Question

In a rhombus ABCD, what needs to be proven to demonstrate that DB bisects both ABC and ADC?

a) Prove Theorem 4-22
b) Show ADB congruent CDB
c) Prove that AC is perpendicular to DB
d) Prove that AB is parallel to CD

Answer 1

The correct method to prove that DB bisects the angles ABC and ADC in a rhombus ABCD is by showing that triangles ADB and CDB are congruent (option b). This confirmation of congruence demonstrates that DB bisects those angles because the properties of a rhombus ensure that its diagonals bisect the angles at the vertices.

Step-by-step explanation:

To demonstrate that DB bisects angles ABC and ADC in a rhombus ABCD, one needs to show that triangle ADB is congruent to triangle CDB (option b). According to the properties of a rhombus, the diagonals bisect the angles of the rhombus. Consequently, if one can show that triangle ADB is congruent to triangle CDB, it directly implies that DB bisects the angles at corners B and D of the rhombus.

By proving congruence of these triangles, for instance, via Side-Angle-Side (SAS) or Angle-Side-Angle (ASA) postulate, we would establish that DB divides the angles into two equal parts, thus confirming that DB bisects the angles ABC and ADC. This approach utilizes the fact that diagonals in a rhombus bisect each other at right angles and, therefore, form congruent triangles with the sides of the rhombus as we compare them pairwise.

The correct option to prove that DB bisects both angles ABC and ADC in a rhombus ABCD is option b): Show that triangles ADB and CDB are congruent.