MATHS
QUERY

We have already discussed about polygons in the Polygon and its Types chapter. There we learnt that a polygon is a closed curve, which is made up of only line segments.

Triangle is one type of polygon which has three sides or line segments. It is a polygon with least number of sides, in other words, a polygon with less than three sides does not exist.

Triangle is a closed figure which is formed by joining three line segments. It has 3 sides, 3 vertices and 3
angles.

Triangle is denoted by Δ, a delta
symbol.

Lets learn about triangles with the following figure with a triangle whose name is written as ΔABC, where Δ is a symbol of triangle and ABC is the name of a triangle, which always includes three vertices of a triangle.

Example of a triangle ΔABC

We can use any three vertices in any order to represent a triangle.

For example, ABC or BAC or CAB or ACB. Therefore, the triangle can also be written as ΔBAC or
ΔCAB or ΔACB.

As you know that triangle always has three sides. These sides are also called the
line segments.

So, in ΔABC, line segments AB, BC and CA are sides of the ΔABC.
These three sides of a triangle are written as \(\overline{AB}\), \(\overline{BC}\) and \(\overline{CA}\).

Vertex of a triangle is that point where any two sides of a triangle meet out or intersect.

In the ΔABC in the above figure, sides AB and AC meet at point A. So, A is the vertex of ΔABC.

Similarly, sides BC and AB meet at point B. So B is a vertex of ΔABC.

Also, sides AC and BC meet at point C. So C is a vertex of ΔABC.

Therefore, A, B and C are the three vertices of ΔABC.

Angles
are formed at the vertex of a triangle. It is a measurement of how slanted two lines are to each other.
The angles are measured in degrees units.

Let’s understand angles in a triangle ΔABC from the above figure.

As we already discussed above, ΔABC has three vertices A, B and C.

Therefore, we can say angles are formed at vertices A, B and C.
Angles are written as ∠ABC, ∠BAC and ∠ACB or in a short form as ∠B,
∠A and ∠C respectively.

In other words, we can say:

∠ABC or ∠B is formed at vertex B of ΔABC.

∠BAC or ∠A is formed at vertex A of ΔABC.

∠ACB or ∠C is formed at vertex C of ΔABC.

Triangle has many types depending upon the length of its sides.

A triangle is said to be an equilateral triangle if all sides of a triangle are of equal length.

ΔABC is equilateral triangle because AB = BC = CA, where AB is length of side AB, BC is length of
side BC and CA is the length of side CA.

Equilateral triangle ΔABC

A triangle is said to be an Isosceles triangle if any two sides of the triangle are of equal length.

ΔABC is Isosceles triangle because AB = AC.

Isosceles triangle ΔABC

A triangle is said to be a scalene triangle if all sides of the triangle are of unequal length.

ΔABC is Scalene triangle because AB ≠ BC ≠ AC.

Scalene triangle ΔABC

Triangle has many types depending upon the length of its angles.

A triangle is said to be an acute angled triangle if each angle of a triangle is acute. Acute angle is that
angle
which is less than 90°.

So, the ΔABC in figure is an acute angled triangle because all angles are less than 90°.
That is, ∠A, ∠B and ∠C are all less than 90°.

Acute angled triangle ΔABC

A triangle is said to be a right angle triangle if one of its angles is right angle. Right angle is that angle
whose measure is 90°.

So, the ΔABC in figure is a right angled triangle because ∠C is equal to 90°.

Right angled triangle ΔABC

A triangle is said to be an obtuse angled triangle if one of its angles is an obtuse angle. Obtuse angle is that
angle
which is greater than 90°.

So, the ΔABC in figure is an obtuse angled triangle because ∠C is greater than 90°.

Obtuse angled triangle ΔABC

Triangle is a plane and closed figure which is formed by three line segments.

There are 3 sides in a triangle.

There are 3 angles in a triangle.

There are three types of a triangle according to its sides.

1. Equilateral triangle

2. Isosceles triangle

3. Scalene triangle

There are three types of a triangle according to its angles.

1. Right angled triangle

2. Obtuse angle triangle

3. Acute angle triangle

Sum of three angles of a triangle is equal to 180^{0}.

**Solution**

In isosceles ΔABC

AB = AC

∴ ∠B = ∠C ( ∵ base angles are equal )

∠C = 30^{0}

In ΔABC,

∠A + ∠B + ∠C = 180^{0}

∠A + 30^{0} + 30^{0} = 180^{0}

∠A + 60^{0} = 180^{0}

∠A = 180^{0} - 60^{0}

∠A = 120^{0}

**Solution**

In right angled triangle ΔABC

∠A + ∠B + ∠C = 180^{0}

∠A + 90^{0} + 30^{0} = 180^{0}

∠A + 120^{0} = 180^{0}

∠A = 180^{0}- 120^{0}

∠A = 60^{0}

**Solution**

In triangle ΔABC

∠A + ∠B + ∠C = 180^{0}

50^{0} + 60^{0} + ∠C= 180^{0}

110^{0} + ∠C= 180^{0}

∠C = 180^{0}- 110^{0}

∠C = 70^{0}

**Solution**

Let the angles be x, 2x and 3x

In triangle, sum of three angles is 180^{0}.

∴ ∠A + ∠B + ∠C = 180^{0}

x + 2x + 3x = 180^{0}

6x = 180^{0}

x = \(\frac{180}{6}

x = 30^{0}

∴ the three angles are:

x = 30^{0}

2x = 2 x 30^{0} = 60^{0}

3x = 3 x 30^{0} = 90^{0}

∴ the three angles are 30^{0}, 60^{0} and 90^{0}