Introduction to Triangles

Definition: A triangle is a closed figure made up of three line segments.

The three line segments are called the sides of the triangle.

Parts of a triangle:

• Vertices — the three corner points where two sides meet. We label them with capital letters like A, B, C.
• Sides — the three line segments joining the vertices. In triangle ABC, the sides are AB, BC, and CA.
• Angles — the three angles formed at each vertex. In triangle ABC, the angles are ∠A, ∠B, and ∠C.

Naming a triangle:

• A triangle is named by its three vertices.
• Triangle with vertices A, B, C is written as △ABC.
• It can also be called △BCA or △CAB — the order does not matter as long as all three vertices are included.

Understanding the parts of a triangle step by step:

1. Mark three points A, B, and C on paper. These points should not be on the same line.
2. Join A to B with a straight line. This gives side AB.
3. Join B to C with a straight line. This gives side BC.
4. Join C to A with a straight line. This gives side CA.
5. You now have a closed figure with three sides — this is △ABC.

Regions of a triangle:

• The space inside the triangle is called the interior of the triangle.
• The space outside the triangle is called the exterior of the triangle.
• The three sides themselves form the boundary of the triangle.

Opposite sides and opposite angles:

• In △ABC, the side opposite to vertex A is side BC.
• The side opposite to vertex B is side CA.
• The side opposite to vertex C is side AB.
• Similarly, the angle opposite to side BC is ∠A.

Triangles can be grouped in two ways:

By sides:

• Equilateral triangle — all three sides are equal. All three angles are also equal (each is 60°).
• Isosceles triangle — two sides are equal. The two angles opposite the equal sides are also equal.
• Scalene triangle — all three sides are different. All three angles are also different.

By angles:

• Acute-angled triangle — all three angles are less than 90°.
• Right-angled triangle — one angle is exactly 90°.
• Obtuse-angled triangle — one angle is greater than 90°.

Important: A triangle can be described using both groupings. For example, a triangle can be an isosceles right-angled triangle (two equal sides and one 90° angle).

Triangles in real life:

• Construction — Roof trusses and bridges use triangles because they are the strongest shape. A triangle does not change shape when force is applied, unlike a rectangle which can collapse into a parallelogram.
• Traffic signs — Warning signs are shaped like triangles (yield sign, caution sign).
• Art and design — Triangles are used in rangoli, tiling, and patterns.
• Navigation — Sailors and pilots use triangles to find distances and directions (called triangulation).
• Sports — Snooker balls are arranged in a triangle. The penalty area in football has triangular corners.
• Nature — Mountains, pyramids, and certain leaf shapes are triangular.