Question

Side angle side - SAS

Answer 1

In geometry, the Side-Angle-Side (SAS) theorem is a fundamental principle used to prove the congruence of two triangles. According to the SAS criterion, if two sides of one triangle are equal in length to two sides of another triangle, and the angles included between those sides are equal, then the two triangles are congruent.

Here's a breakdown of the SAS congruence criterion:

- **Two sides (S)**: Triangle \(ABC\) has side \(AB\) and side \(AC\), and triangle \(DEF\) has side \(DE\) and side \(DF\).
- **Included angle (A)**: The angle between sides \(AB\) and \(AC\) in triangle \(ABC\) is equal to the angle between sides \(DE\) and \(DF\) in triangle \(DEF\).

### Notation:
If:
- \(AB = DE\)
- \(AC = DF\)
- \(\angle BAC = \angle EDF\)

Then:
\[
\triangle ABC \cong \triangle DEF
\]

### Applications:
The SAS criterion is widely used in various geometric proofs and constructions, including:
- Proving that two triangles are congruent.
- Establishing properties of geometric figures.
- Solving problems involving triangle measurements.

### Example:
If you have triangle \(XYZ\) where \(XY = 5\), \(XZ = 7\), and \(\angle YXZ = 60^\circ\), and triangle \(PQR\) where \(PQ = 5\), \(PR = 7\), and \(\angle QPR = 60^\circ\), then according to SAS, \(XYZ \cong PQR\).

This theorem is vital for further studies in geometry and is foundational for other theorems and principles.