Final answer:
Identifying a quadrilateral as a rhombus comes from proving that the diagonals bisect each other and are perpendicular, which results in congruent triangles and equal sides. Furthermore, the diagonals in a rhombus bisect its angles due to the congruency of triangles formed between a diagonal and two sides of the rhombus.
Step-by-step explanation:
If the diagonals of a quadrilateral bisect each other and are perpendicular, then each pair of opposite triangles created by the diagonals are congruent by SAS (Side-Angle-Side). Since the diagonals bisect each other, we know that each half of the diagonals are equal. With the diagonals being perpendicular, we conclude that corresponding angles are right angles. All these congruent triangles imply that all four sides of the quadrilateral are equal, making it a rhombus.
In a rhombus, the diagonals bisect the angles. This can be deduced from the property that the diagonals bisect each other at a perpendicular angle, and that all sides of a rhombus are equal. This implies that the two triangles formed by one of the diagonals and two sides of the rhombus are congruent by the SSS (Side-Side-Side) postulate. Thus, the angles where the diagonal meets the sides are bisected, proving that in a rhombus, the diagonals bisect the angles.