Category: Geometry Chapter 5

Question from Courtney

I’m having trouble with Chapter 5, Lesson 1, Problem 29.
I have worked the problem, I know what the answer should be, but given the figure, I don’t understand how that could be false.
Without the figure, I understand it.

Answer:

You can never assume anything. You only know what you are given in the definition. Don’t trust the figure. If it doesn’t tell you, you don’t know it. So for problem # 29 we know the following:

It’s a line
AB is less than BC
BC is less than CD
CD is less than DE

While the lengths in the figure look similar (or even equal), we don’t know that they really are similar or equal. Don’t trust anything but the defined statements in the problem and marked items in a figure.

Here is the question to think about. Given the definitions AB < BC < …. can you draw the figure such that the last DE is much larger.

Put some numbers in it and think inches.

AB = 1
BC = 2
CD = 3
DE = 500

Does that fit the definition of the problem? (Not the figure – the definition?)

Again – figures are often only ONE example that fits the problem – but they do not show every example.

 

Cathy asked:

Need some help with this problem – Geometry Ch. 5.1 #44:
The answer key for this question states that we can’t draw any definite conclusion about whether angle 3 and 4 are vertical angles.  
We previously proved angles 3 and 4 to be equal; based on the drawing, how could they not be vertical, too?

 

Dr. Callahan Wrote

For #44 note they are asking if you can PROVE. 

Look at the logic. If vertical angles, then <3 = <4 and <1 = <2. And since we know (given) that <2=<4 then we would have to conclude <1=<3 by substitution. 
Note – that is NOT a given. 
The books answer is that you are jumping to conclusions ASSUMING line AC is straight to the end after it intersects with BC. We do not know that. It is easy to be fooled by how a drawing looks. In proofs you have to stick with the logical conclusions you can make from the given facts and not what it looks like. So the book is correct – the answer is no. 
Hope this helps. 

Question from Teresa:

I’m in Jacobs Geometry, Chapter 5, Lesson 2. In problems 32 and 33 looking at the figure I thought the answer was “given” for both of them. What the answer key says doesn’t seem to make sense with the question. Am I just misunderstanding the problem?

Answer from Dr. Callahan:

First – they are not “given” since there was nothing “given” in the problem or in the figure.

So instead he is asking us WHY these items can be determined – which is more definition.

I do realize this sounds silly – that a midpoint is possible because every line segment has one midpoint – but that is the idea he is after. He is trying to nail down that NOTHING can be said without backing it up with reason and fact.

You will find other instances of this in the book (and likely have already) but the trick is when you have a silly question – often it is either GIVEN (in which case look back at the problem and see if it is indeed given) or it is definition.

Hope this helps.

dwc