1. Introduction to Parts of a Triangle
Once we know what a triangle is, the next step is to understand its basic parts. Every triangle has certain fixed components—vertices, sides, angles, and specific regions inside and outside. These parts form the foundation of all triangle-related concepts in geometry.
In this topic, we study each part clearly and simply, like personal class notes.
2. Vertices of a Triangle
The corners of a triangle are called its vertices (singular: vertex). If a triangle is named \( \triangle ABC \), then its three vertices are the points \( A \), \( B \), and \( C \).
Every vertex is the point where two sides of the triangle meet.
3. Sides of a Triangle
The sides of a triangle are the three line segments joining its vertices. For triangle \( ABC \), the sides are:
- \( \overline{AB} \)
- \( \overline{BC} \)
- \( \overline{CA} \)
Each side lies opposite one of the vertices. For example, side \( \overline{BC} \) lies opposite vertex \( A \).
3.1. Naming of Opposite Sides
For convenience, we often name the sides using small letters opposite to the respective vertices:
- Side opposite \( A \) is named \( a \)
- Side opposite \( B \) is named \( b \)
- Side opposite \( C \) is named \( c \)
This naming method becomes useful when writing formulas such as the area or trigonometric relations in triangles.
4. Angles of a Triangle
Every triangle has three angles, each formed at a vertex. In \( \triangle ABC \):
- \( \angle A \) is formed by sides \( \overline{AB} \) and \( \overline{AC} \)
- \( \angle B \) is formed by sides \( \overline{BA} \) and \( \overline{BC} \)
- \( \angle C \) is formed by sides \( \overline{CA} \) and \( \overline{CB} \)
Angles help decide the type of triangle, such as acute, right or obtuse.
4.1. Interior and Exterior Angles
Interior angle: The angle inside the triangle at each vertex is called an interior angle.
Exterior angle: If one side of the triangle is extended, it forms an exterior angle adjacent to the interior angle.
The exterior angle is important because it relates directly to the sum of the two non-adjacent interior angles.
5. Interior and Exterior Regions
A triangle divides the plane into two regions:
- Interior region: The space inside the boundary formed by the three sides.
- Exterior region: Everything outside the triangle.
These regions help in understanding areas, constructions and proofs involving triangles.
6. Putting It All Together
In summary, every triangle has:
- three vertices
- three sides
- three interior angles
- three exterior angles (if sides extended)
- an interior region
- an exterior region
Understanding these parts clearly makes all future triangle concepts much easier to learn.