In triangle ABC, angle B is 23 degrees, angle C is 41 degrees, AD is the bisector, E is a point on AB such that

In triangle ABC, angle B is 23 degrees, angle C is 41 degrees, AD is the bisector, E is a point on AB such that AE = AC. Find angle BDE.

Consider triangles AED and ACD, in them:
AD – common side;
AE = AC (by condition);
∠ EAD = ∠ CAD (AD – bisector).
We conclude that the triangles are equal. It follows from the equality of the triangles that:
∠ AED = ∠ ACD = 41 °.
Find the corner BED adjacent to it:
∠ BED = 180 ° – ∠ AED = 180 ° – 41 ° = 139 °.
In the triangle BED, we find the angle BDE we need, knowing the other two angles:
∠ BDE = 180 ° – (∠ B + ∠ BED) = 180 ° – (23 ° + 139 °) = 180 ° – 162 ° = 18 °.
Answer: The degree measure of the BDE angle is 18 °.