ABCDE is a regular polygon.
a) Find the sum of measures of all the interior angles,
b) What would be the measure of each exterior angle?
c) How many diagonals can be drawn in the polygon? Also name the diagonals formed.
d) Does a polygon with sum of interior angles 270° exist?
e) What is the measure of each interior angle of the polygon?
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Answer:
We know that each interior angle of a regular polygon is given by 180
o
−
n
360
o
, n= number of sides
Here for pentagon, Each interior angle=180
o
−
5
360
o
=108
o
From figure in ΔABE−
A=108
o
And, ∠ABE=∠AEB since AB=AE
⟹∠ABE+∠AEB+∠A=180
o
⟹∠ABE+∠AEB+108
o
=180
o
⟹∠ABE=∠AEB=36
o
Hence,a=36
o
Similarly from ΔBCD−
∠CBD=∠CDB=36
o
Now ∠CBA=∠ABE+∠EBD+∠CBD
⟹108
o
=36
o
+∠EBD+36
o
⟹∠EBD=b=36
o
Now in ΔBCD−
∠CDE=∠CDB+c
⟹108
o
=36
o
+c
⟹c=72
o
Hence c=2a=2b where b=36
o
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