CPCTC

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CPCTC: meaning, method and solved example (Geometry) – Maths PDF

In short: CPCTC, short for corresponding parts of congruent triangles are congruent, is the reason you cite to conclude that matching sides or angles are equal once two triangles are proven congruent.

GeometryGeometryG-CO.B.7

What is CPCTC?

CPCTC is a two-step trick. First prove two triangles congruent with a shortcut such as SAS. After that, every other matching side and angle is guaranteed equal, and the phrase is the written justification for this second step.

It is a tool for drawing conclusions, never for proving the triangles congruent in the first place. It appears as the reason in the row after a congruence statement, and it lets a proof reach segments or angles that were not given as equal, such as the far sides of a bowtie figure.

How to spot it

  • A proof asks you to show two segments or two angles are equal, and the figure contains two triangles that could be matched.
  • The figure looks like a bowtie or a kite, with triangles sharing a vertex or a side.
  • The Prove line talks about sides or angles, not about whole triangles.
  • A table row needs a reason, and the row above it states a congruence.

The method

  1. Find two triangles that contain the segments or angles you need to compare.
  2. Prove those triangles congruent with SSS, SAS, ASA, AAS or HL, writing vertices in matching order.
  3. In the next row, state the pair of corresponding parts and cite CPCTC as the reason.
  4. Check that the parts you named really occupy matching positions in the congruence statement.
ABCDMACBDAMC and BMD congruent by SAS, so AC = BD by CPCTC
Once the two triangles are congruent, the green sides AC and BD are equal by CPCTC.

CPCTC: solved example

Problem

Segments \(\overline{AB}\) and \(\overline{CD}\) cross at \(M\), with \(AM = MB\) and \(CM = MD\). If \(AC = 3x + 1\) and \(BD = 2x + 9\), find \(AC\).

Solution

In triangles \(AMC\) and \(BMD\): \(AM = BM\) (given), \(\angle AMC = \angle BMD\) (vertical angles) and \(CM = DM\) (given). So \(\triangle AMC \cong \triangle BMD\) by SAS.

By CPCTC, \(AC = BD\), so \(3x + 1 = 2x + 9\) and \(x = 8\). Then \(AC = 3(8) + 1 = 25\), and \(BD = 2(8) + 9 = 25\) agrees. The answer is \(AC = 25\).

Try it

Question 1

It is proven that \(\triangle JKL \cong \triangle MNP\). Which angle must equal \(\angle K\), and why?

Show the answer

\(\angle N\), because \(K\) and \(N\) are in the same position of the statement; the reason is CPCTC.

Question 2

Can CPCTC be used to prove that two triangles are congruent?

Show the answer

No. It can only be cited after congruence is already established by SSS, SAS, ASA, AAS or HL.

CPCTC vs. Congruent Triangles

Congruent triangles are the fact: all six matching parts are equal. The abbreviation is the reasoning step that uses that fact to pull out one equal pair from the six. In a proof, you establish the triangles first, then name a single pair and write the abbreviation beside it.

Congruent Triangles

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