In a regular pentagon ABCDE, draw a diagonal BE and then find the measure of:

(i) ∠BAE

(ii) ∠ABE

(iii) ∠BED

(i) Since number of sides in the pentagon is 5,

Each exterior angle = $\dfrac{360°}{n}$

= $\dfrac{360°}{5}$

= 72°

According to the property of polygons, the sum of an interior angle and its corresponding exterior angle is 180°.

⇒ ∠BAE + 72° = 180°

⇒ ∠BAE = 180° - 72°

⇒ ∠BAE = 108°

Hence, the value of ∠BAE is 108°.

(ii) In triangle ABE, AB = AE.

Therefore, ∠ABE = ∠AEB.

We also know that the sum of all the angles in a triangle is 180°.

⇒ ∠BAE + ∠ABE + ∠AEB = 180°

⇒ 108° + ∠ABE + ∠ABE = 180°

⇒ 108° + 2∠ABE = 180°

⇒ 2∠ABE = 180° - 108°

⇒ 2∠ABE = 72°

⇒ ∠ABE = $\dfrac{72°}{2}$

⇒ ∠ABE = 36°

Hence, the value of ∠ABE is 36°.

(iii) Since ∠AED = 108° [As each interior angle of the pentagon is 108°],

⇒ ∠AEB = 36°

Therefore,

∠BED = 108° - 36° = 72°

Hence, the value of ∠BED is 72°.