1. Chapter at a Glance
Take three lines that are not parallel and do not all pass through one point. They cut each other at three points and enclose a triangle — the simplest closed figure made of straight lines. Triangles hold up bridges, roofs and electric towers because a triangle cannot change shape once its sides are fixed. In this chapter you will learn when three lengths can form a triangle, how to construct triangles with ruler, compass and protractor, and the famous fact that the angles of every triangle add up to 180°.
- Parts of a triangle and naming triangles.
- Types of triangles by sides and by angles.
- The triangle inequality: when do three lengths make a triangle?
- Constructing triangles from three sides, from two sides and the angle between them, and from two angles and a side.
- The angle sum property and the exterior angle property.
- Properties of equilateral and isosceles triangles.
2. Parts and Types of Triangles
A triangle ABC has three vertices (A, B, C), three sides (AB, BC, CA) and three angles (∠A, ∠B, ∠C). The side opposite ∠A is BC, and so on.
| By sides | Meaning | By angles | Meaning |
|---|---|---|---|
| Equilateral | All three sides equal | Acute-angled | All angles less than 90° |
| Isosceles | Two sides equal | Right-angled | One angle exactly 90° |
| Scalene | No sides equal | Obtuse-angled | One angle more than 90° |
A triangle can have only one right angle or one obtuse angle — two such angles would already add up to 180° or more.
3. The Triangle Inequality
Can any three sticks make a triangle? Try sticks of 3 cm, 4 cm and 9 cm: the two short sticks together (7 cm) cannot reach across the 9 cm stick. So no triangle forms.
In every triangle, the sum of any two sides is greater than the third side. Three lengths form a triangle exactly when the two shorter lengths add up to more than the longest length.
Why? The straight path between two points is the shortest. Going from B to C directly (side BC) is shorter than going B → A → C, so BC < BA + AC.
Which can be sides of a triangle? (a) 5, 6, 10 (b) 4, 4, 8 (c) 7, 2, 4.
Solution: (a) 5 + 6 = 11 > 10 → yes. (b) 4 + 4 = 8, not greater than 8 → no (the points would lie on a line). (c) 2 + 4 = 6 < 7 → no.
Also, any side is greater than the difference of the other two. If two sides are 6 cm and 10 cm, the third side must be between 10 − 6 = 4 cm and 10 + 6 = 16 cm.
4. Constructing a Triangle from Three Sides (SSS)
Construct ΔABC with AB = 6 cm, BC = 5 cm and CA = 4 cm.
- Draw the base AB = 6 cm.
- With centre A and radius 4 cm (= AC), draw an arc above AB.
- With centre B and radius 5 cm (= BC), draw an arc cutting the first arc at C.
- Join AC and BC. ΔABC is the required triangle.
An equilateral triangle is constructed the same way with all three radii equal to the side. If the arcs do not meet, the lengths break the triangle inequality.
5. Other Constructions (SAS and ASA)
Two sides and the included angle (SAS)
Construct ΔPQR with PQ = 5 cm, PR = 4 cm and ∠P = 60°.
- Draw PQ = 5 cm.
- At P, make an angle of 60° with a protractor and draw a ray.
- Mark R on the ray with PR = 4 cm (use a compass).
- Join QR.
Two angles and the side between them (ASA)
Construct ΔXYZ with XY = 6 cm, ∠X = 50°, ∠Y = 70°.
- Draw XY = 6 cm.
- At X draw a ray at 50°; at Y draw a ray at 70° on the same side.
- The rays meet at Z. (The third angle will measure 180° − 120° = 60°.)
If two given angles add up to 180° or more, the rays never meet on that side — no triangle exists. Three angles alone (AAA) do not fix a triangle's size: many triangles of different sizes have the same angles.
6. The Angle Sum Property
The three angles of any triangle add up to 180°: ∠A + ∠B + ∠C = 180°.
Activity proof: Cut out a paper triangle, tear off its three corners and place them side by side with the vertices touching. They fit exactly along a straight line — a straight angle of 180°.
Reasoning proof: Through vertex A draw a line parallel to BC. The two angles it makes with AB and AC are equal to ∠B and ∠C (alternate angles). Together with ∠A they form a straight line, so ∠A + ∠B + ∠C = 180°.
Two angles of a triangle are 48° and 67°. Find the third.
Solution: 180° − (48° + 67°) = 180° − 115° = 65°.
7. Equilateral and Isosceles Triangles; Exterior Angles
- In an isosceles triangle, the angles opposite the equal sides are equal (base angles).
- In an equilateral triangle all angles are equal, so each is 180° ÷ 3 = 60°.
- In a right triangle, the two acute angles add up to 90°.
Exterior angle property
Extend side BC beyond C to D. The angle ∠ACD outside the triangle is an exterior angle. Since ∠ACD + ∠C = 180° (linear pair) and ∠A + ∠B + ∠C = 180°, we get:
An isosceles triangle has a vertex angle of 40°. Find each base angle.
Solution: The base angles are equal: 2x + 40° = 180°, so x = 70°.
8. Solved Problems (NCERT Style)
The angles of a triangle are in the ratio 2 : 3 : 4. Find them and classify the triangle.
Solution: 2x + 3x + 4x = 180° → x = 20°. Angles: 40°, 60°, 80°. All are less than 90°, so it is acute-angled.
Two sides of a triangle are 8 cm and 13 cm. Between which lengths must the third side lie?
Solution: Greater than 13 − 8 = 5 cm and less than 13 + 8 = 21 cm.
An exterior angle of a triangle is 115° and one interior opposite angle is 45°. Find the other two angles.
Solution: Other interior opposite angle = 115° − 45° = 70°. Third angle = 180° − 115° = 65° (linear pair with the exterior angle). Angles: 45°, 70°, 65°.
