ICSE Class 6 Maths Chapter 20 Triangles

Chapter 20: Triangles

Including Types, Properties and Constructions

20.1 Triangle

A triangle is a plane closed figure bounded by three line segments.

In the adjoining figure, the line segments AB, BC and CA form the triangle ABC.

The three line segments AB, BC and CA are the sides of the triangle ABC.

A triangle is denoted by the Greek letter \(\triangle\) (delta).

Thus, triangle ABC can be written as \(\triangle ABC\).

20.2 Vertex

Vertex of a triangle is a point where any two of its sides meet.

In the figure given above, the sides AB and AC meet at point A.

Therefore, A is a vertex of \(\triangle ABC\).

Similarly, vertex B - the point where the sides BC and AB meet.

And, vertex C - 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 the triangle ABC.

BC is the side opposite to vertex A and A is the vertex opposite to side BC. The same is true for vertex B and side AC, and for vertex C and side AB.

20.3 Angles (Interior Angles) of a Triangle

Every triangle has three angles.

In the triangle ABC drawn alongside, the three angles (interior angles) are: \(\angle BAC\), \(\angle ABC\) and \(\angle ACB\).

An interior angle of a triangle can also be denoted by the letter representing the corresponding vertex. Consider \(\angle ABC\), since it is formed at vertex B, it can be written as \(\angle B\).

Thus, \(\angle ABC = \angle B\), \(\angle BCA = \angle C\) and \(\angle BAC = \angle A\), all are interior angles of the triangle ABC.

The sum of the interior angles of a triangle is always 180 degrees, i.e. two right angles.

In \(\triangle ABC\), \(\angle A + \angle B + \angle C = 180°\)

and in \(\triangle PQR\), \(\angle P + \angle Q + \angle R = 180°\) and so on.

Each triangle has three sides, three vertices and three angles (interior angles).

\(\triangle ABC\) can also be written as \(\triangle BAC\) or \(\triangle CAB\), or \(\triangle ACB\) or \(\triangle CBA\) i.e. the three letters representing a triangle can be written in any order.

Teacher's Note

When you look at a roof's triangular frame or a triangular warning sign, you are seeing real-world examples of how triangles are fundamental structures in construction and design.

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.

20.5 Some Important Results

An exterior angle of a triangle is an adjacent and supplementary angle to the corresponding interior angle of the triangle.

For example:

In the figure given alongside, the side BC of \(\triangle ABC\) is extended up to point D, thus forming an exterior angle ACD. The exterior angle ACD is adjacent and supplementary to the corresponding interior \(\angle ACB\) of the \(\triangle ABC\) i.e. \(\angle ACD + \angle ACB = 180°\).

At each vertex, exterior angle plus interior angle equals 180 degrees.

An exterior angle of a triangle is always equal to the sum of its two opposite interior angles.

For example:

In the given figure, exterior angle ABD is formed by extending the side CB of the triangle ABC.

Therefore, Exterior angle ABD = Sum of interior opposite angles A and C,

i.e. \(\angle ABD = \angle A + \angle C\).

On extending the sides of a triangle, six exterior angles are formed, two at each vertex.

The adjoining figure shows the six exterior angles formed by extending the sides of the triangle ABC.

20.6 Consider the Following Table

Example 1:

For each triangle given below, find the value of x:

Solution:

For part (i):

\(\angle A + \angle B + \angle C = 180°\)

\(3x + 60° + x = 180°\)

\(4x = 180° - 60°\)

\(4x = 120°\)

\[x = \frac{120°}{4} = 30°\]

Sum of the angles of a triangle is 180 degrees.

For part (ii):

\(\angle ACD = \angle A + \angle B\)

\(115° = 2x + 3x\)

\(5x = 115°\)

\[x = \frac{115°}{5} = 23°\]

Exterior angle of a triangle equals the sum of interior opposite angles.

Alternative method:

\(\angle ACD + \angle ACB = 180°\)

\(115° + \angle ACB = 180°\)

\(\angle ACB = 180° - 115° = 65°\)

At each vertex, exterior angle plus interior angle equals 180 degrees.

Now, in \(\triangle ABC\)

\(\angle A + \angle B + \angle C = 180°\)

\(2x + 3x + 65° = 180°\)

\(5x = 180° - 65° = 115°\)

\[x = \frac{115°}{5} = 23°\]

Teacher's Note

Understanding exterior angles helps in analyzing building slopes, ramp angles, and the physics of how objects slide on inclined surfaces in real construction projects.

20.7 Types of Triangles According to Angles

Depending on the sizes of its angles, a triangle can be classified as:

1. Acute-angled triangle 2. Right-angled triangle 3. Obtuse-angled triangle

1. Acute-angled triangle

If each angle of a triangle is acute (less than 90 degrees), it is called an acute-angled triangle.

The adjoining figure shows an acute-angled triangle; each of its angles is less than 90 degrees.

2. Right-angled triangle

If one of the angles of a triangle is a right angle i.e. 90 degrees it is called a right-angled triangle.

The figure given alongside shows a right angled triangle PQR, as \(\angle PQR = 90°\).

Sum of the two acute angles of a right angled triangle is always 90 degrees, i.e. \(\angle P + \angle R = 90°\).

In a right-angled triangle, the side opposite to the right angle is called the hypotenuse. Hypotenuse is the largest side of a right angled triangle.

In the given \(\triangle PQR\), side PR is opposite to angle Q, which is a right angle. Therefore, PR is the hypotenuse. Also, PR > PQ and PR > QR.

Teacher's Note

Right triangles are essential in carpentry - when builders square up corners using the 3-4-5 ratio, they are creating right angles for perfectly aligned walls and frameworks.

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