In the figure below, ABCDEF is a regular hexagon.
Prove that ABCDE is a rectangle.
The given figure is a regular hexagon, so all its angles are equal in measure.
Number of sides of a regular hexagon = 6
We know that the sum of the angles of a polygon with n sides = (n −2) × 180°
Sum of the angles of the given regular hexagon = (6 − 2) × 180°
= 4 × 180°
= 720°
Let the measure of each angle of the given regular hexagon be x°.
⇒ 6x = 720°
⇒ x = 120°
In ΔFEA, by angle sum property, we have:
∠AFE +∠FEA + ∠EAF = 180° …(1)
FE = FA (All sides of a regular hexagon are equal in length)
⇒∠FEA = ∠FAE (Angles opposite to equal sides are equal in measure)
Putting in equation (1):
∠AFE + 2∠FEA = 180°
⇒ 120° + 2∠FEA = 180°
⇒ 2∠FEA = 60°
⇒∠FEA = 30°
Similarly, ∠CDB = ∠CBD = ∠FAE = 30°
As ∠FED = 120°
⇒ ∠FEA + ∠AED = 120°
⇒ 30° + ∠AED = 120°
⇒∠AED = 90°
Similarly, ∠EAB = ∠ABD = ∠BDE = 90°
In quadrilateral ABDE, ∠EAB = ∠ABD = ∠BDE = ∠AED = 90°.
Also, ED = AB
Thus, quadrilateral ABDE is a rectangle.