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CPCTC

CPCTC

"Corresponding parts of congruent triangles are congruent." A theorem stating that if two triangles are congruent, then so are all corresponding parts.

 

 

See also

CPCFC

Worked Example

Problem: In the figure, point M is the midpoint of segment AC and segment BD. Prove that angle A is congruent to angle C. Given: AM = CM, BM = DM, and segments AC and BD intersect at M.
Step 1: Identify what is given. M is the midpoint of AC, so AM = CM. M is the midpoint of BD, so BM = DM.
AM=CM,BM=DMAM = CM, \quad BM = DM
Step 2: Notice that angles AMB and CMD are vertical angles, so they are congruent.
∠AMB≅∠CMD\angle AMB \cong \angle CMD
Step 3: Apply the SAS (Side-Angle-Side) congruence postulate. In triangle AMB and triangle CMD, you have two pairs of equal sides and the included angle between them congruent.
△AMB≅△CMD(by SAS)\triangle AMB \cong \triangle CMD \quad (\text{by SAS})
Step 4: Now apply CPCTC. Since the two triangles are congruent, all their corresponding parts are congruent. Angle A in triangle AMB corresponds to angle C in triangle CMD.
∠A≅∠C(by CPCTC)\angle A \cong \angle C \quad (\text{by CPCTC})
Answer: Angle A is congruent to angle C, justified by CPCTC after proving the two triangles congruent by SAS.

Another Example

Problem: Triangle PQR is congruent to triangle XYZ, with P corresponding to X, Q corresponding to Y, and R corresponding to Z. If PQ = 7 cm, angle Q = 50°, and QR = 10 cm, find XY, angle Y, and YZ.
Step 1: State the congruence and identify corresponding parts. Since triangle PQR is congruent to triangle XYZ, corresponding sides and angles match by their position in the congruence statement.
△PQR≅△XYZ\triangle PQR \cong \triangle XYZ
Step 2: Apply CPCTC to find each corresponding measurement. PQ corresponds to XY, so XY = 7 cm.
PQ=XY=7 cmPQ = XY = 7 \text{ cm}
Step 3: Angle Q corresponds to angle Y, so angle Y = 50°.
∠Q=∠Y=50°\angle Q = \angle Y = 50°
Step 4: QR corresponds to YZ, so YZ = 10 cm.
QR=YZ=10 cmQR = YZ = 10 \text{ cm}
Answer: By CPCTC: XY = 7 cm, angle Y = 50°, and YZ = 10 cm.

Frequently Asked Questions

When do you use CPCTC in a proof?
You use CPCTC after you have already proven two triangles congruent using a method like SSS, SAS, ASA, AAS, or HL. CPCTC is never the first step — it is always a consequence of an established triangle congruence. Think of it as the payoff: you do the work of proving congruence, then CPCTC lets you extract the specific side or angle relationship you need.
Is CPCTC a postulate or a theorem?
CPCTC is a theorem (or a direct logical consequence of the definition of congruent triangles). It follows from the fact that congruent triangles are defined as triangles whose corresponding sides and corresponding angles are all congruent. Once congruence is established by any valid postulate or theorem, CPCTC applies automatically.

CPCTC vs. CPCFC

CPCTC applies to congruent triangles, while CPCFC (Corresponding Parts of Congruent Figures are Congruent) extends the same idea to any congruent figures, not just triangles. CPCTC is the most commonly used version because triangle congruence proofs are fundamental in geometry courses. CPCFC is the more general principle.

Why It Matters

CPCTC is one of the most frequently used justifications in geometry proofs. It bridges the gap between proving triangles congruent and proving individual sides or angles congruent — which is often the actual goal of a proof. Many real-world and advanced geometry problems reduce to showing two triangles are congruent, then using CPCTC to draw the final conclusion.

Common Mistakes

Mistake: Using CPCTC before proving the triangles are congruent.
Correction: CPCTC can only be applied after you have established triangle congruence through SSS, SAS, ASA, AAS, or HL. It is a result, not a starting point.
Mistake: Misidentifying which parts correspond to each other.
Correction: The correspondence is determined by the order of vertices in the congruence statement. In the statement triangle ABC ≅ triangle DEF, vertex A corresponds to D, B to E, and C to F. Side AB corresponds to DE, angle B corresponds to angle E, and so on. Always read the congruence statement carefully.

Related Terms

  • Corresponding — Parts matched by position in congruent figures
  • Congruent — Same shape and size — the key prerequisite
  • Triangle — The geometric figure CPCTC applies to
  • Theorem — CPCTC is classified as a theorem
  • CPCFC — Generalization of CPCTC to all congruent figures
  • SAS — A triangle congruence postulate used before CPCTC
  • SSS — A triangle congruence postulate used before CPCTC
  • ASA — A triangle congruence postulate used before CPCTC