Six measurements fix a triangle, but three of the right kind are enough.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The criteria that work
SSS, SAS, ASA, AAS, and — for right triangles — HL, hypotenuse and one leg.
Why SSA does not work
Two sides and a non-included angle can produce two different triangles. It is not a criterion.
What congruence then gives you
Once triangles are congruent, every corresponding part is equal — which is usually the point.
How little you need
SSS, SAS, ASA and AAS each guarantee congruence from three measurements. You do not need all six. Finding the minimum sufficient information is a genuine mathematical question, and the answer is not obvious in advance.
SSA does not work
Two sides and a non-included angle can produce two different triangles — the ambiguous case. This is why SSA is not a congruence criterion, and it is worth constructing the two triangles once to see the failure for yourself.
AAA gives similarity, not congruence
Three equal angles fix the shape but not the size, so AAA proves similarity only. Confusing it with a congruence criterion is common, and the distinction is exactly what separates the next chapter from this one.
The letter order states the correspondence
Writing △ABC ≅ △DEF claims A↔D, B↔E, C↔F. A congruence statement with the letters in the wrong order is false even when the triangles are genuinely congruent. Matching the vertices correctly is part of the claim.
Step 2: Try It Yourself
Tap and try it out.
- sin — opposite over hypotenuse0.60
- cos — adjacent over hypotenuse0.80
- tan — opposite over adjacent0.75
- The marked angle36.87°
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 Sam Choose a Criterion
Two triangles share a side, and two pairs of angles are known equal.
- Step 1
The shared side is equal to itself, by the reflexive property.
Step 4: Your Turn
Practice makes it stick.
The Criterion
Problem 1 of 2
Three pairs of equal sides. Which criterion is that?
The One That Fails
Problem 2 of 2
Is SSA a valid congruence criterion?
Name the Criterion
1 of 4
Two sides and the angle between them. Which criterion?
2 of 4
Two angles and the side between them. Which criterion?
3 of 4
How many pairs of equal parts do the criteria need?
4 of 4
Congruent triangles, one side is 13. The matching side?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Congruent triangles, one angle is 68°. The matching angle?
Question 2 of 2
Why is SSA not a criterion?
What You Learned
- SSS, SAS, ASA, AAS, and HL prove triangles congruent.
- SSA does not, because two triangles can fit.
- Congruence then makes every corresponding part equal.