In the figure BD = DC and ∠DBC = 25°. What is the value of ∠BAC?

Concept

Two inscribed angles that stand on the same chord from opposite arcs of a circle are supplementary (they add to 180°). Also, in a triangle the angles opposite two equal sides are themselves equal (isosceles base angles).

Application

1. In triangle BDC the sides BD and DC are equal, so the base angles opposite them are equal: ∠DCB = ∠DBC = 25°.

2. The angles of triangle BDC sum to 180°, so ∠BDC = 180° − 25° − 25° = 130°.

3. ∠BAC and ∠BDC both stand on the same chord BC, but A lies on the major arc and D on the minor arc, i.e. on opposite arcs. Such inscribed angles are supplementary: ∠BAC + ∠BDC = 180°.

4. Therefore ∠BAC = 180° − 130° = 50°.

Cross-check

Independent check: the equal chords BD and DC cut off equal arcs, each of arc-measure 2 × 25° = 50° (the inscribed angle is half its arc). Arc BDC (from B to C through D) = 50° + 50° = 100°, so the remaining arc BAC (from B to C through A) = 360° − 100° = 260°. The inscribed angle ∠BDC on arc BAC equals half of it (130°), and ∠BAC on arc BDC equals 100° ÷ 2 = 50°, confirming ∠BAC = 50°.