This worksheet provides a comprehensive guide to understanding and solving CPCTC (Corresponding Parts of Congruent Triangles are Congruent) proofs. We'll cover the fundamentals, work through example problems, and offer solutions to help solidify your understanding. Mastering CPCTC is crucial for success in geometry, so let's dive in!
Understanding CPCTC
Before tackling proofs, let's review the core concept: CPCTC. This theorem states that if two triangles are congruent (meaning their corresponding sides and angles are equal), then their corresponding parts (sides and angles) are also congruent. This seemingly simple statement is the foundation for many geometric proofs.
To use CPCTC effectively, you must first prove that two triangles are congruent. Remember the congruence postulates:
- SSS (Side-Side-Side): If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
- SAS (Side-Angle-Side): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
- ASA (Angle-Side-Angle): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
- AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
- HL (Hypotenuse-Leg - Right Triangles Only): If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
Example CPCTC Proofs
Let's work through some examples to illustrate the process. Each problem will be followed by a step-by-step solution.
Problem 1:
Given: ΔABC ≅ ΔDEF
Prove: ∠A ≅ ∠D
Solution:
- Statement: ΔABC ≅ ΔDEF Reason: Given
- Statement: ∠A ≅ ∠D Reason: CPCTC
Problem 2:
Given: AB ≅ DE, BC ≅ EF, AC ≅ DF
Prove: ∠B ≅ ∠E
Solution:
- Statement: AB ≅ DE, BC ≅ EF, AC ≅ DF Reason: Given
- Statement: ΔABC ≅ ΔDEF Reason: SSS
- Statement: ∠B ≅ ∠E Reason: CPCTC
Problem 3 (More Complex):
Given: ∠X ≅ ∠Z, XY ≅ ZW, ∠Y ≅ ∠W
Prove: XZ ≅ YW
Solution:
- Statement: ∠X ≅ ∠Z, XY ≅ ZW, ∠Y ≅ ∠W Reason: Given
- Statement: ΔXYZ ≅ ΔZWY Reason: ASA
- Statement: XZ ≅ YW Reason: CPCTC
Practice Problems
Now it's your turn! Try these problems and check your answers against the solutions provided below. Remember to clearly state your givens, your goal, and each step in your proof.
Problem 4: (Given Diagram showing two triangles with marked congruent parts) [Insert a diagram here showing two triangles with labeled congruent angles and sides] Prove: Side AB is congruent to Side DE.
Problem 5: (Given Diagram showing two triangles with marked congruent parts) [Insert a diagram here showing two triangles with labeled congruent angles and sides] Prove: Angle C is congruent to Angle F.
Answers to Practice Problems
Problem 4 Solution: (This section should contain a step-by-step solution to problem 4 based on the diagram provided. Remember to clearly state the postulates used.)
Problem 5 Solution: (This section should contain a step-by-step solution to problem 5 based on the diagram provided. Remember to clearly state the postulates used.)
This worksheet provided a foundation for understanding and applying CPCTC proofs. Remember, practice is key! The more proofs you work through, the more comfortable you'll become with identifying congruent triangles and utilizing CPCTC to solve for unknown parts. Good luck!
