Congruent Triangles
“Match just three conditions and two triangles become identical.”
The formula
△ABC ≅ △DEFHow to read it: triangles ABC and DEF are congruent — identical in size and shape, so they overlap exactly
- ≅
- — the congruence sign — can be laid perfectly on top of each other
- △ABC, △DEF
- — the two triangles, written in matching (corresponding) vertex order
The hook
You don't have to measure all three sides and all three angles — match just three things and two triangles are completely identical.
In plain words
Two triangles being congruent means you can rotate or flip one onto the other perfectly. Every corresponding side and angle is equal.
The intuition
A triangle is a 'stubborn' shape. Fix the three side lengths and its shape locks into exactly one form that can't wobble (that's why bridges and towers use it in their frames). It locks in three ways, so there are three tests for congruence.
How it's built
Three tests — SSS (three sides), SAS (two sides and the included angle), ASA (one side and the two angles at its ends). Confirm any one and the rest follow automatically. The key is the position: 'included angle', 'angles at the ends'.
Example
If two sides are 5cm and 7cm and the included angle between them is 60° in both (SAS), the triangles are congruent without measuring anything else.
Common mistake
It seems like three equal angles should mean congruent, but no — AAA gives 'similar', not congruent, since the shape can match while the size differs. Also SSA (a non-included angle) can yield two different triangles, so it isn't a congruence test.
Where it's used
The basic tool of geometric proofs, and the foundation of surveying, drafting, and part specifications — anywhere you 'make or verify identical copies'.
Where it came from
Euclid's Elements already set out SAS, ASA, and SSS as the criteria for congruence, and they've been the starting point of geometric proof for over two thousand years.
Quick check
Which of these is NOT a triangle congruence condition?
- three sides equal (SSS)
- two sides and the included angle equal (SAS)
- three angles equal (AAA)✓
- one side and its two end angles equal (ASA)
Practice
Two triangles have two sides equal to 4 and 6, and the angle between them is also equal. Which congruence condition is this?
Answer: 1
- What's equal: side-angle-side (two sides and the angle between them)
- That arrangement is exactly SAS
Takeaway: When the angle sits 'between' the two sides, it's SAS.
Must two triangles with three equal angles be congruent? If not, write on your notepad what they are instead.
Answer: undefined
- Equal angles don't force equal size
- Example: a small equilateral triangle and a large one — all angles are 60°, but the sizes differ
- So they are similar, not congruent
Takeaway: Equal angles fix the shape but leave the size free — that's similarity.
Are two triangles with one equal side and its two equal end angles congruent? Also name the condition.
Answer: undefined
- Fix one side and the two angles at its ends and the triangle is determined uniquely
- This is the ASA condition
- So yes, they are congruent
Takeaway: One side plus its two end angles (ASA) pins down exactly one triangle.