At four o’clock the minute hand is on the 12 and the hour hand is on the 4.  The angle formed is 4/12 of the total number of degrees in a circle, 360.

4/12 * 360 = 120 degrees

A analog clock is divided up into 12 sectors, based on the numbers 1–12. One sector represents 30 degrees (360/12 = 30). If the hour hand is directly on the 10, and the minute hand is on the 2, that means there are 4 sectors of 30 degrees between then, thus they are 120 degrees apart (30 * 4 = 120).

Because it's an 8 hour clock, each section of the clock has an angle of 45 degrees due to the fact that \(\displaystyle \frac{360}{8}=45\).

When the clock reads 1:30 the hour hand is halfway in between the 1 and the 2, and the minute hand is on the 4 (at the bottom of the clock). Therefore, between the hour hand and the "2" on the clock there are \(\displaystyle 45/2 = 22.5\) degrees and between the 2 and the 4 there are \(\displaystyle 45 * 2 = 90\) degrees. Finally, \(\displaystyle 90 + 22.5 = 112.5\)

If creating a picture helps, draw a circle and place 8 at the top where 12 normally is. Then put 2, 4, and 6 at the positions of 3, 6, and 9 on a normal 12-hour analog clock. 1, 3, 5, and 7 go halfway in between each even number. 

Now, because each section is equally spaced, and because there are 8 sections we simply divide the total number of degrees in a circle (\(\displaystyle 360\)) by the number of sections (8). Thus:

 \(\displaystyle \frac{360}{8}=45\)

When the clock reads 8:15 the minute hand is on the 3 and the hour hand is just past the 8.

Each section of the clock is 

\(\displaystyle \frac{360}{12}=30\) degrees.

From 3 to 8 then there are 150 degrees. However, the hour hand has moved a quarter of the way between the 8 and the 9, or a quarter of 30 degrees. \(\displaystyle 30/4 = 7.5\) and so \(\displaystyle 150 + 7.5 = 157.5\)

Each section of the clock is \(\displaystyle 30$^{\circ}$\), and by \(\displaystyle 2\textup{:}45\) the hour hand has gone three quarters of the way between the \(\displaystyle 2\) and the \(\displaystyle 3\). Thus there are \(\displaystyle 30/4 = 7.5$^{\circ}$\) between the hour hand and the \(\displaystyle 3\) numeral. The minute hand is on the \(\displaystyle 9\), and there are \(\displaystyle 180$^{\circ}$\) between the \(\displaystyle 3\) and the \(\displaystyle 9\). So in total there are \(\displaystyle 180 + 7.5 = 187.5$^{\circ}$\) between the hands

The number of degrees between each numeral on the clock face is equal to the number of degrees in a circle divided by the number of sections:
\(\displaystyle 360$^{\circ}$/6 = 60$^{\circ}$\)

At \(\displaystyle \textup{2:50}\) the \(\displaystyle 2\) hand has gone \(\displaystyle \frac{50}{60}\) way through the \(\displaystyle 60$^{\circ}$\) between the \(\displaystyle 2\)and the \(\displaystyle 3\). Thus there are only \(\displaystyle 10$^{\circ}$\) left between it and the 3. There are 120 degrees between the 3 and the 5, where the minute hand is, so the total amount of degrees between the hands is:

\(\displaystyle 10$^{\circ}$ + 120$^{\circ}$ = 130$^{\circ}$\)

Remember there are \(\displaystyle 30^{\circ}\) in each hour long section of the clockface

\(\displaystyle 360^{\circ}/12 = 30^{\circ}\).

When the clock reads 6:30 the minute hand is on the 6, and the hour hand is halfway between the 6 and 7.

Thus the number of degrees between the hands is 

\(\displaystyle 30^{\circ}/2 = 15^{\circ}\)

When the clock reads \(\displaystyle \textup{1:15}\) on this clock, the hour hand will be \(\displaystyle \frac{1}{4}\) of the way between the \(\displaystyle 1\) and the \(\displaystyle 2\). Since there are \(\displaystyle 2\), evenly sized, sections of this clock each section has:

\(\displaystyle 360^{\circ}/2 = 180^{\circ}\). And \(\displaystyle 180 *\frac{1}{4} = 45^{\circ}\). At \(\displaystyle \textup{1:15}\) the minute hand will be one-quarter of the way around the entire dial. \(\displaystyle 360^{\circ}*\frac{1}{4} = 90^{\circ}\)
Thus the hands are \(\displaystyle 90^{\circ} + 45^{\circ} = 135^{\circ}\) from each other

The entire clock measures 360°. As the clock is divided into 12 sections, the distance between each number is equivalent to 30° (360/12). The distance between the 2 and the 3 on the clock is 30°.  One has to account, however, for the 10 minutes that have passed. 10 minutes is 1/6 of an hour so the hour hand has also moved 1/6 of the distance between the 3 and the 4, which adds 5° (1/6 of 30°). The total measure of the angle, therefore, is 35°.