# A Tale of Three Intersecting Lines

## Overview

This chapter introduces students to the properties and constructions of triangles, emphasizing the relationship between sides and angles. It explores how triangles are formed, when they cannot be formed, and how to construct them using various tools. The chapter also covers the triangle inequality, classification of triangles, angle sum property, and altitudes.

## Key Topics Covered

### 1. Understanding Triangles

* A triangle is a closed figure with **three vertices**, **three sides**, and **three angles**.
* Represented using the symbol ∆ (e.g., ∆ABC).
* Angles are denoted as ∠A, ∠B, and ∠C.

### 2. Equilateral Triangle Construction

* All sides equal in length.
* Construction method using a compass and ruler:

1. Draw a base (e.g., AB = 4 cm).
2. From A and B, draw arcs of radius 4 cm.
3. Intersection point is C. Join AC and BC.

### 3. Constructing Triangles with Given Sides

* Use compass for accuracy.
* Steps:

1. Draw base (e.g., AB = 4 cm).
2. Draw arcs from A and B with given lengths.
3. Intersection gives third point C. Join AC and BC.

### 4. Triangle Inequality Rule

* A triangle can only be formed if the **sum of any two sides is greater than the third**.
* If this condition is not met, triangle cannot exist.
* Example:

* 3, 4, 8 → No triangle (3 + 4 = 7 < 8)
* 4, 5, 8 → Triangle exists (4 + 5 = 9 > 8)

### 5. Triangle Construction Using Sides and Angles

#### Case 1: Two Sides and Included Angle

* Given AB and AC, with ∠A.
* Steps:

1. Draw base AB.
2. Use protractor to mark ∠A.
3. Cut AC on that ray using compass.

#### Case 2: Two Angles and Included Side

* Given AB, ∠A, and ∠B.
* Steps:

1. Draw AB.
2. Construct ∠A and ∠B.
3. Their meeting point is C.

### 6. Angle Sum Property

* **Sum of angles in a triangle = 180°**
* Visual proof using parallel lines and alternate angles.

### 7. Exterior Angles

* An **exterior angle** equals the sum of the two opposite interior angles.
* Example: ∠ACD = ∠A + ∠B

### 8. Altitudes of a Triangle

* Altitude = Perpendicular from a vertex to the opposite side.
* Each triangle has 3 altitudes.
* Construction uses set square and ruler for accuracy.
* Special Case: In right-angled triangles, the side opposite right angle is also an altitude.

### 9. Types of Triangles

#### Based on Sides

* **Equilateral**: All sides equal.
* **Isosceles**: Two sides equal.
* **Scalene**: All sides different.

#### Based on Angles

* **Acute-angled**: All angles < 90°
* **Right-angled**: One angle = 90°
* **Obtuse-angled**: One angle > 90°


## Keywords and Definitions

| Term | Definition |
| -------------------- | ---------------------------------------------------------------------------------------------- |
| Triangle | A closed figure with three sides and three angles. |
| Vertex | A point where two sides of a triangle meet. |
| Side | A line segment connecting two vertices. |
| Angle | The space between two intersecting lines or surfaces at or close to the point where they meet. |
| Equilateral Triangle | A triangle with all three sides equal. |
| Isosceles Triangle | A triangle with two sides equal. |
| Scalene Triangle | A triangle with all sides of different lengths. |
| Triangle Inequality | Rule that the sum of the lengths of any two sides must be greater than the third side. |
| Altitude | A perpendicular dropped from a vertex to the opposite side in a triangle. |
| Exterior Angle | Angle formed when one side of a triangle is extended beyond a vertex. |
| Angle Sum Property | The sum of the three interior angles in any triangle is always 180°. |

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## Practice Problems

### Easy (3 Problems)

1. **Construct an equilateral triangle with side 5 cm.**
**Solution**: Draw AB = 5 cm. Use compass to draw arcs of radius 5 cm from A and B. Intersection is point C. Join AC and BC.

2. **Check if triangle is possible with sides 3 cm, 4 cm, and 5 cm.**
**Solution**: 3+4=7 > 5, 3+5=8 > 4, 4+5=9 > 3 ⇒ Triangle possible ✅

3. **What is the sum of angles in a triangle?**
**Answer**: 180°

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### Medium (2 Problems)

4. **Construct a triangle with AB = 6 cm, AC = 4 cm, and ∠A = 60°.**
**Solution**:

* Draw AB = 6 cm.
* At A, construct 60° angle using protractor.
* Mark point C such that AC = 4 cm.
* Join BC.

5. **Check if triangle is possible with sides 2 cm, 3 cm, and 6 cm.**
**Solution**: 2 + 3 = 5 < 6 ⇒ Triangle not possible ❌

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### Difficult (3 Problems)

6. **Construct a triangle with two angles 45°, 80° and included side 5 cm.**
**Solution**:

* Draw AB = 5 cm.
* At A draw 45°, at B draw 80°.
* Intersection of the arms is C. Triangle ABC formed.

7. **Check triangle existence: 10 cm, 15 cm, 30 cm.**
**Solution**: 10 + 15 = 25 < 30 ⇒ Not possible ❌

8. **Find the third angle of triangle if ∠B = 50° and ∠C = 70°**
**Solution**: ∠A = 180° - (50° + 70°) = 60°

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### Very Difficult (2 Problems)

9. **Construct triangle ABC with AB = 5 cm, ∠A = 45°, ∠B = 140°**
**Solution**: Since ∠A + ∠B = 185° > 180°, triangle not possible ❌

10. **Find if triangle exists with sides: 1.5 cm, 2.5 cm, 4.1 cm**
**Solution**: 1.5 + 2.5 = 4.0 < 4.1 ⇒ Not possible ❌

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