Question

In a rhombus ABCD, the diagonals AC and BD intersect at O. BD is produced to E. ∠ADO : ∠ADE = 1 : 4. Find the angles of ΔAOB.

Answer 1

Given:

• Rhombus ABCD with diagonals AC and BD that intersect at O.
• The ratio of the angles ∠ADO to ∠ADE is 1 : 4.

Step-by-step:

Angle Relationship:

In a rhombus, the diagonals bisect each other at right angles, meaning ∠AOB = 90°.

Angle Proportions:

Since ∠ADO and ∠ADE are in the ratio of 1 : 4, let the smaller angle ∠ADO be x and the larger angle ∠ADE be 4x.

Using the Total Angle at D:

The sum of angles at point D must be 180° (since they are on a straight line).

Therefore, ∠ADO + ∠ADE = 180°.

Substituting the ratio:

x + 4x = 180°

5x = 180°

x = 36°

Calculating the Angles:

So, ∠ADO = 36° and ∠ADE = 144°.

Triangle AOB:

Since the diagonals intersect at right angles, ∠AOB = 90°.

Final Angles:

In triangle AOB:

∠AOB = 90°

∠OAB = 36°

∠OBA = 54° (since the angles of a triangle sum up to 180° and 180° – 90° – 36° = 54°).

Thus, the angles of triangle AOB are 36°, 54° and 90°.