Question

Prove that the diagonals of a rhombus are also angle bisectors. (Theorem 1)

Answer 1

A rhombus has four congruent sides. Without loss of generality, we denote the rhombus by the consecutive points PQRS.

1. Draw diagonal PR.
2. Consider triangle PQR. We conclude that PQ=RQ because a rhombus has four congruent sides. Triangle PRQ is isosceles => angles QPR and QRP are congruent.
3. Similarly, PS=RS. Triangle PSR is isosceles.
4. PR (diagonal) is common side, so triangles PQR is congruent to PSR (SSS).
5. Therefore angles QRP=QPR=SPR=SRP.
6. Since angles QPR=SPR, and SPR=QPR, we conclude that the diagonal PR is an angle bisector.

Finally, using similar arguments, we can show that diagonal QS bisects angles at Q and S.