1. **Problem Statement:** Explain the types of triangles by sides and angles, and important theorems related to triangles. 2. **Types of Triangles by Sides:** - Equilateral: All three sides are equal. - Isosceles: Two sides are equal. - Scalene: All sides are different. 3. **Types of Triangles by Angles:** - Acute-angled: All angles are less than 90°. - Right-angled: One angle is exactly 90°. - Obtuse-angled: One angle is greater than 90°. 4. **Important Theorems:** - Sum of interior angles: The sum of the three interior angles of any triangle is always $$180^\circ$$. - Exterior angle theorem: An exterior angle of a triangle equals the sum of the two opposite interior angles. - Pythagoras theorem (for right-angled triangles): $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse. - Triangle congruence criteria: - SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. - SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent. - ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent. - RHS (Right angle-Hypotenuse-Side): In right-angled triangles, if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of another triangle, the triangles are congruent. 5. These theorems help in solving problems related to triangle properties, proving congruence, and calculating unknown sides or angles.