Point D is the incenter of triangle BCA. If $\angle \mathrm{FDG}$ = $136°$, what is the measure of angle $\angle \mathrm{BCA}$?

• A. $44°$
• B. $136°$
• C. $92°$
• D. $68°$

Answer: (D)

Verified step by step guidance

1. Recall that point D is the incenter of triangle BCA, which means D is the intersection of the angle bisectors of the triangle. Therefore, angle BCA is bisected by the line segment CD.
2. Since D is the incenter, the angle at D related to vertex C is split into two equal parts. This means that the angle BCD is half of angle BCA.
3. Given that \( \angle FDG = 136^\circ \), analyze the geometric configuration around point D, especially how angles around D relate to the angles of triangle BCA. Use the fact that the sum of angles around point D is 360°.
4. Express \( \angle BCA \) in terms of \( \angle FDG \) by using the properties of the incenter and the angle bisector. Set up an equation relating these angles, considering that \( \angle FDG \) is an external angle formed by points F, D, and G.
5. Solve the equation for \( \angle BCA \) by isolating it on one side, using the relationships established in the previous steps, without calculating the final numeric value here.