What does CPCTC stand for? In geometry it means corresponding parts of congruent triangles are congruent. Once you have proved that two triangles are exactly the same size and shape, every matching side and every matching angle must also be equal, and CPCTC is the reason you write to say so.
Students see it in a two column proof, usually in the last step or two. It is not a way to prove triangles congruent. It is what you are allowed to conclude after you already have.
Quick facts
Stands for
Corresponding Parts of Congruent Triangles are CongruentSubject
High school geometryType
Initialism, said letter by letterUsed as
A reason in a two column proofComes after
SSS, SAS, ASA, AAS or HLRelated term
QED, written at the end of a proof
What does CPCTC stand for in geometry, letter by letter?
| Letter | Word | What it means here |
|---|---|---|
| C | Corresponding | In the same position in both triangles |
| P | Parts | Sides and angles |
| C | Congruent | Identical in size and shape |
| T | Triangles | The two shapes you already proved match |
| C | Congruent | Equal in measure |
Notice that congruent appears twice. The first one describes the triangles; the second one is the conclusion about their parts. The small linking words, of and are, get no letters of their own.
When you can use CPCTC in a proof
The order matters. You must first prove the triangles congruent using one of the accepted shortcuts:
- SSS: three pairs of sides match.
- SAS: two sides and the angle between them match.
- ASA: two angles and the side between them match.
- AAS: two angles and a side not between them match.
- HL: the hypotenuse and a leg of two right triangles match.
Only after one of those lines appears in your proof can you write CPCTC to claim that some other side or angle pair is equal. Using it before the triangles are proved congruent is the most common mistake teachers mark.
A short worked example
Problem: Given that AB = DE, BC = EF and angle B = angle E, prove that AC = DF.
| Statement | Reason |
|---|---|
| 1. AB = DE | Given |
| 2. Angle B = angle E | Given |
| 3. BC = EF | Given |
| 4. Triangle ABC is congruent to triangle DEF | SAS |
| 5. AC = DF | CPCTC |
Step 4 sets up the match, and step 5 cashes it in. AC and DF sit opposite the same angle in each triangle, so they correspond. Some students then write QED to close the proof.
Matching parts in the right order
The way you name the triangles tells you which parts correspond. If triangle ABC is congruent to triangle DEF, then A goes with D, B with E and C with F. Side AB matches DE, side BC matches EF and side AC matches DF.
Write the vertices in matching order every time. Naming the second triangle as EDF by accident would pair the wrong parts, and your CPCTC step would no longer be valid.
Pronouncing and writing CPCTC
Say each letter: "see pee see tee see." It never gets blended into a single spoken word. On paper, use capitals and skip the dots. Teachers usually accept CPCTC on its own in the reason column, while some ask for the full phrase on the first proof of the year.
Spoken one letter at a time, it is technically an initialism. Classmates often label it an acronym anyway, which is fine; what is an acronym sorts out the two labels.
Study tips for geometry students
- Circle the congruence statement before you use CPCTC. If you cannot find it, you are not ready.
- Mark matching parts on the diagram with tick marks and arcs.
- Remember that AAA and SSA do not prove congruence.
- Use CPCTC as a step toward other results, such as proving a line bisects an angle.
Memory tricks like this one sit alongside PEMDAS in many math classes. For more abbreviations from class handouts, see school acronyms.
Questions people ask
What does CPCTC mean in a proof?
It means that once two triangles are proved congruent, any pair of matching sides or angles is also congruent. You write CPCTC as the reason for that step.
Can you use CPCTC to prove triangles are congruent?
No. CPCTC works only after the triangles are already proved congruent by SSS, SAS, ASA, AAS or HL. It is a conclusion, not a test.
Is CPCTC a theorem or a definition?
Many textbooks treat it as following directly from the definition of congruent triangles. Others list it as a theorem. Either way, you can cite it as a reason in a proof.
How do you pronounce CPCTC?
Say it letter by letter: C, P, C, T, C. Some teachers simply read the full phrase aloud instead.