Question

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

See the attachment for more details !

Answer 1

ANSWER:

Given:

• An inscribed(or cyclic) hexagon

To Find:

• Sum of Angle A, Angle C and Angle E

Solution:

(Refer the attachment for naming and division)

We divide the hexagon ABCDEF, into 2 cyclic quadrilaterals ABCD and ADEF.

So, Angle A and Angle D are divided by the line AD.

First we take quadrilateral ABCD,

⇒ Angle BAD + Angle BCD = 180° ————(1) (Sum of opposite angles of a cyclic quadrilateral is 180°)

Now we take quadrilateral ADEF,

⇒ Angle FAD + Angle DEF = 180° ————(2) (Sum of opposite angles of a cyclic quadrilateral is 180°)

Now adding (1) & (2),

⇒ Angle BAD + Angle BCD + Angle FAD + Angle DEF = 180° + 180°

⇒ Angle BCD + Angle DEF + (Angle BAD + Angle FAD) = 360°

⇒ Angle BCD + Angle DEF + Angle BAF = 360°

⇒ Angle C + Angle E + Angle A = 360°

⇒ Angle A + Angle C + Angle E = 360°

This is an important result, i.e. Sum of alternate angles of a cyclic hexagon is 360°.

Answer 2

• The formula for calculating the sum of interior angles is ( n − 2 ) × 180 ∘ where is the number of sides. All the interior angles in a regular polygon are equal.
• So, the sum of the interior angles of a hexagon is 720 degrees. All sides are the same length (congruent) and all interior angles are the same size (congruent).
• In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.