# Understanding Triangles: A Comprehensive Guide to Their Classification and Properties

Triangles, with their three sides and three angles, are one of the most fundamental shapes in geometry. But did you know not all triangles are created equal? Depending on the lengths of their sides and the measures of their angles, triangles can be classified into different types, each with its own unique set of properties. This guide will provide a comprehensive breakdown of these classifications, making the intricate world of triangles easier to navigate.

## Step-by-step Guide: Classifying Triangles

**Based on Sides**:

**Equilateral Triangle**:- All three sides are of equal length.All three angles measure \(60^\circ\).

**Isosceles Triangle**:- Two sides are of equal length.The angles opposite these equal sides are also equal.

**Scalene Triangle**:- No sides are of equal length.None of the angles are equal.

**Based on Angles**:

**Acute Triangle**:- All three angles are less than \(90^\circ\).

**Right Triangle**:- One angle measures exactly \(90^\circ\).

**Obtuse Triangle**:- One angle measures more than \(90^\circ\).

### Examples:

**Example 1**:

If a triangle has sides measuring \(5 \text{cm}\), \(5 \text{cm}\), and \(7 \text{cm}\), what type of triangle is it?**Solution**: The triangle is an **Isosceles Triangle** since it has two sides of equal length.

**Example 2**:

Identify a triangle with sides of \(3 \text{cm}\), \(4 \text{cm}\), and \(5 \text{cm}\).**Solution**: The triangle is a **Scalene Triangle** as all its sides have different lengths.

**Example 3**:

A triangle has angles measuring \(60^\circ\), \(60^\circ\), and \(60^\circ\). What type of triangle is it based on its angles?**Solution**: The triangle is an **Acute Triangle** since all its angles are less than \(90^\circ\).

**Example 4**:

Identify a triangle with angles of \(90^\circ\), \(45^\circ\), and \(45^\circ\).**Solution**: The triangle is a **Right Triangle** due to the presence of a \(90^\circ\) angle.

### Practice Questions:

- Classify a triangle with angles measuring \(80^\circ\), \(60^\circ\), and \(40^\circ\).
- What type of triangle has sides measuring \(6 \text{cm}\), \(6 \text{cm}\), and \(6 \text{cm}\)?
- Identify a triangle with one angle measuring \(100^\circ\) and the remaining two angles totaling \(80^\circ\).

**Answers**:

- Acute Triangle.
- Equilateral Triangle.
- Obtuse Triangle.

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