1. **Problem Statement:** We need to construct a pentagon with exactly one right angle (90 degrees) and exactly one acute angle (less than 90 degrees). 2. **Understanding Angles in a Pentagon:** The sum of interior angles in any pentagon is given by the formula: $$\text{Sum of interior angles} = (5-2) \times 180 = 540 \text{ degrees}$$ 3. **Key Rules:** - Exactly one angle must be $90^\circ$ (right angle). - Exactly one angle must be less than $90^\circ$ (acute angle). - The other three angles must be neither right nor acute, so they must be obtuse (greater than $90^\circ$) or possibly reflex (greater than $180^\circ$) but typically obtuse for a simple pentagon. 4. **Constructing the Angles:** Let the angles be $A_1, A_2, A_3, A_4, A_5$ with: - $A_1 = 90^\circ$ (right angle) - $A_2 < 90^\circ$ (acute angle) - $A_3, A_4, A_5 > 90^\circ$ (obtuse angles) 5. **Example Values:** Choose $A_2 = 80^\circ$ (acute), then the sum of the other three angles is: $$A_3 + A_4 + A_5 = 540 - 90 - 80 = 370^\circ$$ 6. **Distributing the Obtuse Angles:** We can pick values such as: $$A_3 = 120^\circ, \quad A_4 = 125^\circ, \quad A_5 = 125^\circ$$ These add up to $370^\circ$ and are all obtuse. 7. **Summary:** - The pentagon has angles $90^\circ, 80^\circ, 120^\circ, 125^\circ, 125^\circ$. - This satisfies exactly one right angle and exactly one acute angle. 8. **Shape Construction:** - Start by drawing a vertex with a right angle. - Adjacent to it, create a vertex with an acute angle. - The remaining vertices have obtuse angles. - Connect the vertices to form a closed pentagon. This method ensures the pentagon meets the angle requirements exactly.