Prove that a triangle is isosceles if one of the angles is 40 degrees and one of the outer angles is 110 degrees.

1. Let us introduce the designation of the vertices of the triangle by the symbols A, B, C, and the angles by the symbol ∠.

∠ А = 40 °. Outside apex angle B = 110 °.

2. We calculate the value of the inner angle at the vertex B, adjacent to the outer one, equal to 110 °:

∠В = 180 ° – 110 ° = 70 °.

3. We calculate the value of ∠С, based on the fact that the total value of all internal angles of the triangle is 180 °:

∠С = 180 ° – ∠А and ∠В = 180 ° – 40 ° – 70 ° = 70 °.

4. Angles B and C at the base of BC are equal. Consequently, the triangle ABC is isosceles, as required.