The figure shows a square, a regular pentagon and a regular hexagon put together. How much is ∠BAC?

We know that, all angles of square are 90°.

In regular pentagon,

Sum of the angles of n-sided polygon = (n – 2) × 180°

⇒ S = (5 – 2) × 180°

⇒ S = 3 × 180°

⇒ S = 540°

Measure of each interior angle

⇒

In regular hexagon,

Sum of the angles of n-sided polygon = (n – 2) × 180°

⇒ S = (6 – 2) × 180°

⇒ S = 4 × 180°

⇒ S = 720°

Measure of each interior angle

⇒

Let us assume the two points adajacent A as P and L.

∠ BAP + ∠ BAC + ∠ CAL + ∠ PAL = 360° (pair of angles at a vertex)

⇒ 120° + 108° + 90° + ∠ BAC = 360°

⇒ 318° + ∠ BAC = 360°

⇒ ∠ BAC = 360° - 318°

⇒ ∠ BAC = 42°