Geometry Proof question

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The Velociraptors

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I guess I am wondering how I could write a proof on this. I know about the vertical angles theorem, where vertical angles are always congruent. Also I know that angle 1 and 2 are supplementary because they form a linear pair. Angles 3 and 4 also form a linear pair, so they are supplementary. So would I write for the proof,
angle 1 is equal to angle 2 because they for a linear pair. Linear pairs are supplementary, so the must equal 180.
angle 4 and 3 also form a linear pair, so they must be supplementary.
I also know that angle 1= 180 - angle 2 which equals angle 3.
So angle 3 is equal to angle 1. This is what I’m thinking, but I don’t know if I am thinking correctly, or if this is even the correct way to write a proof.
 
The Velociraptors said:
View attachment 23164
I guess I am wondering how I could write a proof on this. I know about the vertical angles theorem, where vertical angles are always congruent. Also I know that angle 1 and 2 are supplementary because they form a linear pair. Angles 3 and 4 also form a linear pair, so they are supplementary. So would I write for the proof, angle 1 is equal to angle 2 because they for a linear pair. Linear pairs are supplementary, so the must equal 180.
angle 4 and 3 also form a linear pair, so they must be supplementary. I also know that angle 1= 180 - angle 2 which equals angle 3.
So angle 3 is equal to angle 1. This is what I’m thinking, but I don’t know if I am thinking correctly, or if this is even the correct way to write a proof.
Click to expand...
In almost all axiomatic geometry textbooks the authors preferer the paragraph type proof that you have given above. In fact of the seven or more geometry I have that date back to the early 1960's, not one of them uses a two column type proof.
Maybe someone else here knows a set of rules for two column proofs.
You have given a good modern proof.
 
But angle 1 does not necessarily equal angle 2. Your proof is flawed.

Let the measures of angles 1, 2, and 3 in degrees be x, y, and z respectively.

Angle 1 and angle 2 are supplementary angles by definition.

Therefore x + y = 180, which entails that x = 180 - y.

Similarly z = 180 - x.

Thus, x = z because things equal to the same thing are themseves equal.
 
JeffM said:
But angle 1 does not necessarily equal angle 2. Your proof is flawed.

Let the measures of angles 1, 2, and 3 in degrees be x, y, and z respectively.

Angle 1 and angle 2 are supplementary angles by definition.

Therefore x + y = 180, which entails that x = 180 - y.

Similarly z = 180 - x.

Thus, x = z because things equal to the same thing are themseves equal.
Click to expand...
You’re right. Angle 1 wouldn’t be equal to angle 2. I’m looking back, and I did not mean that. I just meant they were supplementary, I don’t know why I wrote they were equal. I guess somewhere I meant angle 1 is equal to angle 3.
 
pka said:
In almost all axiomatic geometry textbooks the authors preferer the paragraph type proof that you have given above. In fact of the seven or more geometry I have that date back to the early 1960's, not one of them uses a two column type proof.
Maybe someone else here knows a set of rules for two column proofs.
You have given a good modern proof.
Click to expand...
Sorry, I never meant angle 1 and 2 are equal. I’m pretty sure that in this case, that they aren’t.
 
The Velociraptors said:
You’re right. Angle 1 wouldn’t be equal to angle 2. I’m looking back, and I did not mean that. I just meant they were supplementary, I don’t know why I wrote they were equal. I guess somewhere I meant angle 1 is equal to angle 3.
Click to expand...
Yes, but that is what you are trying to prove. A proof that uses a bit of algebra is clearer.
 
Isn’t your first statement what you are trying to prove?

If so, you cannot start with it. That is circular reasoning.
 
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