Angle Measure in Regular Polygons, Similarity, Congruence
A polygon is said to be regular is all sides and all angles have the same measure.
There is a relationship between the number of sides and the sum of the measures of the
interior angles of a polygon. This relationship can be discovered by subdividing a
polygon into triangles using the diagonals from a single vertex. If we want to measure
the sum of angles v1  v2  v3  v4  v5 in the regular pentagon pictured below
Note that
a  b  c  v1
g  f  v4
e  d  v3
So it must be that v1  v2  v3  v4  v5 = 3(1800 )  5400
The validity of this argument does not depend on the fact a regular pentagon was used.
Also this same argument can be used on any polygon leading to the following theorem.
Theorem: The sum of the interior angles of a polygon with n sides is (n  2)1800
The next table summarizes the results for regular polygons
Congruence
Two polygons are congruent if the have exactly the same size and shape.
For example, if quadrilateral CAMP is congruent to quadrilateral SITE, then their four
pairs of corresponding angles and four pair of corresponding sides would also be
congruent. Also we would write
Similarity
Polygons that have the same shape but not necessarily the same size are said to be
similar.
In mathematics you can think of similar objects as enlargements or reductions of one
another.
What makes polygons similar? Hexagon PQRSTU is an enlargement of hexagon
ABCDEF --- they are similar and we write PQRSTU ABCDEF.
R
D
Q
C
S
E
P
B
A
F
T
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Measure all corresponding angles. How do they compare?
Measure all corresponding sides
Find the ratios of all corresponding sides. . How do the ratios compare?
This investigation leads us to conclude the following:
Two polygons are similar if and only if the corresponding angles are congruent and
the corresponding ides are proportional.
Next we examine some shortcuts for establishing when one triangle is congruent to
another.
Is AA a Similarity Shortcut?
If two angles of one triangle are congruent to two angles of another triangle, does that
mean the two triangles are similar?
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Draw and triangle ABC
Construct another triangle DEF, with D  A and E  B . What will be true
about C and F ? Why?
Carefully measure the lengths of the sides of both triangles. Compare the
corresponding sides. Are the proportions equal?
Compare your results with those near you. State a conjecture.
U
AA Similarity Conjecture
If two angle of one triangle are congruent to two angles of another triangle then the
triangles are similar.
Is SSS a Similarity Shortcut?
Construct a triangle ABD. Then construct a second triangle DEF whose sides are a
multiple of the original triangle. Compare the corresponding angles of the two triangles.
Compare your results with the results of others near you and state a conjecture.
SSS Similarity Conjecture
If the three sides of one triangle are proportional to the three sides of another triangle,
then the two triangles are similar.
Is SAS a Similarity Shortcut?
Construct two triangles with corresponding sides having the same ratio and the included
angles being congruent. Compare the ratios of corresponding sides and the measures of
corresponding angles. Share your results with others near you. What do you conclude?
SAS Similarity Conjecture
If two sides of one triangle are proportional to two sides of another triangle and the
included angles are congruent, then the triangles are similar.