Triangles
🎯 What will you learn on this page?
👉 Identify a triangle and its basic parts.
👉 Classify triangles according to their sides.
👉 Classify triangles according to their angles.
👉 Use the angle-sum property of a triangle.
👉 Find unknown angles and solve simple triangle problems.
A triangle is a closed plane figure formed by three line segments.
The point where two sides meet is called a vertex. A line segment joining two vertices is called a side of the triangle.
A triangle has three vertices, three sides and three interior angles.
Triangle
△ABC
Vertices: A, B, C
Sides: AB, BC, CA
📚 The sum of the three interior angles of every triangle is 180°.
∠A + ∠B + ∠C = 180°🔹 Types of Triangles Based on Sides
According to their sides, triangles are of three types
Scalene Triangle
Isosceles Triangle
Equilateral Triangle
Scalene Triangle
AB ≠ BC, BC ≠ CA, CA ≠ AB
All three sides have different lengths.
All three angles are also different.
Isosceles Triangle
AB = AC
∠B = ∠C
Two sides are equal, and the angles opposite the equal sides are equal.
Equilateral Triangle
AB = BC = CA
∠A = ∠B = ∠C = 60°
All three sides and all three angles are equal.
🔹 Types of Triangles Based on Angles
According to their angles, triangles are of three types
Acute-angled Triangle
Right-angled Triangle
Obtuse-angled Triangle
Acute-angled Triangle
∠A < 90°, ∠B < 90°, ∠C < 90°
All three angles are less than 90°.
Right-angled Triangle
One angle = 90°
The other two angles are acute and together add up to 90°.
Obtuse-angled Triangle
One angle > 90°
Only one angle of a triangle can be greater than 90°.
📚 Important Facts About the Angles of a Triangle
🔹 The three interior angles of a triangle always add up to 180°.
🔹 Every triangle has at least two acute angles.
🔹 A triangle cannot have two right angles.
🔹 A triangle cannot have two obtuse angles.
🔹 All three angles of a triangle cannot be less than 60°, because their sum would then be less than 180°.
🔹 An equilateral triangle is always acute-angled because each angle is 60°.
🔹 A Triangle Can Be Classified in Two Ways
The same triangle can be classified according to both its sides and its angles.
For example:
👉 A triangle with angles 45°, 45°, 90° is an isosceles right-angled triangle.
👉 A triangle with angles 60°, 60°, 60° is an equilateral acute-angled triangle.
👉 A triangle with angles 30°, 60°, 90° can be a scalene right-angled triangle.
✏️ Solved Examples
✍️ Example 1: Find the Third Angle
Two angles of a triangle are 52° and 68°. Find the third angle.
The sum of the angles of a triangle is 180°.
Third angle = 180° - (52° + 68°) = 180° - 120° = 60°Therefore, the third angle is 60°.
✍️ Example 2: Classify a Triangle by Its Sides
The side lengths of a triangle are 6 cm, 6 cm, 9 cm. What type of triangle is it?
Two sides are equal.
Therefore, it is an isosceles triangle.
✍️ Example 3: Classify a Triangle by Its Angles
The angles of a triangle are 35°, 55°, 90°.
Since one angle is 90°, it is a right-angled triangle.
✍️ Example 4: Find Angles Using an Equation
The three angles of a triangle are x°, 2x° and 3x°. Find all the angles.
x + 2x + 3x = 180° 6x = 180° x = 30°Therefore, the angles are:
30°, 60°, 90°So the triangle is a right-angled triangle.
✍️ Example 5: Is Such a Triangle Possible?
Can a triangle have two angles measuring 95° and 100°?
95° + 100° = 195°The sum of just these two angles is already greater than 180°.
Therefore, such a triangle is not possible.
🧩 Practice Questions
✍️ 1. Two angles of a triangle are 46° and 72°. Find the third angle.
✍️ 2. The sides of a triangle are 7 cm, 7 cm, 10 cm. Classify the triangle according to its sides.
✍️ 3. Classify a triangle with angles 60°, 60°, 60° according to both its sides and its angles.
✍️ 4. The angles of a triangle are 42°, 58°, 80°. What type of triangle is it according to its angles?
✍️ 5. The angles of a triangle are x°, 2x°, 3x°. Find x and all three angles.
✍️ 6. One acute angle of a right-angled triangle is 37°. Find the other acute angle.
✍️ 7. Can 100°, 50°, 30° be the angles of a triangle? If yes, classify the triangle according to its angles.
✍️ 8. Explain why a triangle cannot have two obtuse angles.
1. 62°
2. Isosceles triangle
3. Equilateral and acute-angled triangle
4. Acute-angled triangle
5. x = 30°; angles = 30°, 60°, 90°
6. 53°
7. Yes. Their sum is 180°; it is an obtuse-angled triangle.
8. Each obtuse angle is greater than 90°, so two obtuse angles would already have a sum greater than 180°.
⚠️ Common Misconceptions
🔹 An isosceles triangle only has two equal angles.
This is incomplete. It has two equal sides, and the angles opposite those equal sides are also equal.
🔹 An equilateral triangle can be right-angled.
No. Every angle of an equilateral triangle is 60°.
🔹 A triangle can have two obtuse angles.
No. Two obtuse angles would have a sum greater than 180°.
🔹 You can always decide the type of a triangle just by looking at its diagram.
Not always. Use the given side lengths or angle measures for a correct classification.
✅ Quick Revision
🔹 A triangle is a closed figure with three sides, three vertices and three angles.
🔹 The sum of the three interior angles of a triangle is 180°.
🔹 By sides, triangles are scalene, isosceles or equilateral.
🔹 By angles, triangles are acute-angled, right-angled or obtuse-angled.
🔹 Every angle of an equilateral triangle is 60°.
🔹 In an isosceles triangle, the angles opposite the equal sides are equal.
🔹 Every triangle has at least two acute angles.
🔹 A triangle can have at most one right angle or one obtuse angle.
🔹 Congruence and other triangle theorems can be studied in greater detail on later pages.
