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## Class 7 (Old)

### Course: Class 7 (Old)>Unit 12

Lesson 2: Criteria for congruence of triangles

# Why SSA isn't a congruence postulate/criterion

There are some cases when SSA can imply triangle congruence, but not always. This is why it's not like the other triangle congruence postulates/criteria. Created by Sal Khan.

## Want to join the conversation?

• He didn't have a video on SSA BEFORE this one...now i'm super confused about what he's talking about. What other video was BEFORE this one? and What's a postulate? •  A postulate is kind of like a definition or theorem, but it is something you have to accept without any "proof." For example, 2+2 is something you have to accept as 4. The 2 could be 2 apples which have billions and billions of atoms and that doesn't really equal 4 apples. Numbers are relative.
• What's the difference between an axiom and a postulate? •  The terms "postulates" and "axioms" can be used interchangeably: just different words referring to the basic assumptions - the "building blocks" taken as given (assumptions about what we take to be true), which together with primitive definitions, form the foundation upon which theorems are proven and theories are built.

The choice to use one particular term rather than the other is largely a function of the historical development of a given branch of math. E.g., geometry has roots in ancient Greece, where "postulate" was the word used by the Pythagoreans, et.al.

So it's largely a matter of history, and context, and the word favored by the mathematicians that introduced or made explicit their "axioms" or "postulates." "Postulate" was once favored over "Axiom", with the development of analytic philosophy, particularly logical positivism, the term "axiom" became the favored term, and its prevalence has persisted since. Perhaps I should be corrected: As Peter Smith points out, it is "more likely" that the "uptake" of the term "axiom" can be attributed to "mathematicians like Hilbert (who talks of axioms of geometry), Zermelo (who talks of axioms of set theory), etc.".

Neither term is more formal than the other. I personally prefer "postulate" over "axiom", since a "postulate" transparently conveys (or connotes - as in connotation) that what we are calling a postulate is "postulated" as a "supposition", from which we agree to work in building theorems or a theory. In contrast, to me, the connotation of an "axiom" is that of a "law" of some sort, which MUST be followed or MUST be true, though it is no stronger than, or different from, a postulate. But again, this is simply a personal observation and preference, and the term "axiom" seems to have more "uptake" at this point in history
• What is SSA? Or where can I find the video on it? Thanks! • SSA is not a postulate and you can find a video, More on why SSA is not a postulate: This IS the video.This video proves why it is not to be a postulate.

Nonetheless, SSA is side-side-angles which cannot be used to prove two triangles to be congruent alone but is possible with additional information.
• Is Hypotenuse Leg Therorem the same as RSH? • Isn't SSA actually SAS , because in my school we are studying it that way. And is there a video on how to find which property to use in the triangles .. Thank you.what is the meaning of "Wwf" • No, SSA and SAS are two different things! The order of the letters matters a lot. Both of these two postulates tell you that you have two congruent sides and one congruent angle, but the difference is that in SAS, the congruent angle is the one that is formed by the two congruent sides (as you see, the "A" is between the two S), whereas with SSA, you know nothing about the angle formed by the two congruent sides: you only know that the angle formed by the second congruent side and the third side (whose length you don't know) is congruent.

I fear I'm not being very clear (it would be easier to draw it and show you but I can't), but feel free to ask other questions if you need further explanation.
• Why do we learn about SSA if it is not a congruence postulate? • Today in class, my friend and I had a debate about whether or not SSA created congruent triangles. She proved that it did not using the example above, but I proved that it did. I said that if you know two sides and one angle across from one of the sides, you can use the law of sines to find the angle across from the other side. From there, you can find the final angle, and use the law of sines again to find the third side. From that, I concluded that SSA created congruent triangles. Can someone explain what I did wrong? • You only considered one of the two possible cases you could have when you have SSA, you showed it was okay for one case, but ignored the other case because the sin (Θ) = sin (180 - Θ) which should also work with the law of sines. What you did prove is SSS congruence.
As a Math teacher, I enjoy these kinds of debates and the deeper thinking that goes behind them.
• At Sal says that you do not want to use SSA as a postulate. In my proofs, how can I reference it in the special case where the given angle is obtuse? Thanks! • A-S-S acronym substitute sounds all right.  