It is given that AB and AC are the equal sides of an isosceles ΔABC, in which…

Concept

The three interior angles of any triangle sum to 180°, and angles lying along a straight line also sum to 180°. In an isosceles triangle the base angles (opposite the two equal sides) are equal, and every interior angle of an equilateral triangle equals 60°.

Step-by-step application

1. Since AB = AC in ΔABC, the base angles are equal: ∠ABC = ∠ACB. Call this common base angle β. Since ΔDEF is equilateral, every angle of ΔDEF equals 60°.

2. Points A, D, B are collinear because D lies on side AB, so the angles at D along that line satisfy ∠ADE + ∠EDF + ∠FDB = 180°. Substituting ∠ADE = b and ∠EDF = 60° gives ∠FDB = 120° − b.

3. Points B, F, C are collinear because F lies on side BC, so the angles at F along that line satisfy ∠BFD + ∠DFE + ∠EFC = 180°. Substituting ∠BFD = a and ∠DFE = 60° gives ∠EFC = 120° − a.

4. In ΔBDF, the interior angles sum to 180°: ∠DBF + ∠BFD + ∠FDB = 180°. Substituting β, a and (120° − b) gives β = 60° + b − a.

5. In ΔCEF, the interior angles sum to 180°: ∠ECF + ∠CEF + ∠EFC = 180°. Substituting β, c and (120° − a) gives β = 60° + a − c.

6. Both expressions equal the same base angle β, so 60° + b − a = 60° + a − c. Simplifying gives b + c = 2a, i.e. a = (b + c) / 2.

Cross-check

Taking b = 50° and c = 30° as a numeric check, the relation gives a = (50° + 30°) / 2 = 40°. Then β = 60° + b − a = 70° from the B side, and β = 60° + a − c = 70° from the C side — the two independently computed values of β agree, confirming the relation.

So the relationship connecting the three marked angles is a = (b + c) / 2.