Name: Sum of Exterior Angles of a Polygon
Uploaded: 2019-07-19
Duration: 5 min 7 s
Description: Everyone talks about the interior angles of a polygon, but what about the external ones? In this video, you will learn about the sum of exterior angles of a...

Question

A B C D E F

is a regular hexagon. Find each angle of

△ A B E

.

Accepted Answer

Given that $A B C D E F$ is a regular hexagon. So, all sides of it are equal in length.

Here, we have, $∠ A B C = ∠ B A F = ∠ A F E = \frac{(6 - 2) \times 180 °}{6} = \frac{4 \times 180 °}{6} = 120 °$ -----(i) [The measure of each interior angle of an $n$ -sided polygon is $\frac{2 (n - 2) \times 180 °}{n}$ ]

Since all the sides of the hexagon are equal in length, we have, $A F = F E$ . So, $△ A E F$ is an isosceles triangle and so, we get, $∠ E A F = ∠ A E F$ . -----(ii) [If two sides in a triangle are equal in length, then it is an isosceles triangle and in an isosceles triangle, the angles opposite to the equal sides are equal]

Now, in $△ A E F$ , we have, $∠ E A F + ∠ A E F + ∠ A F E = 180 °$ [The sum of angles in a triangle is $180 °$ ]

$△ A B E$

$\Rightarrow 2 ∠ E A F = 180 ° - 120 °$

$\Rightarrow 2 ∠ E A F = 60 °$

$\Rightarrow ∠ E A F = \frac{60 °}{2} = 30 °$ .-----(iii)

Now, $∠ B A E = ∠ B A F - ∠ E A F = 120 ° - 30 ° = 90 °$ [From (i) and (iii)]

The diagonal $B E$ bisects $∠ A B C$ . So, we have, $∠ A B E = \frac{120 °}{2} = 60 °$ .

Now, using the angle sum property of a triangle on $△ A B E$ , we obtain, $∠ A B E + ∠ B A E + ∠ A E B = 180 °$

$\Rightarrow 60 ° + 90 ° + ∠ A E B = 180 °$

$\Rightarrow ∠ A E B = 180 ° - 60 ° - 90 °$

$\Rightarrow ∠ A E B = 30 °$ .

Therefore, the angles in $△ A B E$ are $∠ A E B = 30 °, ∠ A B E = 60 °$ and $∠ B A E = 90 °$ .

$A B C D E F$ is a regular hexagon. Find each angle of $△ A B E$ .

Important Questions on Understanding Quadrilaterals

Learn and Prepare for any exam you want !

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

Important Questions on Understanding Quadrilaterals

Learn and Prepare for any exam you want !

$A B C D E F$ is a regular hexagon. Find each angle of $△ A B E$ .