Solve the following geometry problems involving angles. Problem 1: A right angle is divided into three angles: 2x, 3x, and 45 degrees. Find x. Problem 3: Two lines intersect, forming angles d, e, f, a

$$5x = 90^\circ - 45^\circ$$ $$5x = 45^\circ$$

$$x = \frac{45^\circ}{5}$$ $$x = \boxed{\text{9}^\circ}$$

:
The image shows two intersecting lines, forming angles $d$, $e$, $f$, and $78^\circ$.

Angles $e$ and $78^\circ$ are vertically opposite angles, which means they are equal.

$$e = \boxed{\text{78}^\circ}$$

Angles $d$ and $78^\circ$ form a linear pair, meaning they add up to $180^\circ$.

$$d + 78^\circ = 180^\circ$$ $$d = 180^\circ - 78^\circ$$ $$d = \boxed{\text{102}^\circ}$$

Angles $f$ and $78^\circ$ form a linear pair, so $f + 78^\circ = 180^\circ$.
Alternatively, angles $f$ and $d$ are vertically opposite angles, so $f = d$.

$$f = \boxed{\text{102}^\circ}$$