Triangles Class 6 ICSE — Maths Chapter 20
Triangles Class 6 ICSE introduces the triangle, its parts, and the rules that govern its angles. In this chapter of your Mathematics book, you learn the notation used for triangles, the meaning of a vertex, the interior angle sum, exterior angles, and how to classify triangles. The content follows the CISCE board syllabus. Each section below follows the chapter in order, with worked patterns using fresh numbers.
Download Triangles Class 6 ICSE Mathematics Chapter 20 PDF
Download the Triangles Class 6 ICSE Mathematics chapter PDF to revise offline or print for practice. It is the full chapter as printed in the book — 7 pages, 8 sections, 21 questions.
What is inside Chapter 20
7 pages · 8 sections · 5 worked examples · 21 exercise questions
| Section | Title |
|---|---|
| 20.1 | TRIANGLE |
| 20.2 | VERTEX |
| 20.3 | ANGLES (INTERIOR ANGLES) OF A TRIANGLE |
| 20.4 | EXTERIOR ANGLE OF A TRIANGLE |
| 20.6 | CONSIDER THE FOLLOWING TABLE |
| 20.7 | TYPES OF TRIANGLES ACCORDING TO ANGLES |
| 20.8 | TYPES OF TRIANGLES ACCORDING TO SIDES |
| 20.9 | CONSTRUCTION OF TRIANGLES |
- EXERCISE 20(A): 9 questions
- Revision Exercise (Chapter 20): 12 questions
20.1 TRIANGLE
A triangle is a plane closed figure bounded by three line segments. In triangle ABC, the three line segments AB, BC and CA form the triangle. These segments are the sides of the triangle.
A triangle is denoted by the Greek letter Δ (delta). Triangle ABC is written as Δ ABC. The order of the letters does not matter. Δ ABC can also be written as Δ BAC, Δ BCA, Δ CAB, Δ ACB or Δ CBA. All of these name the same triangle.
Every triangle has three sides, three vertices and three angles. This is the foundation of this ICSE Class 6 chapter: a triangle is a closed shape with exactly three straight sides. You will use this notation in every angle and construction question that follows in the rest of the Class 6 Maths book.
20.2 VERTEX
A vertex of a triangle is a point where any two of its sides meet. In Δ ABC, the sides AB and AC meet at point A, so A is a vertex. In the same way, B is the point where the sides BC and AB meet, and C is the point where the sides AC and BC meet. The plural of vertex is vertices.
Thus A, B and C are the three vertices of triangle ABC. Each side lies opposite a vertex. Side BC is opposite vertex A, side AC is opposite vertex B, and side AB is opposite vertex C. This opposite pairing matters when you apply the exterior angle rule later in the chapter.
20.3 ANGLES (INTERIOR ANGLES) OF A TRIANGLE
Every triangle has three interior angles. In Δ ABC, the interior angles are ∠BAC, ∠ABC and ∠ACB. An interior angle can be named by its vertex letter alone. Since ∠ABC is formed at vertex B, it can be written as ∠B. So ∠ABC = ∠B, ∠BCA = ∠C and ∠BAC = ∠A.
The central rule of this section is the angle sum property: the sum of the interior angles of a triangle is always 180°, that is, two right angles. In Δ ABC, \angle A + \angle B + \angle C = 180^\circ. The same holds for any triangle. This rule is why a triangle can never have two right angles or two obtuse angles.
Worked pattern. Suppose two angles of a triangle are 48° and 73°. The third angle is 180^\circ - (48^\circ + 73^\circ) = 180^\circ - 121^\circ = 59^\circ. Check the result: 48^\circ + 73^\circ + 59^\circ = 180^\circ. Always keep this check in mind when you solve angle questions in the Class 6 ICSE syllabus.
20.4 EXTERIOR ANGLE OF A TRIANGLE
When any side of a triangle is extended, the angle formed outside the triangle is called an exterior angle. The chapter gives two properties of an exterior angle.
First, an exterior angle is adjacent and supplementary to the corresponding interior angle. If side BC of Δ ABC is extended to D, the exterior angle ACD and the interior angle ACB lie on the same straight line, so \angle ACD + \angle ACB = 180^\circ.
Second, an exterior angle equals the sum of its two interior opposite angles. If side CB is extended to form exterior angle ABD, then \angle ABD = \angle A + \angle C. When you extend all three sides, you form six exterior angles, two at each vertex.
Common mistake: confusing the adjacent interior angle with the two interior opposite angles. The adjacent interior angle shares a vertex with the exterior angle and is supplementary to it. The two interior opposite angles are the pair the exterior angle equals, not the one it adds up with.
20.6 CONSIDER THE FOLLOWING TABLE
The chapter gathers the six exterior angles in a table. For each exterior angle it lists the adjacent interior angle, the interior opposite angles, the supplementary relation and the equality relation.
| Exterior angle | Adjacent interior angle | Interior opposite angles | Supplementary relation | Equality relation |
|---|---|---|---|---|
| \angle 1 | \angle A | \angle B and \angle C | \angle 1 + \angle A = 180^\circ | \angle 1 = \angle B + \angle C |
| \angle 2 | \angle A | \angle B and \angle C | \angle 2 + \angle A = 180^\circ | \angle 2 = \angle B + \angle C |
| \angle 3 | \angle B | \angle A and \angle C | \angle 3 + \angle B = 180^\circ | \angle 3 = \angle A + \angle C |
| \angle 4 | \angle B | \angle A and \angle C | \angle 4 + \angle B = 180^\circ | \angle 4 = \angle A + \angle C |
| \angle 5 | \angle C | \angle A and \angle B | \angle 5 + \angle C = 180^\circ | \angle 5 = \angle A + \angle B |
| \angle 6 | \angle C | \angle A and \angle B | \angle 6 + \angle C = 180^\circ | \angle 6 = \angle A + \angle B |
Worked pattern 1 (angle sum rule). In a triangle, the three angles are x, 2x and 30^\circ. By the angle sum rule, x + 2x + 30^\circ = 180^\circ. So 3x = 150^\circ, and x = 50^\circ.
