Geometry Question:
Prove the symmetric Property for congruence of triangles.
Given: ∆ABC≅ ∆DEF
Prove: ∆DEF≅ ∆ABC
The radius of the base of a cylinder is 4x cm is 27 cm and its height is 33 cm. Find the surface area of the cylinder in terms of 3.14
To prove the symmetric property for congruence of triangles, we need to show that if ∆ABC is congruent to ∆DEF, then ∆DEF is also congruent to ∆ABC.
To prove congruence between two triangles, we often use a combination of congruence postulates, such as side-angle-side (SAS), angle-side-angle (ASA), or side-side-side (SSS).
Here is one possible proof using the SAS congruence postulate:
Given: ∆ABC ≅ ∆DEF (Given that triangle ABC is congruent to triangle DEF)
To prove: ∆DEF ≅ ∆ABC
Proof:
1. ∆ABC ≅ ∆DEF (Given)
2. AB ≅ DE (By CPCTC: Corresponding Parts of Congruent Triangles are Congruent)
3. BC ≅ EF (By CPCTC)
4. ∠B ≅ ∠E (By CPCTC)
5. ∠C ≅ ∠F (By CPCTC)
6. ∆DEF ≅ ∆ABC (By SAS (Side-Angle-Side) Congruence Postulate: 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)
Therefore, we have proven that if ∆ABC is congruent to ∆DEF, then ∆DEF is also congruent to ∆ABC, thus confirming the symmetric property of congruence of triangles.
Note: CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent." It is a commonly used theorem in triangle congruence proofs.