Category: Geometry Chapter 3

Question from Anna:

Geometry Chapter 3 Lesson 5 Number 11. How do we figure this out?

Answer from Cassidy Cash:

Start with what we know:

North East West South make 90-degree angles so <NOE = <EOS = <SOW = 90 degrees

A = 50. So to this point, draw with N = 0, A = 50 (given), E = 90, S = 180, W = 270

Now you have to figure out the rest. Here are a few ways.

1. You know <NOE = 50 = <DOS since these are linear pairs. So add 50 to the 180 to get 230 at point D. Keep doing the vertical angles.
2. Or you also know the lines of the map (north, east, south, west) bisect those angles < AOB and the like. So If <NOA = 50 and you know <NOE = 90 so <AOE = 40. And since they bisect, you now know <EOB=40, <BOS=50.

Either route you take, you just will see a pattern and you start figuring out one from the other.

So in the end the coordinates (angles) are 0,50,90, 130,180,230,270, and 310

 

Question from Kimberly:

p. 113 “Sun Directions”Says, “If a circular protractor…the coordinate of OA is 50.”What are we missing?  Seems like it should be 45, not 50.

 

Answer from Dr. Callahan:

This is a given – not a question. This is telling how the sun moves through the year. So take it as  given and then work the problem from there. 

 

Question from Mary:

Geometry Chapter 3 Lesson 4 Numbers 19-23. Help please?

Answer from Cassidy Cash:

Let’s start at the top.

#19) This one is 45 degrees, because it is half of 90.

You are right that half of 360 is 180, and you are also right that the figure is a square. You are further right that all the angles in a square equal 360. So congratulations! You have lots of things right. However, the question here is wanting to know about a specific angle. Let me see if I can draw you a picture.

 

#20) Here, they are continuing to bisect angles,only this time they are cutting in half angle ACD. Since we found in number 19 that angle ACD is 45 degrees, we know that angle FCD is 45/2, or 22.5 degrees.

 

 

 

#21) Here they are choosing a larger angle

that while not a bisecting angle, IS an angle we can figure out based on what we have found in numbers 19 and 20 (Note that all of these problems are building on each other intentionally).

Notice that if angle FCD is half of angle ACD, then the other half: Angle ACF must be half of ACD also. This means that <FCD = <ACF, both are 22.5 degrees. Why is this important? Well look at this figure to the right.

 

# 22) DCE is another addition angle. Meaning we have to add together smaller angles to figure it out (You could also subtract, I’ll show you that in a minute).

Option 1 :Let’s start by labeling these angles 1, 2, 3, and 4 for the sake of brevity

 

DCE = <1 + <2 + <3 = 22.5 + 45 (based on what we’ve found in 19-21). That makes DCE = 67.5 degrees

Option 2 is subtraction.  You know the “whole” angle (the original right angle) equals 90 degrees. We also know by process of deduction that <4 = 22.5 degrees. So we can subtract 90 – 22.5 = 67.5 degrees. Either way is correct.

#23) Last , but never least, is angle DFC. This one asks you to apply what you know about triangles.

Notice that Angle DFC is the top of a triangle. I will highlight the triangle in this figure to the right.

We know that the three inside angles of a triangle add together to equal 180 (as you so correctly pointed out originally). So we use that information here. We know that the corner angle of a square equals 90 degrees. Number 19 found for us that the small angle of this triangle, <FCD (aka <1) Is 22.5 degrees. We can now calculate the value of the remaining third angle, angle DFC.

180 – (22.5 + 90) = 180 – 112.5 = 67.5 degrees.

There are other ways to solve this problem, using many geometry theorems that will work. I have shown you the way that I would approach the problem.

 

Question from Amy:

Geometry Chapter 3 Lesson 2 Number 47

Answer from Cassidy Cash:

For Number 47 you are basically copying the proof on page 86 of the text, but replacing some of the values to account for the change in inequalities. If you look at the number line diagram on page 86, your brain most likely assumes that the point “a” is smaller than the point “c”, because we naturally read left to right.

That particular proof where a < b < c, supports that assumption and the following proof treats the number line as if the values are less on the left side, and increasingly greater as you move to the right. However, in number 47 of this same lesson they are asking you to approach the problem as if the larger values are on the left and things get progressively smaller as you move right. That’s what the phrase “a > b > c” is meant to indicate.

From there you are just walking through the proof exactly as it is written on page 86, but switching the placement of the numbers specifically in step 3 where you have the ruler postulate. To get a positive answer to a subtraction problem, the first number in the problem has to be larger than the second. Therefore, in this step for number 47 we have to actually say “AB = a –b” that means, that if we were to add values to this number line it would look like this: Notice that in this image I have inverted the ruler so that the larger numbers are on the left and the smaller numbers are towards the right, as indicated by the problem that the scenario should be.