Worked pattern 2 (exterior angle property). An exterior angle of 100^\circ equals the sum of two interior opposite angles 3x and 2x. So 5x = 100^\circ, and x = 20^\circ.
Alternative method. First use the supplementary relation: 100^\circ plus the adjacent interior angle equals 180^\circ, so the adjacent interior angle is 80^\circ. Then in the triangle, 3x + 2x + 80^\circ = 180^\circ, giving 5x = 100^\circ and x = 20^\circ. Both routes give the same answer.
20.7 TYPES OF TRIANGLES ACCORDING TO ANGLES
Depending on the sizes of its angles, a triangle is acute-angled, right-angled or obtuse-angled.
An acute-angled triangle has each angle less than 90°. A right-angled triangle has one angle of exactly 90°, and the other two angles are acute. Their sum is always 90°, so \angle P + \angle R = 90^\circ. An obtuse-angled triangle has one angle greater than 90°.
The side opposite the right angle in a right-angled triangle is called the hypotenuse. The hypotenuse is the largest side of the triangle. In Δ PQR with ∠Q = 90°, side PR is the hypotenuse, and PR is longer than both PQ and QR.
Reason for the rule: the largest angle always faces the longest side. Since the right angle is the biggest angle in a right-angled triangle, the side opposite it must be the longest.
20.8 TYPES OF TRIANGLES ACCORDING TO SIDES
On the basis of sides, a triangle is isosceles, equilateral or scalene.
An isosceles triangle has at least two equal sides. The angles opposite the equal sides are equal, so if PQ = PR, then ∠R = ∠Q. An equilateral triangle has all three sides equal, and all three angles are equal. Since the sum is 180°, each angle is \frac{180^\circ}{3} = 60^\circ. A scalene triangle has all three sides of different lengths.
Memory aid: every equilateral triangle is also isosceles, because it has at least two equal sides. But the reverse is not always true: an isosceles triangle need not have all three sides equal.
| Type | Side condition | Angle property |
|---|---|---|
| Isosceles | at least two sides equal | angles opposite the equal sides are equal |
| Equilateral | all three sides equal | each angle is 60° |
| Scalene | all three sides unequal | no special equal-angle property |
EXERCISE 20(A) begins here with 9 questions on finding angles and classifying triangles.
20.9 CONSTRUCTION OF TRIANGLES
A triangle can be constructed when any one of three conditions is given: the lengths of the three sides; the lengths of two sides and the angle included between them; or two angles and the included side, that is, the side common to both angles.
The step pattern is the same each time. First draw a rough sketch so you know where each part goes. Then draw one side; when the three sides are given, draw the longest side first. Use a compass to cut arcs for the other sides, and join the points where the arcs meet.
Construction 1 (three sides). To construct a triangle with sides 5 cm, 6 cm and 7 cm, draw the longest side 7 cm first. Set the compass to 6 cm and draw an arc from one end. Set it to 5 cm and draw an arc from the other end. The arcs meet at the third vertex. Join both ends to that point.
Construction 2 (two sides and the included angle). Given two sides of 4 cm and 5 cm with an included angle of 45°, draw the 5 cm side first. At one end, construct the 45° angle with the compass. Mark 4 cm along the new ray, then join to the other end.
Construction 3 (two angles and the included side). Given a side of 6 cm with angles of 50° and 40° at its ends, draw the side, construct each angle at its end, and extend the rays. The rays meet at the third vertex. This ICSE chapter closes with ten construction practice questions after the worked examples. If you need to revise drawing angles with a compass, see the construction of angles chapter.
What to practise from Triangles Class 6 ICSE
This chapter closes with two practice blocks that test every rule and construction method covered above.
- EXERCISE 20(A): 9 questions. These ask you to find marked unknown angles, decide whether three given angles can form a triangle, solve for x from angle expressions, and classify triangles by angle and by side.
- Revision Exercise (Chapter 20): 12 questions. These mix angle finding, ratio problems, and constructions that include drawing angle bisectors and perpendiculars.
Master the angle sum and exterior angle rules first, then attempt the constructions with a ruler and compass.
Triangles Class 6 ICSE frequently asked questions
What is the sum of the angles of a triangle in the triangles class 6 icse chapter?
The sum is always 180°, which equals two right angles. For any triangle, \angle A + \angle B + \angle C = 180^\circ.
How is an exterior angle of a triangle related to its interior opposite angles in this ICSE chapter?
An exterior angle equals the sum of its two interior opposite angles. It is also supplementary to its adjacent interior angle, so the two add up to 180°.
What are the three conditions under which a triangle can be constructed in Class 6 ICSE Maths?
The three conditions are: the lengths of the three sides; the lengths of two sides and the included angle; and two angles with the included side.
Which side is the hypotenuse in a right angled triangle in this ICSE chapter?
The hypotenuse is the side opposite the right angle. It is the largest side of the right-angled triangle.
In Class 6 ICSE Maths, is every equilateral triangle also isosceles?
Yes, because an equilateral triangle has at least two equal sides. But the converse is not always true: an isosceles triangle need not be equilateral.
Reference: ICSE Class 6 Mathematics textbook material, chapter 20: Triangles — CISCE-aligned.