Points, Lines, Angles, and Planes
Introduction to :
You thought you were done with algebra. Unfortunately, math continually builds upon 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.
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 since 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 are 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 (as they should be!). That's our answer.
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.
An angle measures 15° more than its supplement. What is its measure?
An angle is one fifth as large as its complement. What is its measure?
Suppose AB = 2x + 3, BC = 3x – 2, and AC = x + 10. If B is between A and C, what is the length of AC?
Suppose AB = 2x + 3, BC = 3x – 2, and AC = x + 10. If B is between A and C, is B the midpoint of AC?
An angle's measure is 6x + 9. If its complement is 27°, what is the value of x?
Two complementary angles, ∠ABC and ∠CBD, have measurements of x + 7 and 2x + 5 degrees, respectively. What are the measurements of these angles?
Vertical angles have measurements of 3x – 2 and 2x + 28. What are the measurements of these angles?
Two supplementary angles have measurements of x + 30 and 3x⁄2. What are the measurements of both angles?
A segment is bisected into pieces measuring x + 2 and 42 – 3x. How long is the segment?
An angle is eight times larger than its complement. What are the measures of both angles?