Here’s a complete lesson on Triangles and Their Types designed for Classes 6, 7, and 8 with definitions, examples, diagrams (descriptions), and exercises.

📘 Lesson: Triangles and Types of Triangles

🔹 What is a Triangle?

A triangle is a closed figure formed by three line segments.
It has:

• 3 sides
• 3 angles
• 3 vertices (corners)

👉 Example: A triangle with vertices A, B, and C is written as △ABC.

🔹 Classification of Triangles

Triangles can be classified based on sides and based on angles.

1️⃣ Based on Sides

1. Equilateral Triangle

• All three sides are equal.
• All angles are 60°.
• Example: △ABC where AB = BC = CA = 6 cm.

2. Isosceles Triangle

• Two sides are equal.
• Angles opposite equal sides are also equal.
• Example: △PQR where PQ = PR = 5 cm, QR = 7 cm.

3. Scalene Triangle

• All three sides are different.
• All angles are also different.
• Example: △XYZ where XY = 4 cm, YZ = 6 cm, XZ = 5 cm.

2️⃣ Based on Angles

1. Acute-angled Triangle

• All angles are less than 90°.
• Example: △LMN where ∠L = 50°, ∠M = 60°, ∠N = 70°.

2. Right-angled Triangle

• Has one right angle (90°).
• The side opposite the right angle is called the hypotenuse.
• Example: △DEF where ∠E = 90°.

3. Obtuse-angled Triangle

• Has one angle greater than 90°.
• Example: △UVW where ∠U = 120°, ∠V = 30°, ∠W = 30°.

🔹 Important Properties of Triangles

1. Angle Sum Property:
   The sum of all angles in a triangle is always 180°.
   👉 Example: If ∠A = 50°, ∠B = 60°, then ∠C = 70° because 50° + 60° + 70° = 180°.
2. Exterior Angle Property:
   An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
   👉 Example: If ∠A = 50°, ∠B = 60°, then exterior angle at C = 50° + 60° = 110°.

🔹 Examples

Example 1:

△ABC has ∠A = 40°, ∠B = 70°. Find ∠C.
👉 Solution:
∠C = 180° – (40° + 70°) = 70°
So, △ABC is an Isosceles Acute-angled Triangle.

Example 2:

If sides of a triangle are 5 cm, 5 cm, and 8 cm, find its type.
👉 Solution:
Two sides are equal → Isosceles Triangle.

Example 3:

△XYZ has angles 30°, 60°, and 90°. Find its type.
👉 Solution:
One angle is 90° → Right-angled Triangle.

🔹 Exercises (for practice)

1. Name the triangle:
   a) Sides: 7 cm, 7 cm, 7 cm
   b) Angles: 40°, 70°, 70°
   c) Angles: 120°, 30°, 30°
2. In △PQR, ∠P = 55°, ∠Q = 65°. Find ∠R.
3. A triangle has sides 6 cm, 8 cm, and 10 cm. Is it scalene, isosceles, or equilateral?
4. Draw and label an acute triangle and a right triangle in your notebook.