Question

in the figure, triangle BDE has the following angle measure:angle B=40 degrees angle D=60 degress and angle E=80. investing a pattern between triangle BDE and ABC and make a conjecture about the measure of angle C

Answer 1

I don’t have the figure, so I’ll need one quick clarification: where are D and E placed relative to triangle ABC (for example, are D and E points on sides AC and AB respectively, with B a common vertex of the two triangles)? Could you upload or describe the diagram?

A likely/common configuration and conjecture
- If triangle BDE is the small triangle inside ABC with the same vertex B, D lying on AC, and E lying on AB, then the natural correspondence is
B (big) ↔ B (small), C ↔ D, A ↔ E.
- Under that correspondence the angles match: angle B = 40°, angle A = angle E = 80°, and angle C = angle D = 60°. So the conjecture is
angle C = 60°.

Why: triangle BDE would be similar to triangle ABC (same angle at B, and the vertices D and E lie on the sides that correspond to C and A), so corresponding angles are equal; angle C equals angle D = 60°. The three angles 40° + 80° + 60° sum to 180°, as required.

If your diagram is different, upload or describe it and I’ll adapt the reasoning to the actual placement of D and E.