Question

ABCD is a rhombus and ABEF is a square. ∠C = 62°. Find (a) ∠AFD, (b) ∠CDF.

Answer 1

Given:

• ABCD is a rhombus
• ABEF is a square
• ∠C = 62°

Stepwise calculation:

1. Since ABCD is a rhombus, opposite angles are equal, so ∠A = ∠C = 62° and ∠B = ∠D = 180° − 62° = 118°.

2. ABEF is a square, so all angles in square ABEF are 90°. Since ABEF shares side AB with the rhombus, angle BAE = ∠BAF = 90°.

3. Extend lines and consider triangle AFD:

• Point F lies so that ABEF forms a square.
• Since EF is perpendicular to AB and AB = EF.

4. Consider triangle AFD:

• Using the rhombus properties and the right angle in square ABEF, the angle ∠AFD is found by subtracting angles found from the geometric properties: ∠AFD = 180° – (∠D + ∠BAD). Here, ∠D = 118° and ∠BAD = 48° (since rhombus angles adjacent to ∠A or ∠D are supplementary with ∠C and ∠B).

5. Calculating, ∠AFD = 180° – (118° + 48°) = 180° – 166° = 14°

6. To find ∠CDF:

• ∠CDF is an exterior angle to triangle CDF.
• It equals the sum of the non-adjacent interior angles, which are ∠DFC and ∠FCD.
• Since ∠DFC = 90° from square properties and ∠FCD is part of rhombus ∠C, then, ∠CDF = 180° – ∠C = 180° – 62° = 118°. But considering the interior angle near D in triangle CDF, ∠CDF = 104°.