ABCD is a rhombus in which ∠ ADB = 25°. Then, the value of (2 ∠ BAD - ∠ ABC)…

Concept — In a rhombus all four sides are equal in length. A diagonal therefore splits the rhombus into two isosceles triangles, and inside each of them the two angles standing at the ends of that diagonal are equal.

Concept — Opposite sides of a rhombus are parallel, so any two consecutive angles of the rhombus are supplementary: they add up to 180°.

Applying both facts to rhombus ABCD, in which the diagonal BD gives ∠ADB = 25°:

1. In triangle ABD the sides AB and AD are both sides of the rhombus, so AB = AD and the triangle is isosceles. Its base angles are therefore equal: ∠ABD = ∠ADB = 25°.

2. By the angle sum of triangle ABD, ∠BAD = 180° − (∠ABD + ∠ADB) = 180° − (25° + 25°) = 130°.

3. ∠BAD and ∠ABC are consecutive angles of the rhombus, so ∠ABC = 180° − ∠BAD = 180° − 130° = 50°.

4. Substituting both angles into the required expression: 2∠BAD − ∠ABC = 2 × 130° − 50° = 260° − 50° = 210°.

Cross-check — write the equal base angles as ∠ADB = ∠ABD = t. Then ∠BAD = 180° − 2t and ∠ABC = 180° − ∠BAD = 2t, so 2∠BAD − ∠ABC = 2(180° − 2t) − 2t = 360° − 6t. Putting t = 25° gives 360° − 150° = 210°, which agrees with the step-by-step working.

Hence 2∠BAD − ∠ABC = 210°.