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Mathematics › 🔍 Grade 10 › Triangle Congruence & an Introduction to Proof
🔍 Grade 10 · Lesson 2 of 12

Triangle Congruence & an Introduction to Proof

Three carefully chosen matching parts are enough to know two triangles are identical — if you choose the right three.

Grade 10Geometry
Triangle Congruence & an Introduction to Proof — illustration
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The big idea: Two triangles are congruent when one can be moved exactly onto the other, matching every side and angle. You never need to check all six parts: five criteria — SSS, SAS, ASA, AAS and HL — each use just three pieces of information to pin down a triangle uniquely. Two look-alikes, SSA and AAA, don't work, and understanding <em>why</em> they fail sets up ideas you'll meet again soon: the ambiguous case of the Law of Sines, and triangle similarity.
🎯 By the end, you'll be able to
  • State the five triangle congruence criteria: SSS, SAS, ASA, AAS and HL
  • Identify which side or angle must be included for SAS, ASA and AAS to apply
  • Explain why SSA and AAA are not valid congruence criteria
  • Write a short proof that two triangles are congruent using CPCTC
📎 You should already know
  • Triangle angle sum (180°)
  • Naming and labeling triangle parts

What it means for triangles to be congruent

Two triangles are congruent when they are exactly the same size and the same shape — one can be slid, turned, or flipped onto the other so every part lines up. When triangles are congruent, every pair of corresponding sides and every pair of corresponding angles is equal; this fact goes by the name CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

But you don't have to check all six parts (three sides and three angles) to be sure. A handful of shortcuts let you confirm two triangles are congruent from just three carefully chosen pieces of information.

🔑 The five congruence criteria
Two triangles are congruent if any of these match: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), or HL (hypotenuse and a leg, right triangles only). Each one supplies just enough information to determine the whole triangle — there is only one triangle you could possibly build from those parts.

Included vs. non-included

An angle is included between two sides if it sits at the vertex where those two sides meet. A side is included between two angles if it connects their two vertices. This distinction is the whole game in SAS, ASA and AAS — the letters tell you not just which parts you know, but exactly where they sit relative to each other.

⚠️ Included matters — don't swap it
In SAS, the known angle must be the one between the two known sides — not just any angle in the triangle. In ASA, the known side must be the one between the two known angles. AAS is the odd one out: it uses a side that is not between the two angles. Mixing these up doesn't just break the label — in the SSA case below, it can mean the triangle isn't determined at all.

Why SSA is not a criterion

Knowing two sides and a non-included angle (SSA) sounds like just as much information as SAS, but it isn't enough. Picture a fixed angle at vertex A and a fixed side AB. The second known side, starting at B, can often swing like a hinge and still reach the third side at two different points — producing two triangles with the exact same SSA measurements but different shapes. Because the information doesn't pin down a single triangle, SSA is never listed as a congruence criterion.

✨ A preview: the hinge comes back
This same swinging behaviour reappears later in this module, in the ambiguous case of the Law of Sines: given SSA measurements, the Law of Sines can likewise produce zero, one, or two valid triangles. What you're noticing here about congruence is the same geometric fact that makes that later lesson tricky.
\[ AB + BC > AC \qquad AB + AC > BC \qquad BC + AC > AB \]
The triangle inequality: each side must be shorter than the sum of the other two, or the three lengths can't close up into a triangle at all.

AAA: same shape, not necessarily same size

What about three equal angles (AAA)? Matching all three angles forces the triangles to have the same shape, but says nothing about size — you could scale the whole triangle up or down and every angle would stay the same. AAA guarantees similarity, not congruence, and that distinction is exactly where the next lesson picks up.

🔑 HL: why the right angle matters
HL (hypotenuse-leg) looks like SSA — it gives two sides and an angle that isn't between them — and SSA is exactly the case that fails. But HL only applies to right triangles, and that right angle is precisely what removes the ambiguity: once the hypotenuse and one leg are fixed, the Pythagorean theorem pins down the third side uniquely, so the swinging-hinge problem can't occur.
🎮 Congruence Criteria Lab LIVE
Drag two triangles' sides and angles to test which combinations of SSS, SAS, ASA, AAS and HL lock the triangles together.
📝 Worked example: Triangle ABC has AB = 7, BC = 9, and the angle between them, &ang;B = 40°. Triangle DEF has DE = 7, EF = 9, and &ang;E = 40°. Are the two triangles congruent, and by which criterion?
  1. In ▵ABC, ∠B sits at the vertex where sides AB and BC meet, so it is the angle included between those two sides. The same is true for ∠E between DE and EF in ▵DEF.
  2. Both triangles supply two sides (7 and 9) and the angle included between them (40°) — that's SAS information.
  3. Two sides and the included angle determine a triangle uniquely, so the two triangles must be identical in shape and size.
✓ &triangle;ABC &cong; &triangle;DEF by <strong>SAS</strong>.
📝 Worked example: In isosceles triangle ABC, AB = AC, and M is the midpoint of BC. Prove that &triangle;ABM &cong; &triangle;ACM.
  1. AB = AC, since ▵ABC is isosceles (given).
  2. BM = CM, since M is the midpoint of BC (given).
  3. AM = AM, since it is a side shared by both smaller triangles (reflexive property).
  4. All three pairs of sides match, so ▵ABM ≅ ▵ACM by SSS.
✓ &triangle;ABM &cong; &triangle;ACM by <strong>SSS</strong>; by CPCTC, &ang;AMB = &ang;AMC, and since they are supplementary, AM is also perpendicular to BC.

Check your understanding

1. Which criterion requires the known angle to be included between the two known sides?
SAS names the order exactly: side, angle, side — the angle must sit between the two given sides.
2. Why is SSA (two sides and a non-included angle) not a valid congruence criterion?
The side not touching the given angle can often swing to two different positions and still fit the given lengths and angle, producing two distinct triangles from the same SSA data.
3. HL (hypotenuse-leg) is valid only for which triangles, and why?
HL is really SSA plus a guaranteed right angle. That right angle lets the Pythagorean theorem fix the third side uniquely, so the usual SSA ambiguity can't happen.
4. Triangle ABC has &ang;A = &ang;D, &ang;B = &ang;E, &ang;C = &ang;F, but AB = 4 while DE = 8. What can you conclude?
Matching all three angles (AAA) only guarantees the same shape. With AB ≠ DE the triangles are different sizes, so they're similar but not congruent.
5. Can a triangle have sides of length 4, 5, and 10?
The triangle inequality requires each side to be shorter than the sum of the other two. Here 4 + 5 = 9, which is less than 10, so the three lengths can't close up into a triangle.
✅ Key takeaways
  • Congruent triangles are identical in size and shape; CPCTC says every corresponding side and angle then matches.
  • SSS, SAS, ASA, AAS and HL each supply just enough information to determine a triangle uniquely.
  • In SAS the angle must be included between the two sides; in ASA the side must be included between the two angles; AAS is the non-included case.
  • SSA is not a valid criterion — the non-included side can swing into two different triangles, foreshadowing the ambiguous case of the Law of Sines.
  • HL works only for right triangles, where the right angle removes SSA's ambiguity; AAA gives similarity, not congruence.