Question 753795: I am completely lost on how to do a proof. Is there any way you could just explain to me how to do one? And define what ASA, AAS, and SAS are? Thank you very much for your help!
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! A proof is an explanation of the reasoning that takes you from the given statement to the statement you have to prove.
It may be easy to explain and defend your points while discussing it with another person.
Unfortunately, written proofs are often required, and those are more difficult, because you may forget to state some intermediate conclusions that seem very obvious to you, but from the teacher's point of view they need to be proven.
Different teachers have different requirements for proofs (and for geometry problems in general), and different codes and abbreviations are expected and accepted by different teachers.
In the USA (or at least where my children went to school) teachers favor what they call the two-column proof, and they use acronyms like ASA, AAS, SAS, and CPCTC that are expected and often required in problems and proofs.
ACRONYMS:
The codes ASA, AAS, and SAS are used to show that two triangles are congruent (they are the same triangle, moved over, and/or rotated, and/or flipped) because comparing their sides and angles, you find that "corresponding" sides and angles are comngruent (have the same measure).
ASA is short for angle-side-angle, and it's easier to show the meaning with a drawn example.
The way I drew the triangles below, are really the same triangle, flipped over. One triangle looks as the other triangle flipped, or as the other triangle's reflection on the green line.
In geometry, we say that the two triangles are congruent.
You can think of "congruent" as meaning "with the same measures".
I could state that "ABC is congruent to YXZ"
The names of the vertices (and sides are listed so that congruent parts in both triangles are listed in the same order.
Although the drawing looks that way, you cannot conclude that the triangles are congruent just from the way they look. You can only draw conclusions from information given to you.
If you are told that in the triangles ABC and XYZ above:
sides AB and XY congruent,
angles CAB and XYZ are congruent, and
angles ABC and ZYX are congruent,
you can conclude that the two triangles are congruent.
You may state your conclusion as "ABC is congruent to YXZ by ASA"
Each teacher may want it stated a specific way. Maybe you should say "by the ASA postulate" or "by ASA congruency". There may be a symbol for congruent that you are expected to use instaed of writing "is congruent to".
ASA is an abbreviated way of saying that there is one side with a certain length in each of the two triangles (corresponding congruent sides), and the angles on either side of the congruent sides have the same measure.
In a similar way, SAS means that the parts that are the same in two triangles are one of the angles and the two sides forming that angle.
AAS would mean that what is the same is two angles and one side that is not between those two angles.
The acronym CPCTC is short for Congruent Parts of Congruent Triangles are Congruent.
A PROOF:
Knowing that that in triangles ABC and XYZ below, sides AB and XY congruent,
angles CAB and XYZ are congruent, and angles ABC and ZYX are congruent,
prove that sides BC and ZX are congruent
A two column proof may list sytatements and reasons like this:
...... Statement ................................................................. Reason
1.... sides AB and XY congruent................................ given
2.... angles CAB and XYZ are congruent............. given
3.... angles ABC and ZYX are congruent............. given
4.... triangles ABC and YXZ are congruent...... ASA congruency (see 1, 2, and 3)
5.... sides ABC and ZX congruent........................... by CPCTC
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