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Geometry Ch5.1 #44

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.