A triangle has six parts, but three well-chosen ones are enough to fix it completely.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
SSS
Three matching sides force congruence. Three side lengths build exactly one triangle.
SAS
Two sides and the angle between them force congruence. The angle must be the included one.
ASA and AAS
Two angles and any side work, because the third angle follows from the 180° total.
Why SSA fails
Two sides and a non-included angle can describe two different triangles. It is the ambiguous case.
Why AAA fails
Three matching angles give the same shape at any size. That is similarity, not congruence.
Four criteria
SSS, SAS, ASA and AAS each guarantee congruence from three measurements. You never need all six, and finding the minimum sufficient set is a real mathematical question.
SSA is not a criterion
Two sides and a non-included angle can produce two different triangles. Constructing both once is more convincing than being told, and it explains why the ambiguous case exists in trigonometry.
AAA gives similarity only
Three equal angles fix the shape and not the size. That is similarity, not congruence, and the distinction is what separates this chapter from the similarity work that follows in later courses.
Why three measurements suffice
Each criterion corresponds to a construction that can be completed in only one way. Triangles are rigid in a way other polygons are not, which is why they appear in bridges and roof trusses.
Step 2: Try It Yourself
Tap and try it out.
The hypotenuse is not given. It comes from 4² + 3² = 25, whose square root is 5.
Step 3: Watch an Example
One step at a time.
Watch Diego Choose a Criterion
Two triangles share two pairs of equal sides, with the equal angle between those sides.
- Step 1
Diego counts what is given: two sides and one angle in each triangle.
Step 4: Your Turn
Practice makes it stick.
The Criterion
Problem 1 of 2
Three pairs of equal sides. Which criterion? 1 SSS, 2 SAS, 3 ASA.
The Failure
Problem 2 of 2
Two sides and a non-included angle. Does this prove congruence? 1 yes, 0 no.
Prove It
1 of 8
Two angles and the side between them. 1 SSS, 2 SAS, 3 ASA.
2 of 8
Two sides and the included angle. 1, 2 or 3?
3 of 8
Does AAA prove congruence? 1 yes, 0 no.
4 of 8
How many parts does a triangle have in total, sides and angles?
5 of 8
How many matching parts are needed for congruence?
6 of 8
Angles 50 and 60 are given. What is the third, in degrees?
7 of 8
Sort each combination by whether it proves congruence.
Tap something to move it.
- Empty
- Empty
8 of 8
Two angles and a non-included side. Does it prove congruence? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
Three pairs of equal sides. Which criterion? 1 SSS, 2 SAS, 3 ASA.
Question 2 of 2
Why does SSA fail to prove congruence?
What You Learned
- SSS, SAS, ASA and AAS each prove triangle congruence.
- SSA is ambiguous and AAA gives only similarity.
- Three well-chosen parts fix a triangle completely.