1. Let's start by stating the problem: We want to understand the theorems related to isosceles and equilateral triangles. 2. **Isosceles Triangle Theorem:** This theorem states that in an isosceles triangle, the angles opposite the equal sides are also equal. 3. To explain, an isosceles triangle has at least two sides of equal length. If we label the triangle as $\triangle ABC$ with $AB = AC$, then the angles opposite these sides, $\angle C$ and $\angle B$, are equal. 4. **Formula/Rule:** If $AB = AC$, then $\angle B = \angle C$. 5. **Equilateral Triangle Theorem:** This theorem states that all sides and all angles in an equilateral triangle are equal. 6. In an equilateral triangle, each side has the same length, and each angle measures exactly $60^\circ$. 7. **Formula/Rule:** If $AB = BC = CA$, then $\angle A = \angle B = \angle C = 60^\circ$. 8. These theorems help us understand the properties of these special triangles and are fundamental in geometry. 9. To summarize: - Isosceles triangle: two equal sides, two equal opposite angles. - Equilateral triangle: three equal sides, three equal angles of $60^\circ$ each. This explanation covers the key points of the isosceles and equilateral triangle theorems.