In triangle ABC the sides AB and AC are equal. The point D lies on the side BC and the point E lies on the side AC, chosen so that AD = AE. If ∠BAD = 30°, what is the measure of ∠CDE?

Write the equal base angles as β, so ∠BAC = 180° − 2β and ∠DAC = 180° − 2β − 30° = 150° − 2β. In triangle ADC the angle at D is ∠ADC = 180° − β − (150° − 2β) = 30° + β. Since AD = AE, triangle ADE is isosceles with apex angle ∠DAE = ∠DAC = 150° − 2β, so its base angles give ∠ADE = (180° − 150° + 2β)/2 = 15° + β. Subtracting, ∠CDE = ∠ADC − ∠ADE = (30° + β) − (15° + β) = 15°. The β cancels, so the answer is always half of ∠BAD, whatever the apex angle. The trap 30° copies ∠BAD across, 60° doubles it, and 75° treats triangle ADE as having 30° at its apex.