The problem requires us to find the size of angle $r$ and angle $t$ based on the given diagram. We know that the angles on a straight line add up to $180^{\circ}$.

1. Problem Description

The problem requires us to find the size of angle $r$ and angle $t$ based on the given diagram. We know that the angles on a straight line add up to $180^{\circ}$.

2. Solution Steps

a) To find angle $r$, we know that the angles $31^{\circ}$, $r$, and $77^{\circ}$ lie on a straight line. Therefore, their sum must be $180^{\circ}$.

$31^{\circ} + r + 77^{\circ} = 180^{\circ}$

$r + 108^{\circ} = 180^{\circ}$

$r = 180^{\circ} - 108^{\circ}$

$r = 72^{\circ}$

b) To find angle $t$, we know that the angles $t$ and $77^{\circ}$ are vertically opposite angles. Vertically opposite angles are equal.

$t = 77^{\circ}$

3. Final Answer

a) $r = 72^{\circ}$

b) $t = 77^{\circ}$