哪些因素能够唯一确定一个三角形（直至全等）？

There are many characteristics that a triangle has: location, orientation, angle measures, side lengths, area, inradius, circumradius, exradii, altitude lengths, and many, many more.

Here, we will discard location and orientation and just determine a triangle up to congruence.

So what do we need? Here are some (of course not all) of the ways to determine a triangle:

all three side lengths

two angles and one side

two sides and the included angle

all three altitude lengths

all three median lengths

all three angle bisector lengths

one side, one angle, area

all three exradii

Also, some triangles have some special ways. For example, two sides and an angle (not necessarily included) are enough for a right triangle but not for other triangles.

In many cases, a triangle will need three to four measurements to be determined. In general, the other methods for determining a triangle essentially boil down to the main three methods: SSS, SAS, and AAS (including ASA).

If two triangles are Congruent it means they are identical, both in shape and size. If two triangles are Similar it means they are only identical in shape.

Similarity

All you need to know if you are trying to figure out if two triangles are similar is two identical angles.

Example:

What you know:

Triangle A has got one 50 degree and one 30 degree angle, and triangle B has got one 30 degree and one 100 degree angle

You need to prove the triangles are similar:

You should know you the sum of all three angles of a triangle is always 180. So you can calculate the third angle from triangle A by subtracting 50 and 30 from 180, which gives you 100. Now you can see triangle A has got one angle of 30 and one angle of 100 degrees, just like triangle B. The same goes for triangle B: 180 - 100 - 30 = 50. Now you know both triangles have the same angles which means they are similar

Q.E.D.

Congruence

To find out if two triangles are congruent there are a couple of ways to proof it. Five things you will need to remember are: SSS, SAS, ASA, SAA and SSN. An S means the length of the two triangles are the same, an A means the angles of two triangles are the same and the N stands for "Ninety degree angle". You can only prove two triangles are congruent if you can prove one of those five theses.

Example:

What you know:

Again we've got two triangles, A and B. From Triangle A you know it has 2 sides which are 3 cm long, in between these sides is a ninety degree angle. From Triangle B you know it has two 45 degree angles and the in between these sides is a √18 cm side

You need to proof these two triangles are congruent:

From what you know you can figure out both triangles have two sides of the same length and one 90 degree angle. SSN is one of the five theses you can to use to prove congruence.

Q.E.D.

Here is a great triangle drawing tool that can help you out in this topic.

(Video) What are the Triangle Congruence Theorems?

by MashUp Math

Here is a summary of all the congruence rules. Watch the full video here $\\$ Remember, the congruence rules that are valid: $\\$ SSS $\\$ SAS $\\$ ASA $\\$ AAS $\\$ HL $\\$

The congruence rules that are NOT valid: $\\$ AAA $\\$ SSA $\\$