In triangle ABC, angle C is 75 degrees, and the outside angle at apex A is 105 degrees. Prove that triangle ABC is isosceles.

1. The external angle of a triangle (∠O), according to its properties, is equal to the total value of two internal angles that are not adjacent to it:

∠О = ∠А + ∠С = 105 °.

∠А + 75 ° = 105 °.

∠А = 105 ° – 75 ° = 30 °.

2. Considering that the total value of all internal angles of the triangle is 180 °, we calculate the value of ∠В:

∠В = 180 ° – ∠А – ∠С = 180 ° – 30 ° – 75 ° = 75 °.

3. ∠С and ∠В adjacent to the BC side are equal. The ABC triangle is isosceles.