Euclid's Proposition 4: If two sides of two triangles are equal and the contained angle is equal, the two triangles are equal.

Euclid's statement of proposition 4: If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.

Construction

  • Construct any triangle. Mark the vertices of the triangle A, B, and C.
  • Construct another triangle with A'B' = AB, A'C' = AC and the angle BAC equal to BAC.

Proof

  1. Place the triangle ABC on top of the triangle A'B'C' with the point A on top of point A', and the line segment AB on top of A'B'.
  2. The point B will be on top of B', because BC is the same length as B'C'.
  3. The line AC will be on top of A'C' because the angle BAC equals the angle B'A'C'.
  4. This means that the point C is on top of C', because AC equals A'C'.
  5. Since B is on top of B' and C is on top of C', the line segment BC is on top of B'C', so BC is equal to B'C'.
  6. This means that the entire triangle ABC equals the entire triangle A'B'C'.
  7. QED.

More Information

Euclid's Elements

http://babbage.clarku.edu/~djoyce/java/elements/toc.html