)
Octagon
(8-gon)
Decagon
(10-gon)
n-gon
Did you notice patterns in the table above? Can you describe these patterns? Use
your pattern to develop a formula for finding the sum of the interior angles of a
polygon. Put this formula in the last row of your table. Use this formula to find the
sum of the interior angles of a 20-gon.
Problems:
1. Find the sum of the numbered angles in the figure below.
2
3
4
1
5
2. Solve for x, find the measure of each angle and the sum of the angles.
B
120
(2x-5)
A
(3x-25)
2x
E
2x
D
C
3. Find x and y in the picture to the right.
B
y
E
A
85
x
55
C
D
4. What is the sum of the two acute angles in every right triangle. Find the
value of ABC below.
A
22
C
5. Find the measure of BDE and BEF and CBD
B
A
C
ABE=63
EBD=84
AC FD
F
E
D
B
What type of polygon is a stop sign?
Are all the sides congruent?
Are all the angles congruent?
In Washington D.C, there is another famous regular polygon. Can you name it?
You now know how to find the sum of the interior angles of any polygon. Can you
find a formula for the measure of one interior angle of a regular polygon? Now
check/verify your formula. Explain how you would do this. Justify your thinking.
6. Find the measure of one angle of a 20-gon.
7. If the measure of one interior angle of a regular polygon is 144 , how many
sides does the regular polygon have?