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Points, Lines, Angles, and Planes - Numbers in Geometry

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Numbers in Geometry

Addition and subtraction aren't just for numbers anymore. As it turns out, we can add and subtract segments and angles. In fact, the ancient Greeks were so fond of geometry that they defined their arithmetic in terms of adding and subtracting segments. Good thing we aren't ancient Greeks, right?

  • 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.

    Sample Problem

    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.

    Sample Problem

    Suppose AB = 10 and BC = 20. What is AC?

    Since B is between A and C in the alphabet, it's 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.

    Sample Problem

    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.

    Sample Problem

    In the above figure, if ∠AOC  has a measure of 70° and ∠AOB has a measure of 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.

    Sample Problem

    In the figure, ∠1 measures 54°. What are the measures of the other three numbered angles?

    Since ∠1 and ∠2 come from splitting up a straight line, we know they're supplementary and thus add up to 180°. That means ∠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.

  • Algebra in Geometry

    You thought you were done with algebra. Unfortunately, math continually builds on itself like an endless dance of the Hokey Pokey, and geometry is no exception. Now shake it all about.

    We might be asked to find the measure of an angle or line segment, and in addition to understanding the geometric relationships between the parts of the picture, we'll also need to flaunt our algebra skills (if you've got it, flaunt it!) to finish the problem off for good.

    Typically our plan of attack will be something like this:

    • Draw a picture (if we don't have one already).
    • Use geometry to figure out a relationship between some of the unknown quantities.
    • Form an equation or two (or seven) using this relationship.
    • Solve the equations for whatever variables appear. Here's where algebra sneaks into the picture. What a creeper.
    • If necessary, plug in the variables to get whatever the question is asking us for. One commonly overlooked but essential thing to do is to answer the question that's being asked.
    • Check to make sure that our answers actually make sense within the geometrical relationships. This will make sure our answer is right.

    Armed with this strategy, we can run through a few examples.

    Sample Problem

    In the picture, m∠ABC = 2x + 7 and m∠DBE = 4x – 14. What is the measure of ∠ABC?

    Now, you could be a smart-aleck and say, "Duh! Angle ABC equals 2x + 7," but this answer probably won't give you many points. Who knows what that x is? If you told a carpenter to cut a piece of wood at a 2x + 7 degree angle, the carpenter might cut off your thumbs with a hacksaw. Instead, we should probably solve the problem.

    Since ∠ABC and ∠DBE are opposite each other and come from intersecting lines, they're vertical (and therefore congruent) angles. This means m∠ABC = m∠DBE. Now we're getting somewhere: we can plug in what we know about the angles to get:

    2x + 7 = 4x – 14

    Now, we move all the variables to one side and the numbers to the other to get:

    21 = 2x

    Dividing both sides by 2 gives us:

    x = 10.5

    It's tempting to draw a box around it and say, "Look, there's x. We're done. Woohoo!" But does that answer the question? We wanted to find the measure of angle ∠ABC. That means we should plug x = 10.5 back into 2x + 7.

    m∠ABC = 2x + 7
    m∠ABC = 2(10.5) + 7
    m∠ABC = 28°

    Just to check that everything makes sense, we should calculate m∠DBE by plugging the x we found into 4x – 14.

    m∠DBE = 4(10.5) – 14 = 28°

    Their measures are equal, just like they should be. That's our answer.

    Sample Problem

    An angle measures 25 degrees more than its complement. What is its measure?

    Since this problem didn't even extend the common courtesy to draw us a picture, we can draw our own. We should have two complementary angles, so let's draw it as a right angle split into two parts (remember, it's right to give compliments):

    What else do we know? The only other piece of information is that one of the angles is 25° bigger than the other one. If we label the smaller angle x, the larger one would have to be x + 25.

    We know complementary angles add up to 90°, so that means:

    x + (x + 25) = 90

    We can add those x's together.

    2x + 25 = 90

    Solving for x gives us:

    x = 32.5

    Now we can fix our picture to include the actual measurements of the angles:

    Now, what's the question asking for? Quoth the problem, "Nevermore."

    Oops. We meant, quoth the problem, "An angle measures 25 degrees more than its complement. What is its measure?" Since the angle the question cares about is bigger than its complement, that means the answer to the question is 32.5 + 25 = 57.5°. Finally, we quickly check that 32.5 + 57.5 = 90°, so the angles are actually complements after all.

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