We are given a diagram with two intersecting lines. We are asked to find the size of angle $f$ in part a, and then use that result to find the size of angle $g$ in part b. The given angles are $31^\circ$ and $73^\circ$.

1. Problem Description

We are given a diagram with two intersecting lines. We are asked to find the size of angle $f$ in part a, and then use that result to find the size of angle $g$ in part b. The given angles are $31^\circ$ and $73^\circ$.

2. Solution Steps

a) The angles on a straight line add up to $180^\circ$. The angles $31^\circ$, $f$, and $73^\circ$ lie on a straight line. Therefore, we can write the equation:

$31^\circ + f + 73^\circ = 180^\circ$

$f = 180^\circ - 31^\circ - 73^\circ$

$f = 180^\circ - 104^\circ$

$f = 76^\circ$

b) The angles $g$ and $31^\circ$ are vertically opposite angles, so they are equal. Similarly, the angle opposite to $73^\circ$ is $73^\circ$. Finally, the angle opposite to $f=76^\circ$ is $76^\circ$.

Alternatively, the angles $g$, $f$, and $73^\circ$ lie on a straight line. Therefore, we can write the equation:

$g + f + 73^\circ = 180^\circ$

Since we found that $f = 76^\circ$, we can substitute this value into the equation:

$g + 76^\circ = 31^\circ + 73^\circ$ because $g$ and $31^\circ$ are vertically opposite angles. The other vertically opposite angle pair is given by the combination of $73^\circ$ and $f$.

This gives that $f = 73^\circ$

Then $g+31 + 73 = 180$

And similarly $g+76+73=180$

The angles $31^\circ$ and $g$ are vertically opposite. Therefore, $g = 31^\circ$

3. Final Answer

a) $f = 76^\circ$

b) $g = 31^\circ$