(i) Vertically opposite angles at intersecting lines are equal.

\(\begin{aligned}∠a &= ∠c\\\\ ∠b &= ∠d \end{aligned}\)

(ii) The sum of adjacent angles at intersecting lines is \(180^{\circ}\).

\(\begin{aligned}∠a+\angle d &= 180^\circ\\\\ ∠d+\angle c&= 180^\circ\\\\ ∠c+\angle b&= 180^\circ\\\\ ∠b+\angle a&= 180^\circ \end{aligned}\)

(iii) If two intersecting lines are perpendicular to each other, then all angles at the intersecting lines have the same size of \(90^{\circ}\).

The following diagram shows \(PSQ\), \(RSTU\) and \(PTV\) are straight lines.

Calculate the values of \(x\)\(\) and \( y\).

Noted that the sum of adjacent angles at intersecting lines is \(180^{\circ}\).

So,

\(\begin{aligned}x + 135^{\circ} &= 180^{\circ} \\\\x&=180^\circ-135^\circ \\\\x&=55^\circ. \end{aligned}\)

Also, the vertically opposite angles at intersecting lines are equal.

Thus, \(y = 62^{\circ}\).