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Question

ABCDEF is a regular hexagon. Find each angle of ABE.

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Solution

Given that ABCDEF is a regular hexagon, so all sides of it are equal in length.
Sum of the angles of a n sided polygon =(n2)×180
Hence,
Sum of all angles of a hexagon
=(62)×180
=720
And,
Each angle of a regular hexagon
=7206
=120.......(1)
Quadrilateral ABEF contains half the angle measure of the hexagon.
Hence, its interior angle measure
=720/2=360
AFE=120.......(ii) [ from (1)]
In AFE,
AF=EF [ since, they are a part of regular hexagon].......(iii)
FAE=FEA
[ Angles opposite to equal sides are equal]
In AFE,
FAE+FEA+AFE=180
[ sum of all angles of a triangle is 180o]
substituting (ii) & (iii) in above equation,
FAE=FEA=30o
FAE+EAB=120o [ Angle of a regular hexagon]
EAB=90o.......(iv)
ABE=1/2×120o [ABE bisects ABC]
=60o
In ABE,
AEB+ABE+EBA=180o
[ sum of all angles of a triangle]
AEB=180o60o90o=30o [ from (iv) & (v) ]
ABE=60oEAB=90oAEB=30o

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