Understanding Triangle Congruence
When two triangles are congruent, it means they have the same shape and size. If triangle XYZ is congruent to triangle QPR, it follows certain properties.
Key Properties of Congruent Triangles:
- Equal Corresponding Sides:
- The lengths of sides XY, YZ, and ZX are equal to the lengths of sides QR, RP, and PQ, respectively.
- Equal Corresponding Angles:
- The angles ∠X, ∠Y, and ∠Z are equal to angles ∠Q, ∠R, and ∠P, respectively.
Verification of Statements
To assess whether specific statements about triangles XYZ and QPR are correct, consider the following:
- Statement 1: "If XY = QR, then YZ = RP."
- This is incorrect without additional information. All sides must be equal for congruence.
- Statement 2: "If ∠X = ∠Q, then ∠Y = ∠R."
- This is also incorrect by itself. All angles must be equal to confirm triangle congruence.
- Statement 3: "If all sides of XYZ are equal to all sides of QPR, then they are congruent."
- This is correct. This is known as the Side-Side-Side (SSS) criterion for congruence.
Conclusion
To determine the correctness of statements regarding congruence, ensure that all corresponding sides and angles match. Remember that congruence can be proved through various criteria, including SSS, SAS (Side-Angle-Side), and ASA (Angle-Side-Angle).