Key takeaway: CPCTC stands for corresponding parts of congruent triangles are congruent, a reason used after two triangles are proved congruent to claim that any matching sides or angles are equal.
Grade 10 – GeometryG-CO.B.7Geometry and Measurement
Definition of CPCTC
CPCTC is a justification used in proofs. First you prove two triangles congruent with a shortcut such as SSS or SAS. Then you may state that any pair of matching sides or matching angles is congruent, including pairs you never had direct information about.
The order is essential. It is the step that comes after congruence, never the way to prove it. It lets a proof about triangles reach conclusions about segments and angles, such as showing two segments are equal or two lines are parallel.
Formula and notation
In a two-column proof, the reason column simply reads “CPCTC.” The logic is:
\[ \triangle ABC \cong \triangle DEF \ \Rightarrow \ AB = DE,\ BC = EF,\ AC = DF,\ \angle A = \angle D,\ \angle B = \angle E,\ \angle C = \angle F \]
Use only the pairs that match by position in the congruence statement.
Key facts
- It can be cited only after the triangles are proved congruent.
- Corresponding means matching positions, as listed in the congruence statement.
- It turns triangle congruence into facts about single sides and angles.
- A diagram that merely looks congruent is not enough to use it.
CPCTC example
Segments \(\overline{AC}\) and \(\overline{BD}\) cross at \(X\), with \(AX = XC = 6\) and \(BX = XD = 9\). If \(AB = 4k + 3\) and \(CD = 7k – 12\), find \(AB\).
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\(\angle AXB\) and \(\angle CXD\) are vertical angles, so they are congruent. With the two pairs of equal sides, \(\triangle AXB \cong \triangle CXD\) by SAS.
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By CPCTC, the sides across from the vertical angles are equal: \(AB = CD\).
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So \(4k + 3 = 7k – 12\), giving \(15 = 3k\) and \(k = 5\). Then \(AB = 4(5) + 3\).
\(AB = \) 23, and \(CD = 7(5) – 12 = 23\) as well.
Common mistake
Watch out
In a proof, Dario writes “\(\angle A \cong \angle D\) by CPCTC” in row 2, before showing that the triangles are congruent.
Why it is wrong
This reasoning is circular. The citation may only be used after the triangles are proved congruent by SSS, SAS, ASA, AAS or HL. First establish congruence in earlier rows, then cite CPCTC for the matching parts.
Related terms and lessons
- Congruent Triangles
- Triangle Congruence (SSS, SAS, ASA, AAS)
- Two-Column Proof
- HL Theorem (Hypotenuse-Leg)
- Vertical Angles
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