Same Math › Geometry · grades 9–10 › Triangle congruence and CPCTC
What are the triangle congruence shortcuts?
Two triangles are congruent if all six parts match, but you only need to check three. SSS (three sides), SAS (two sides and the angle between them), ASA (two angles and the side between them), AAS (two angles and a side not between them) and HL (hypotenuse and leg of right triangles) all work. SSA and AAA don't. Once triangles are congruent, CPCTC says every other matching part is congruent too.
- Tick marks show matching sides.
- Arcs show matching angles.
- CPCTC: corresponding parts of congruent triangles are congruent.
Why schools teach it
Three facts settle all six
Geometry students explain why SSS, SAS and ASA prove triangles congruent (HSG.CO.B.8) and use congruence to prove other facts (HSG.SRT.B.5).
The order of letters matters: in SAS the angle is between the two sides. Kids mark diagrams with ticks and arcs, then read the shortcut off the marks.
At the kitchen table
What to say, and what to skip
Say this
How many sides are marked? How many angles?
Is the angle between the two sides?
SAS, not SSA.Is there a shared side?
That's a free match by the Reflexive Property.Once they're congruent, what else matches?
CPCTC.
Skip this
- Accepting SSA.Two sides and a non-included angle can make two different triangles.
- Accepting AAA.Same angles means the same shape, not the same size. That's similar, not congruent.
- Using CPCTC before proving the triangles congruent.CPCTC comes after.
See it in action
Naming the shortcut
Marks: 1 side, 2 sides, and the angle between them
SASMarks: 2 angles and the side between them
ASAMarks: 2 sides and an angle not between them
Not enough (SSA)
Vertical angles and shared sides often give the third match for free.
Free practice sheet
Make a triangle congruence sheet
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Parents also ask
What's the difference between ASA and AAS?
In ASA the side is between the two angles. In AAS it isn't. Both work.
Why does HL work if SSA doesn't?
HL is a special SSA for right triangles. The right angle fixes the shape, so only one triangle fits.
How do you say CPCTC?
Teachers say the letters, or “corresponding parts of congruent triangles are congruent.”
Does the order in △ABC ≅ △DEF matter?
Yes. A matches D, B matches E, C matches F.