Problem 19 - Entrance Test

1. **Analyze triangle ABC:** Given that triangle ABC is isosceles with AB = AC, and angle A = 40 degrees. The base angles are equal: angle B = angle C. Sum of angles in a triangle is 180 degrees: Angle B + Angle C + Angle A = 180 degrees 2 * Angle C + 40 degrees = 180 degrees 2 * Angle C = 140 degrees Angle C = 70 degrees. So, Angle ABC = Angle C = 70 degrees. 2. **Analyze triangle BCD:** Point D is on AC such that BD = BC. This means triangle BCD is an isosceles triangle. The base angles are equal: angle BDC = angle BCD. We know angle BCD is the same as angle C of the larger triangle, so angle BCD = 70 degrees. Therefore, angle BDC = 70 degrees. Now, find angle CBD in triangle BCD: Angle CBD + Angle BCD + Angle BDC = 180 degrees Angle CBD + 70 degrees + 70 degrees = 180 degrees Angle CBD + 140 degrees = 180 degrees Angle CBD = 40 degrees. 3. **Find angle ABD:** We want to find angle ABD. We know that angle ABC is composed of angle ABD and angle CBD. Angle ABC = Angle ABD + Angle CBD. We found Angle ABC = 70 degrees and Angle CBD = 40 degrees. 70 degrees = Angle ABD + 40 degrees Angle ABD = 70 degrees - 40 degrees Angle ABD = 30 degrees.