Can a triangle have angles 95°, 50° and 40°?
Solution: 95 + 50 + 40 = 185° ≠ 180°. Not possible.
9. Master Formula Table
| Property | Statement |
|---|---|
| Triangle inequality | a + b > c (for every pair) |
| Third side range | |a − b| < c < a + b |
| Angle sum | ∠A + ∠B + ∠C = 180° |
| Exterior angle | = sum of interior opposite angles |
| Equilateral triangle | each angle 60° |
| Isosceles triangle | base angles equal |
| Right triangle | two acute angles add to 90° |
| Constructions that fix a triangle | SSS, SAS, ASA |
10. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Types | By sides and by angles | Classify |
| Inequality | Two shorter sides > longest | Possible or not |
| Constructions | SSS, SAS, ASA | Steps |
| Angle sum | 180° | Missing angle |
| Exterior angle | Sum of opposite interior angles | Quick calculations |
| Isosceles/equilateral | Equal sides ↔ equal angles | Base angles |
11. Mind Map
- Vertices, sides, angles
- Types
- Triangle inequality
- Third-side range
- SSS with arcs
- SAS, ASA
- Sum 180°
- Exterior angle
- Isosceles, equilateral
12. Tips and Tricks
- For the triangle inequality, check only the two smallest sides against the largest.
- Use the exterior angle property to skip a step when finding angles.
- In constructions, always draw the longest given side as the base — the figure fits the page better.
- Accepting lengths like 4, 4, 8 (equality is not enough — the sum must be greater).
- Changing the compass radius between arcs.
- Thinking a triangle can have two obtuse angles.
13. Important One-liners
- A triangle has 3 sides, 3 angles and 3 vertices.
- The sum of any two sides of a triangle is greater than the third side.
- The angles of a triangle add up to 180°.
- Each angle of an equilateral triangle is 60°.
- The base angles of an isosceles triangle are equal.
- An exterior angle equals the sum of the two interior opposite angles.
- A triangle can have at most one right or one obtuse angle.
- Three sides (SSS) fix a triangle; three angles (AAA) do not fix its size.
14. Exam-pattern Questions with Detailed Answers
Which set can form a triangle?
- 2, 3, 6
- 3, 4, 7
- 5, 7, 11
- 1, 1, 3
Show answer
Only 5 + 7 = 12 > 11 works. The others fail: 5 < 6, 7 = 7, 2 < 3.
Answer: (C) 5, 7, 11
Find the third angle of a triangle whose angles are 90° and 35°.
Show answer
180° − 125° = 55°.
Answer: 55°
What is each angle of an equilateral triangle?
Show answer
All three angles are equal and add up to 180°, so each is 60°.
Answer: 60°
In ΔABC, ∠A = 2∠B and ∠C = 3∠B. Find all angles.
Show answer
2x + x + 3x = 180° → x = 30°. ∠B = 30°, ∠A = 60°, ∠C = 90°.
Answer: 60°, 30°, 90°
An exterior angle of a triangle is 130°. The two interior opposite angles are equal. Find them.
Show answer
Each is 130° ÷ 2 = 65°.
Answer: 65° each
Two sides of an isosceles triangle are 4 cm and 9 cm. What is its perimeter?
- 17 cm
- 22 cm
- 17 cm or 22 cm
- 13 cm
Show answer
If the equal sides were 4, 4, 9: 4 + 4 = 8 < 9, impossible. So the sides are 9, 9, 4, and the perimeter is 22 cm.
Answer: (B) 22 cm
Can you construct a triangle with ∠X = 110° and ∠Y = 75°? Why?
Show answer
110° + 75° = 185° > 180°, leaving no room for a third angle. The rays will not meet.
Answer: No
In a right triangle, one acute angle is 4 times the other. Find them.
Show answer
The acute angles add to 90°: x + 4x = 90° → x = 18°. Angles: 18° and 72°.
Answer: 18° and 72°
Show that a triangle cannot have two right angles.
Show answer
Two right angles already make 180°, leaving 0° for the third angle — impossible, since every angle of a triangle is more than 0°. (Also, two lines perpendicular to the same side would be parallel and never meet.)
Answer: The third angle would be 0°
How many triangles with whole-number sides have a perimeter of 9 cm? List them.
Show answer
Look for a ≤ b ≤ c with a + b + c = 9 and a + b > c: (1, 4, 4), (2, 3, 4), (3, 3, 3). Others like (1, 3, 5), (2, 2, 5), (1, 2, 6) fail.
Answer: 3 triangles: (1, 4, 4), (2, 3, 4), (3, 3, 3)
In ΔPQR, the exterior angle at R is 120° and ∠P : ∠Q = 1 : 2. Find ∠P, ∠Q and ∠R.
Show answer
∠P + ∠Q = 120° (exterior angle property): x + 2x = 120° → x = 40°. ∠P = 40°, ∠Q = 80°, ∠R = 180° − 120° = 60°.
Answer: 40°, 80°, 60°
A triangle has angles (x + 10)°, (2x − 20)° and (3x + 10)°. Find x and classify the triangle.
Show answer
x + 10 + 2x − 20 + 3x + 10 = 180 → 6x = 180 → x = 30. Angles: 40°, 40°, 100°.
Two equal angles → isosceles; one angle 100° → obtuse.
Answer: x = 30; an obtuse-angled isosceles triangle
15. Quick Revision Summary
- Triangles: equilateral, isosceles, scalene; acute, right, obtuse.
- Triangle inequality: the two shorter sides must add to more than the longest.
- Construct with SSS (arcs), SAS or ASA.
- Angle sum = 180°; exterior angle = sum of the two interior opposite angles.
- Equal sides face equal angles; equilateral angles are 60°.