Question

e of the missing angles.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. The angle opposite to the $105^{\circ}$ angle is also $105^{\circ}$. Let's find angle $d$.
We know that the sum of angles around a point is $360^{\circ}$, and also, we can use the fact that linear - pair angles (angles on a straight line) sum to $180^{\circ}$.
For the linear - pair with the $105^{\circ}$ angle and $d$, we have $105 + d=180$.
$d = 180 - 105=75^{\circ}$

Step2: Use linear - pair for angle $e$

We know that one of the angles in the linear - pair with $e$ is $32^{\circ}+d$. Since $d = 75^{\circ}$, the angle adjacent to $e$ is $32 + 75=107^{\circ}$.
Using the linear - pair property ($e+(32 + d)=180$), we get $e=180-(32 + 75)=73^{\circ}$

Step3: Use vertical - angle for angle $f$

Angle $f$ is vertical to the angle composed of $32^{\circ}$ and $d$. Since vertical angles are equal, $f = 32 + 75=107^{\circ}$

Answer:

$d = 75^{\circ}$, $e = 73^{\circ}$, $f = 107^{\circ}$