Thank you.
But how do you get 180-30, where do the 30 comes from? Why do we need to divide 150 by 2?

Realize the triangle PSV is isosceles triangle. Angle PSV = 30. Therefore, the other two angle for isosceles triangle is 150÷2 =75°
FYI, SR=PS as there are square for PQRS.

Yes. PS = SR (sides of the same square)

But SR = SU
(sides of the same equilateral triangle)


Therefore PS = SU

So triangle PSU is isosceles. And ∠SUP = ∠SPU


∠PSU =∠PSR (one of the square's right angles) - ∠USR(one of the angles of the equilateral triangle)

= 90° - 60°
= 30°

∠SUP + ∠SPU
= 180° - ∠PSU (angle sum of triangle PSU is 180°)

= 180° - 30°
= 150°

Since ∠SUP = ∠SPU,
2 ∠SPU = 150°
∠SPU = 150° ÷ 2 = 75°


Presenting as (180° - 30°)/2 is just combining these into 1 step.

Each of the sides of the equilateral triangle are equal to each side of the square.

As for why ∠RPS is 45°, realise that ∆PSR is isosceles since PS = SR (sides of the same square).

So ∠RPS = ∠PRS (base angles of isosceles triangle)

Since ∠PSR = 90° , (all 4 angles of a square are right angles)

∠RPS + ∠PRS = 180° - ∠PSR (angle sum of triangle PSR)

= 180° - 90°
= 90°


Since ∠RPS = ∠PRS,
2 ∠RPS = 90°
∠RPS = 90° ÷ 2 = 45°

Noted. Thank you so much.

Edited the first comment for more details.