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CPCFC

CPCFC

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

 

 

See also

CPCTC

Worked Example

Problem: Quadrilateral ABCD is congruent to quadrilateral EFGH, where A↔E, B↔F, C↔G, and D↔H. If AB = 5 cm, BC = 8 cm, and angle B = 110°, find EF, FG, and angle F.
Step 1: State the congruence relationship. We are given that quadrilateral ABCD ≅ quadrilateral EFGH.
ABCD≅EFGHABCD \cong EFGH
Step 2: Identify the corresponding parts using the vertex correspondence A↔E, B↔F, C↔G, D↔H. Side AB corresponds to side EF, side BC corresponds to side FG, and angle B corresponds to angle F.
AB↔EF,BC↔FG,∠B↔∠FAB \leftrightarrow EF, \quad BC \leftrightarrow FG, \quad \angle B \leftrightarrow \angle F
Step 3: Apply CPCFC. Since the figures are congruent, all corresponding parts are congruent. Therefore each corresponding pair shares the same measurement.
EF=AB=5 cm,FG=BC=8 cm,∠F=∠B=110°EF = AB = 5 \text{ cm}, \quad FG = BC = 8 \text{ cm}, \quad \angle F = \angle B = 110°
Answer: By CPCFC, EF = 5 cm, FG = 8 cm, and angle F = 110°.

Another Example

Problem: Regular pentagon PQRST is congruent to regular pentagon VWXYZ. Diagonal PR = 9.4 cm. Find the length of diagonal VX.
Step 1: Establish the correspondence from the congruence statement: P↔V, Q↔W, R↔X, S↔Y, T↔Z.
PQRST≅VWXYZPQRST \cong VWXYZ
Step 2: Identify the corresponding diagonal. Diagonal PR connects the 1st and 3rd vertices of the first pentagon. The matching diagonal in the second pentagon connects V (1st) and X (3rd), so PR corresponds to VX.
PR↔VXPR \leftrightarrow VX
Step 3: Apply CPCFC. Corresponding parts of congruent figures are congruent, and this extends to diagonals — not just sides and angles.
VX=PR=9.4 cmVX = PR = 9.4 \text{ cm}
Answer: By CPCFC, diagonal VX = 9.4 cm.

Frequently Asked Questions

What is the difference between CPCFC and CPCTC?
CPCTC (Corresponding Parts of Congruent Triangles are Congruent) applies only to triangles. CPCFC is the broader version that applies to any congruent figures — triangles, quadrilaterals, pentagons, and so on. CPCTC is actually a special case of CPCFC.
Does CPCFC apply to diagonals and other internal segments, or just sides and angles?
CPCFC applies to all corresponding parts of the congruent figures, including sides, angles, diagonals, medians, and any other geometric element that can be defined by corresponding vertices. If the figures are congruent, every matching measurement is equal.

CPCFC vs. CPCTC

CPCTC is restricted to congruent triangles and is the version most commonly used in two-column proofs about triangles. CPCFC generalizes the same idea to all congruent figures — quadrilaterals, pentagons, hexagons, and beyond. Whenever you work with congruent polygons that have more than three sides, CPCFC is the appropriate principle to cite. In a triangle-specific proof, CPCTC and CPCFC are interchangeable.

Why It Matters

CPCFC lets you extract detailed information about individual parts of a figure once overall congruence has been established. In proofs involving quadrilaterals, regular polygons, or complex figures, it serves as the key reasoning step that connects a congruence statement to specific side lengths, angle measures, or diagonal lengths. Without it, you would need to prove each pair of parts congruent separately, making many proofs far longer and more difficult.

Common Mistakes

Mistake: Matching corresponding parts based on appearance rather than the congruence statement's vertex order.
Correction: Always use the order of letters in the congruence statement to determine which parts correspond. In ABCD ≅ EFGH, A corresponds to E, B to F, and so on. Drawing the figures in different orientations can make visual matching unreliable.
Mistake: Thinking CPCFC only applies to sides and angles, ignoring diagonals or other internal segments.
Correction: CPCFC covers every corresponding part defined by corresponding vertices. Diagonals, medians, and other segments within the figure are all included, as long as you correctly identify which vertices correspond.

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