Ok, now we have to realize what all of these letters are trying to accomplish. The term “AB” means “Tell me how long the segment that starts at A and ends at B is in terms of units”, in order to answer that question, we can either use a ruler to measure, or we can use the values of point “a” and point “b” to determine the length. Point “a” in my diagram above is sitting on number 15, and point “b” is sitting on number 11. Common sense tells us this means that segment AB is 4 units long, but if we did not know the values of “a” and “b” (as we do not know them in problem 47) we would have to find another way to express this process. We could say “subtract a from b” as we did in the example on page 86. However, if we look at the diagram and try to apply that logic, we’ll find that 11-­‐15 = -­‐ 4. That doesn’t make sense, given that length cannot be measured in negative values. If something is “So many units long” the ‘so many units” cannot be a negative number.

What else can we do? Well, subtraction will end up postive if we place the larger value first, and then subtract the smaller value. In the scenario where a > b > c, “b” is the smaller value. So we replace the “AB = b – a” that they had on page 86, with “AB = a – b” for number 47. It helps to think of the situation as opposite, which it is in fact, the turned around version of the same idea.

Does that help? Then with the segment BC, we apply the same process, and say that since c is smaller than b, to get the length of the segment BC we would have to subtract the smaller value from the larger value, so BC = b – c. From there, it is simply substitution.

We are trying to prove that AB + BC = AC using the scenario where a > b > c.  A more logical way to say this is that the lengths of the line segment are the same no matter which side of the ruler you are looking at. If you walk from your house to your mailbox, and then turn around to walk from your mailbox to your house it does not matter if you were headed in the “Positive” direction (a < b < c) or the negative direction (a > b > c) the number of steps you took (which corresponds to the length of the segment) is exactly the same both times. So in order to prove that AB + BC = AC, we have to first show what AB + BC equals in terms of the equations we came up with above. The final step, then, will be to show that AC equals the same as the AB + BC expression, and then we can make the conclusion the AB + BC = AC.

That process looks like this: AB + BC = (a -­‐ b) + ( b – c) This is simply substitution. We simplify this equation to find that (a -­‐ b) + ( b – c) = a –c

By the same process that told us AB = a –b , and that BC = b –c, we can also state that AC = a –c So since AC = a –c AND AB + BC = a – c

We can conclude that AC = AB + BC, thus proving the theorem.

Question from Teresa:

How do I explain how to answer 35 and 36 in chapter 3.4. My son is > working this independently of me. Looking at the answer book, is he to apply > what he has learned in the chapters to demonstrate the statements. Can you > eleboarate how he would approach a proof? I am at a loss.

Answer from Cassidy Cash:

Proofs often cause problems because they are way more broken down than what most people take the time to actually say. For instance, If I were to tell you to tell me WHY 2 + 2 = 4, you might be at a loss to explain that. “It just is” you might say, “that’s what 2 and 2 IS”. But a math book is looking for you to say “that is the rule of addition; if you add 2 and 2, the result is 4”

This same type of being very direct, very—almost simplistic and patronizing in the way a proof actually defines every single step— But when you are approaching these problems the key here is to remember that they ARE asking for every single step to be defined.

In number 34, They start by giving you a statement, which they put the words “by hypothesis” after the given statements, but you should just read that as “Given”, meaning “you don’t have to do anything to this step”, the word “given” or “by hypothesis” already defines those steps. After the given step of “AB = CD”, They say “AB + BC = BC + CD” and ask you to tell why. All that “why” means is that they want you to tell them what math property is happening here to make that statement true. If it is hard to see right away what math property you are dealing with, you can try replacing these letters with numbers, to make it clearer.

For instance, If we say AB = 6 and CD = 6, then we know that 6 = 6 So then If BC = 4, Then we can add 4 to both sides of “6=6” and we will have 6 + 4 = 4 + 6 Which with letters is the “AB + BC = BC + CD” they are asking us to explain. But when we have 6 + 4 = 4 + 6 It is easier to see that this statement is true because that is addition. If I add something to both sides of an equation (which remember “equation” just means “contains an equal sign”), then we are not changing the value of that equation. The rule of addition is the reason that step is true. So “addition” is all you need to write there.

The next step they ask about says “AB + BC = AC and BC + CD = BC” which is the “between-ness of points theorem” which is covered on page 86, and if you flip back there and look at that, you will see that it is pretty straightforward since they are even using the same letters here as they did on pg. 86, so it is fairly easy to pick out what they are looking for there. (granted you may have to do some searching, but actually page 741 lists all the postulates and theorems that are covered in the text all in one place, so you can flip there and search when you need to and it is a little easier).

Then the final step is just substitution because we already said that “AB + BC ” is the same thing as “BC + CD”, so then since AB + BC = AC and BC + CD = BC, then it only makes sense that AC = BC See how that works? Try number 35 and write me back if you need some more help. (Don’t forget to compare with the solutions manual if you get stuck)