Question

Find the sum of three angles, in an inscribed hexagon .

See the attachment for more details !​

Answer 1

If you do this for a hexagon, you will find that you can create 4 triangles. You should know that the sum of the interior angles in a triangle is 180 degrees.

So 4 x 180 = 720 degree.

Now divide this by the number of sides in a hexagon:

720/6 = 120 degrees (the interior angle).

This works for all regular polygons with 4 sides or more.

Answer 2

Aim :-

• To find the sum of the 3 angles inscribed in the circle.

Answer :-

Concept applied :-

Since the shape is a hexagon, when we divide it, we get 2 cyclic quadrilaterals. Cyclic quadrilaterals are those whose opposite angles add up to 180°. A Quadrilateral inscribed inside a circle is said to be cyclic.

Solution :-

Let the angles of the hexagon be ∠A, ∠B, ∠C, ∠D, ∠E and ∠F respectively.

Method 1 :-

(refer to attachment m1)

In Quadrilateral ADEF :-

∠EDA + ∠F = 180° (Opposite angles of a cyclic quadrilateral add up to 180°)

In Quadrilateral ABCD :-

∠CDA + ∠B = 180° (Opposite angles of a cyclic quadrilateral)

Now that we have the sum of these, the sum of the given 3 angles will be :-

➡ ∠F + ∠EDA + ∠B + ∠CDA

➡ 180° + 180° (as we already know that ∠F + ∠EDA = 180° and ∠B + ∠CDA = 180°)

➡ 360°

Therefore, the sum of the 3 angles :- 360°

Method 2 :-

(refer to attachment m2)

In Quadrilateral ABEF :-

∠A + ∠FEB = 180° (opposite angles of a cyclic quadrilateral)

In Quadrilateral BCDE :-

∠C + ∠DEB = 180° (cyclic quadrilateral opposite angles)

Hence by adding these :-

➡ ∠A + ∠FEB + ∠C + ∠DEB

➡ 360°

Important points to note :-

• The angles of a Hexagon add up to 720°.
• The angles of a Quadrilateral add up to 360°
• Any shape inscribed inside a circle is said to be cyclic (the properties differ)