Well recently I did some unique triangle stuff and I didn't understand it. My teacher said there's stuff like AAA, SAS, etc. And that a triangle can be unique. But what I don't understand is, if you have a triangle with say 2 angles and like 3. Couldn't you just double the 3 to a 6 and keep the same angles?
The first thing is to understand the difference between similar and congruent (identical). Two triangles are similar if their shapes are similar, but their sizes may be different. An equilateral triangle is similar to another equilateral triangle, but one could have sides of length 1 and the other sides of length 2. These are *similar*. But for them to be congruent both their angles and their sides would all have to be the same.
Like you said, if two triangles had 2 angles the same (and therefore the 3rd angle must be known to be the same since they all add up to 180), then you've only proven them *similar*. A side could be 3 units long, but if you doubled them all, you would still have a *similar* triangle, but it wouldn't be *congruent*.
A concept commonly taught in high school mathematics is that of proving the "angle" and "side" theorems, which can be used to define two triangles as similar or congruent.
In each of these three-letter acronyms, A stands for equal angles, and S for equal sides. For example, ASA refers to an angle, side and angle that are all equal and adjacent, in that order.
AAA - Angle-Angle-Angle. If two triangles share three common angles, they are *similar*. (Obviously, this means that the side lengths are locked in a common ratio, but can vary proportionally, making the triangles similar.) Additionally, since the interior angles of a triangle have a sum of 180°, having two triangles with only two common angles (sometimes known as AA) implies similarity as well.
Two triangles are *congruent* if their corresponding sides and angles are equal. Usually it is sufficient to establish the equality of three corresponding parts and use one of the following results to conclude the congruence of the two triangles:
SAS (Side-Angle-Side): Two triangles are *congruent* if a pair of corresponding sides and the included angle are equal.
SSS (Side-Side-Side): Two triangles are *congruent* if their corresponding sides are equal.
ASA (Angle-Side-Angle): Two triangles are *congruent* if a pair of corresponding angles and the included side are equal.
While the AAS (Angle-Angle-Side) condition also guarantees *congruence*, SSA (Side-Side-Angle) does not, as there are often two dissimilar triangles with a pair of corresponding sides and a non-included angle equal. This is known as the ambiguous case. Of course, AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence shows only *similarity* and not congruence.
However, a special case of the SSA condition is the HL (Hypotenuse-Leg) condition. This is true because all right triangles (which this condition is used with) have a congruent angle (the right angle). If the hypotenuse and a certain leg of a triangle are congruent to the corresponding hypotenuse and leg of a different triangle, the two triangles are congruent.