CPCTC
The CPCTC theorem states that when two triangles are congruent, their corresponding parts are equal. The CPCTC is an abbreviation used for 'corresponding parts of congruent triangles are congruent'.
| 1. | What is CPCTC? |
| 2. | CPCTC Triangle Congruence |
| 3. | CPCTC Proof |
| 4. | FAQs on CPCTC |
What is CPCTC?
The abbreviation CPCTC is for Corresponding Parts of Congruent Triangles are Congruent. The CPCTC theorem states that when two triangles are congruent, then every corresponding part of one triangle is congruent to the other. This means, when two or more triangles are congruent then their corresponding sides and angles are also congruent or equal in measurements. Let us understand the meaning of congruent triangles and corresponding parts in detail.
Congruent Triangles
Two triangles are said to be congruent if they have exactly the same size and the same shape. Two congruent triangles have three equal sides and equal angles with respect to each other.
Corresponding Parts
Corresponding sides mean the three sides in one triangle are in the same position or spot as in the other triangle. Corresponding angles mean the three angles in one triangle are in the same position or spot as in the other triangle.
In the given figure, △ABC ≅ △LMN. It means that the three pairs of sides and three pairs of angles of △ABC are equal to the three pairs of corresponding sides and three pairs of corresponding angles of △LMN.
In these two triangles ABC and LMN, let us identify the 6 parts: i.e. the three corresponding sides and the three corresponding angles. AB corresponds to LM, BC corresponds to MN, AC corresponds to LN. ∠A corresponds to ∠L, ∠B corresponds to ∠M, ∠C corresponds to ∠N. And if △ABC ≅ △LMN, then as per the CPCTC theorem, the corresponding sides and angles are equal, i.e. AB = LM, BC = MN, AC = LN, and ∠ A = ∠L, ∠B = ∠M, ∠C = ∠N.
CPCTC Triangle Congruence
CPCTC states that if two triangles are congruent by any criterion, then all the corresponding sides and angles are equal. Here, we are discussing 5 congruence criteria in triangles.
| Criterion | Explanation | CPCTC |
|---|---|---|
| SSS | All the 3 corresponding sides are equal | All the corresponding angles are also equal |
| AAS | 2 corresponding angles and the non included side are equal | The other corresponding angles and the other 2 corresponding sides are also equal |
| SAS | 2 corresponding sides and the included angle are equal | The other corresponding sides and the other 2 corresponding angles are also equal |
| ASA | 2 corresponding angles and the included sides are equal | The other corresponding angles and the other 2 corresponding sides are also equal |
| RHS / HL | The hypotenuse and one leg of one triangle are equal to the corresponding hypotenuse and a leg of the other | The other corresponding legs and the other two corresponding angles are equal |
CPCTC Proof
To prove CPCTC, first, we need to prove that the two triangles are congruent with the help of any one of the triangle congruence criteria. For example, Consider triangles ABC and CDE in which BC = CD and AC = CD are given.
Follow the points to prove CPCTC
- BC = CD and AC = CD (Given)
- ∠ACB = ∠EDC (Vertically opposite angles are equal)
- Thus, △ABC ≅ △EDC; By SAS (side-angle-side) criterion
- Now the two triangles are congruent, therefore, using CPCTC, AB = DE, ∠ABC = ∠EDC and ∠BAC = ∠DEC.
Important Notes
Given below are some important notes related to CPCTC. Have a look!
- Look for the congruent triangles keeping CPCTC in mind.
- Before using CPCTC, show that the two triangles are congruent.
Related Articles on CPCTC
Check out these interesting articles to know more about CPCTC and its related topics.
CPCTC Examples
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Example 1: Observe the figure given below and find the length of LM using the CPCTC theorem, if it is given that △ EFG ≅ △LMN.
Solution: Given that △ EFG ≅ △LMN. So, we can apply the ASA congruence rule to it which states that if two corresponding angles and the included side are equal in two triangles, then the triangles will be congruent. Here, two angles are given which are 30 degrees and 102 degrees such that ∠EFG = ∠LMN and ∠FEG = ∠MLN. So, by applying the CPCTC theorem we can identify that FE and ML are the corresponding sides of two congruent triangles △ EFG and △LMN. Therefore, FE = ML. Hence, the length of side LM is 3 units.
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Example 2: Observe the figure given below in which PR = RS and QR is perpendicular to PS. Find y using the CPCTC theorem.
Solution: First let us prove that △PQR ≅ △SQR,
- PR = RS (given)
- QR = QR (common side)
- ∠QRP = ∠QRS (as QR is perpendicular to PS)
- Therefore, △PQR ≅ △SQR (SAS criterion)
- PQ = QS ( By CPCTC)
Now as PQ = QS
Therefore, 4y = 28
Answer ⇒ y = 7 units
FAQs on CPCTC
Is CPCTC a Theorem?
Yes. CPCTC is a theorem that says corresponding parts of congruent triangles are congruent. Corresponding means angles and sides that are in the same respective position in the two triangles.
What is CPCTC for Similar Triangles?
CPCTC for similar triangles is not true. So, we cannot apply the CPCTC theorem for similar triangles. Corresponding angles of the two similar triangles are equal, whereas, corresponding sides of the triangles are not equal, but proportional.
How do you Prove CPCTC?
After showing the proposed triangles are congruent, we can immediately say that the corresponding parts of congruent triangles are congruent. It can be justified by superimposing triangles on each other and then by observing the corresponding angles and side lengths.
What does CPCTC Stand for?
CPCTC stands for corresponding parts of congruent triangles are congruent. Sometimes, it is also called CPCT which means corresponding parts of congruent triangles.
What is an Example of CPCTC?
The theorem CPCTC tells that when two triangles are congruent then their corresponding sides and angles are also said to be congruent. For example, triangle ABC and triangle PQR are congruent triangles therefore according to the theorem the sides AB = PQ, BC = QR, and CA = RP. Also ∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R.
How do you Prove CPCTC Using SSS Criterion?
In SSS triangle congruence all the three corresponding sides are equal. In other words, the two triangles are said to be congruent if all corresponding sides of one triangle are equal to the sides of another triangle. Thus, when two triangles are congruent then according to CPCTC all the corresponding angles are also equal.
How do you Prove CPCTC Using SAS Criterion?
In SAS triangle congruence the two corresponding sides and the included angle are equal. In other words, the two triangles are said to be congruent if two corresponding sides and the included angle are equal. Thus, when two triangles are congruent then according to CPCTC the other corresponding side and the other two corresponding angles are also equal.