The figure shown a regular pentagon and a regular hexagon put together. How much is ∠PQR?

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 point opposite to Q as S.

∠ PQS + ∠ RQS + ∠ PQR = 360° (pair of angles at a vertex)

⇒ 120° + 108° + ∠ PQR = 360°

⇒ 228° + ∠ PQR = 360°

⇒ ∠ PQR = 360° - 228°

⇒ ∠ PQR = 132°