7.5. In a certain triangle, one segment was drawn. It turned out that on the resulting drawing, all known types of triangles can be seen: equilateral, isosceles, scalene, right, acute, and obtuse.

a) Provide an example of such a triangle and the segment drawn in it.

b) Determine what values the angles of such a triangle can take. Provide all possible answers and prove that there are no others.
Solution: Let's immediately solve part b). After drawing one segment in the triangle, either 2 or 3 triangles will be formed. The 3 triangles case only occurs if the segment connects a vertex to a point on the opposite side. Since we need a right, acute, and obtuse triangle, this last case must be the one that applies. The drawn segment cannot be an altitude (as this would result in 2 right triangles). This leaves 2 options. 1) The original triangle was obtuse, and a segment was drawn from the vertex of the obtuse angle, dividing the obtuse angle into a right angle and an acute angle. However, this would not result in an acute triangle. 2) The original triangle was a right triangle. In this case, to form the required triangles, one of its angles must be $60^{\circ}$, and the drawn segment must be the median from the right angle. This situation satisfies our conditions.

Answer: b) $90^{\circ}, 30^{\circ}$ and $60^{\circ}$.

Recommendations for checking:

| included in the work | points |
| :--- | :--- |
| Correct example of a triangle and correct proof that triangles with other angles do not exist | 7 points |
| Not justified that the drawn segment connects a vertex to a point on the opposite side | points not deducted |
| Correctly completed two out of three points (see criteria for 2 points) | 4 points |
| Correctly completed one of the points: 1) provided a correct example of a triangle and showed which segment was drawn; 2) justified that the original triangle cannot be acute; 3) justified that the original triangle cannot be obtuse | point |
| Answer without justification and/or incorrect answer |  |