In the ICSE Class 9 Mathematics curriculum, the topic of triangles is a fundamental part of geometry. Here is a summary of the key concepts covered:

1. **Types of Triangles**: Triangles are classified based on their sides and angles. They can be classified as equilateral (all sides are equal), isosceles (two sides are equal), or scalene (no sides are equal). Based on angles, triangles can be acute (all angles are less than 90 degrees), obtuse (one angle is greater than 90 degrees), or right-angled (one angle is exactly 90 degrees).

2. **Properties of Triangles**: Various properties of triangles are discussed, including the sum of interior angles (which is always 180 degrees), the exterior angle property (an exterior angle is equal to the sum of the two interior opposite angles), and the Pythagorean theorem (in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides).

3. **Congruence of Triangles**: Two triangles are congruent if their corresponding sides and angles are equal. Various criteria for triangle congruence are discussed, including Side-Angle-Side (SAS), Side-Side-Side (SSS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS).

4. **Similarity of Triangles**: Two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion. The concept of similarity is important for solving problems involving indirect measurements and scale drawings.

5. **Theorems Related to Similarity**: Theorems such as the Basic Proportionality Theorem (Thales theorem) and the Angle Bisector Theorem are discussed in the context of similar triangles.

6. **Application Problems**: Real-life problems involving triangles, such as finding the height of a flagpole, the distance between two inaccessible points, or the area of a triangle, are solved using the concepts of triangle properties and similarity.

Understanding these concepts is crucial for further studies in geometry and trigonometry.