1. **Problem Statement:** Define and describe right-angled triangles and equilateral triangles. 2. **Right-angled Triangle:** A right-angled triangle is a triangle in which one of the angles is exactly $90^\circ$. 3. **Properties:** The side opposite the right angle is called the hypotenuse, and it is the longest side. 4. **Formula:** The Pythagorean theorem applies: $$a^2 + b^2 = c^2$$ where $a$ and $b$ are the legs and $c$ is the hypotenuse. 5. **Equilateral Triangle:** An equilateral triangle has all three sides equal in length. 6. **Properties:** All three internal angles are equal, each measuring $60^\circ$. 7. **Formula:** If each side is $s$, the perimeter is $3s$ and the area is $$\frac{\sqrt{3}}{4}s^2$$. 8. **Summary:** Right-angled triangles have one $90^\circ$ angle and satisfy the Pythagorean theorem. Equilateral triangles have all sides and angles equal, each angle being $60^\circ$.