Question

Prove that a diagonal of a rhombus bisects the angles at the vertices.

Answer 1

Given:

Let ABCD be a rhombus so AB = BC = CD = DA. 

Let AC be one diagonal.

To Prove:

Diagonal AC bisects ∠A and ∠C i.e., ∠BAC = ∠CAD and ∠BCA = ∠DCA.

Proof [Step-wise]:

1. Draw diagonal AC.

Consider triangles ΔABC and ΔCDA.

2. In a rhombus, all four sides are equal.

So, AB = CD and BC = DA.

Also, AC = AC   ...(Common side)

3. Therefore, AB = CD, BC = DA and AC = AC.

By SSS, ΔABC ≅ ΔCDA

4. From congruence, corresponding angles are equal.

So, ∠BAC = ∠CAD. 

Hence, AC bisects ∠A.

5. Similarly, the congruence gives ∠BCA = ∠DCA.

So, AC bisects ∠C.

Diagonal AC bisects the angles at vertices A and C of rhombus ABCD.

Thus, each diagonal of a rhombus bisects the vertex angles.