5. On a standard graph paper, an angle is drawn (see figure). Find its measure without using measuring instruments. Justify your answer.
Answer: $45^{\circ}$.

Solution. Connect the two "extreme" points with a segment (as shown in the figure). The resulting triangle is isosceles because two of its sides, $A B$ and $B C$, are diagonals of three-cell rectangles. Diagonal $A B$ divides the angle of the rectangle at vertex $B$ into two angles that complement each other to a right angle. Triangles $A D B$ and $C E B$ are equal by two legs, so their corresponding angles are equal. Therefore, angle $C B E$ complements angle $A B E$ to a right angle. Thus, triangle $A B C$ is 

![](https://cdn.mathpix.com/cropped/2024_05_06_9a263354b38dd7e4b9ffg-3.jpg?height=465&width=511&top_left_y=327&top_left_x=1412)
isosceles and right-angled. Its angles $A$ and $C$ at the base $A C$ are equal by the property of isosceles triangles and have a measure of $45^{\circ}$ by the theorem on the sum of the angles of a triangle.

Comment. If we connect point $B$ to the midpoint of $A C$, we also get an isosceles right-angled triangle. The reasoning is similar.

## Grading Criteria.

- A correct and justified solution - 7 points.
- Generally correct reasoning with minor errors that do not fundamentally affect the solution, and a correct answer - $\mathbf{5}$ points.
- Additional constructions and markings on the diagram that clearly show the solution process, a correct answer, but no reasoning provided - 3 points.
- A correct answer without justification or with incorrect justification - $\mathbf{0}$ points.