Points, Lines, Angles, and Planes
Arithmetic in Geometry
Say you're going to school, and you plan on stopping by Taco Bell on the way. Your house, Taco Bell, and school are all on the same street, and Taco Bell is between you and school. They're collinear, so on a map it would look something like this.
Now, if the distance from your house to Taco Bell is 2 miles and the distance from Taco Bell to school is 2 miles, how far is it from your house to school? Wishing the school were millions of light-years away won't help, since it's only 2 + 2 = 4 miles away. Going to school is the same as going to Taco Bell and then school (as long as you don't stop to get a burrito and miss first period).
More generally, if B is between A and C, then AC = AB + BC. That's called the Segment Addition Postulate. In fact, the converse of this statement is also true: if AC = AB + BC, then B is between A and C. That's all there is to adding segments in geometry.
Suppose A is between Z and P, and say you know that AZ = 10 and ZP = 56. What is AP?
In situations like this, where the problem is just given to you in words, it's sometimes useful to draw a picture to organize your thoughts. Since we know A is between Z and P, we can draw a picture like this:
Then we can write down the numbers we know:
Now, it should be clear that we have a segment addition question. We can set it up as 10 + AP = 56, and subtracting 10 from each side gives us AP = 46.
Suppose AB = 10 and BC = 20. What is AC?
Now, since B is between A and C in the alphabet, it is tempting to draw a picture like this:
But, for all we know it could look like this:
or even this:
Since we don't know which point (if any) is between the other two, we don't have enough information to solve the problem.
When a segment is split into two halves, pessimists say it's half empty and optimists say it's half full. Mathematicians just say that it's bisected, and call the point in the middle the midpoint. If something cuts something else in half, we sometimes call the thing doing the cutting a bisector.
In the image below, line l4 is a bisector of segment AH. What is the midpoint of AH?
Since l4 cuts AH in half, we know that AE is congruent to EH. That's because point E is on l4. This means that E is the midpoint of AH.
Adding angles is a bit more complicated since there's no clear-cut idea of "betweenness" for angles. Instead, we typically add adjacent angles according to the Angle Addition Postulate.
Since ∠AOB and ∠BOC are adjacent, we have m∠AOC = m∠AOB + m∠BOC. The idea is that if we split something two smaller angles and then put them back together again, we should get the same thing. Still, splitting an angle in two and splitting your dad's Porsche in two will probably have very different outcomes.
In the above figure, if m∠AOC = 70° and m∠AOB = 45°, what is m∠COB?
Since ∠COB is just another name for ∠BOC, we have 70° = 45° + m∠COB, so that means m∠COB = 25°.
Just in case we didn't already have enough terminology for angles, we have special words for pairs of angles whose measures add up to certain numbers. If two angles add up to 90°, we call them complementary. Such angles often arise from splitting a right angle into two smaller angles.
On the other hand, sometimes complementary angles are two angles completely unrelated to each other that just happen to add up to a right angle.
Similarly, if two angles add up to 180°, we call them supplementary. Supplementary angles usually come from splitting up a straight line into two angles.
Just like with complementary angles, they can also come from other configurations.
Sometimes it's easy to confuse complementary and supplementary. We remember them by saying, "It is right to give your friends compliments." This helps you advance not only your geometry skills but also your social life so that you won't have to take your cousin to school dances anymore. Those slow dances can get real awkward.
In the figure, m∠1 = 54°. What are the measures of the other three numbered angles?
Since ∠1 and ∠2 come from splitting up a straight line (line l), we know they are supplementary and thus add up to 180°. That means m∠2 has to be 180 – 54 = 126°. Angles ∠3 and ∠1 are vertical (and therefore congruent!), so m∠3 = 54°. Similarly, ∠2 and ∠4 are congruent so m∠4 = 126° also.