The answer is, it depends! As it turns out there are two rules. 
Case 1 proves the subtraction of angles rule. Case 2 describes when addition is needed.
 
Case 1: Central Angle ADB = Angle A - Angle B
Both towns are north or south of the sun's zenith.
The three yellow lines are parallel. That's because they represent the sun's rays which are assumed to be parallel.
Sun Angle A = <ADC If you have two parallel lines (the sun's rays) intersected by a straight line, then alternate interior  angles are equal. A and F are such angles and they are equal. 
Sun Angle B = <BDC For the same reason as above.
That means that   A - B = 
<ADC - <BCD
By subtraction
But <ADC - <BCD = <ADB (which is the central angle) By substitution
So A - B = <ADB What this means is that the central angle) = A (sun angle #1) - B (sun angle #2) by substitution.
 
Case 2: Central Angle ADB = Angle A + Angle B
The towns are on opposite sides of the sun's zenith.
The three yellow lines are parallel. That's because they represent the sun's rays which are assumed to be parallel.
Sun Angle A = <ADC If you have two parallel lines (the sun's rays) intersected by a straight line, then alternate interior  angles are equal. A and F are such angles and they are equal. 
Sun Angle B = <BDC For the same reason as above.
That means that   A + B = <ADB .
 
Return to story