Question

13) ABCDEF is a regular hexagon. Find each angle of triangle ABE.
​Please tell

Answer 1

Step-by-step explanation:

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 =(n−2)×180

Hence,

Sum of all angles of a hexagon

=(6−2)×180

=720

And,

Each angle of a regular hexagon

= 720/6

=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 180 °

]

substituting (ii) & (iii) in above equation,

∠FAE=∠FEA=30°

∠FAE+∠EAB=120 °

[ Angle of a regular hexagon]

∴∠EAB=90 °

.......(iv)

∠ABE=1/2×120 °

[∵∠ABE bisects ∠ABC]

=60 °

In △ABE,

∠AEB+∠ABE+∠EBA=180 °

[ sum of all angles of a triangle]

∠AEB=180 ° −60 −90 °

=30 °

[ from (iv) & (v) ]

∠ABE=60 °

∠EAB=90 °

∠AEB=30 °