Introduction
We have already discussed about polygons in the
Curve, Polygon and Types of Polygon
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.
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.
Sides of triangle
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
AB,
BC,
and
CA.
Vertex of triangle
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 of triangle
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.
Types of triangle on the basis of length of sides
Triangle has many types depending upon the length of its sides.
Equilateral triangle
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.
Isosceles triangle
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.
Scalene triangle
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.
Types of triangle on the basis of size of angles
Triangle has many types depending upon the length of its angles.
Acute angled triangle
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°.
Right angled triangle
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°.
Obtuse angled triangle
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°.
List of types of triangles on the basis of sides
| Name of triangle | Number of equal sides |
|---|---|
| Equilateral triangle | All 3 sides are equal |
| Isosceles triangle | Any 2 sides are equal |
| Scalene triangle | No sides are equal |
List of types of triangles on the basis of angle
| Name of triangle | Measure of angle |
|---|---|
| Acute angled triangle | All three angles are acute angles |
| Right angled triangle | One angle is 90° |
| Obtuse angled triangle | One angle is obtuse angle |
Frequently Asked Questions
1) What is a triangle?
Triangle is a plane and closed figure which is formed by three line segments.
2) How many sides are there in a triangle?
There are 3 sides in a triangle.
3) How many angles are there in a triangle?
There are 3 angles in a triangle.
4) What are the types of a triangle according to its sides?
There are three types of a triangle according to its sides.
1. Equilateral triangle
2. Isosceles triangle
3. Scalene triangle
5) What are the types of a triangle according to its angles?
There are three types of a triangle according to its angles.
1. Right angled triangle
2. Obtuse angle triangle
3. Acute angle triangle
6) What is the sum of three angles of a triangle?
Sum of three angles of a triangle is equal to 1800.
Solved Examples
1) Name the following triangles from the names scalene triangle, isosceles triangle, equilateral triangle, acute triangle, obtuse triangle and right angled triangle.
-
Solution
In ΔABC, all sides are equal in length
i.e. AB = BC = AC = 3 cm
Therefore, ΔABC is an equilateral triangle. -
Solution
In ΔABC, two sides are equal in length
i.e. AB = BC = 4 cm
Therefore, this is an isosceles triangle. -
Solution
In ΔABC, ∠ABC = 110°, which is an obtuse angle.
Therefore, this is an obtuse angled triangle. -
Solution
In ΔABC, ∠A and ∠B are equal
i.e. ∠A = ∠B = 80°
∴ CA = CB (∵ in a triangle, sides opposite to equal angles are equal)
Therefore, this is an isosceles triangle. -
Solution
In the ΔABC, AB = 5 cm, BC = 3 cm and CA = 4 cm
As, length of three sides of ΔABC are unequal
Therefore, ΔABC is a scalene triangle. -
Solution
In ΔABC, ∠A = 70°, ∠B = 50° and ∠C = 60°
As, all the three angles ∠A, ∠B and ∠C has measure less than 90°
Therefore, ΔABC is an acute triangle. -
Solution
In ΔABC, ∠B = 90°
As, one angle ∠B in ΔABC is 90°
Therefore, ΔABC is a right angled triangle.
2) In isosceles triangle ΔABC, with AB=AC, if base angle = 300, find the other angles.
Solution
In isosceles ΔABC
AB = AC
∴ ∠B = ∠C ( ∵ base angles are equal )
∠C = 300
In ΔABC,
∠A + ∠B + ∠C = 1800
∠A + 300 + 300 = 1800
∠A + 600 = 1800
∠A = 1800 - 600
∠A = 1200
3) In right angled triangle ΔABC at B, if ∠C = 300, find ∠A.
Solution
In right angled triangle ΔABC
∠A + ∠B + ∠C = 1800
∠A + 900 + 300 = 1800
∠A + 1200 = 1800
∠A = 1800- 1200
∠A = 600
4) In triangle ΔABC, if ∠A = 500, ∠B = 300, find ∠C.
Solution
In triangle ΔABC
∠A + ∠B + ∠C = 1800
500 + 300 + ∠C= 1800
800 + ∠C= 1800
∠C = 1800- 800
∠C = 1000
5) If the angles of ΔABC are in ratio 1 : 2 : 3, find all angles.
Solution
Let the angles be x, 2x and 3x
In triangle, sum of three angles is 1800.
∴ ∠A + ∠B + ∠C = 1800
x + 2x + 3x = 1800
6x = 1800
x =
180
6
x = 300
∴ the three angles are:
x = 300
2x = 2 x 300 = 600
3x = 3 x 300 = 900
∴ the three angles are 300, 600 and 900
Fill in Blanks Worksheet
| Type: | Blanks |
| Count: | 1 |
- A triangle in which none of the sides are equal, is a ___________ triangle.
- A triangle with 90° angle is ___________ triangle.
- If three sides of a triangle are equal, then all of the three angles of the triangle are ___________.
- If two sides of a triangle are equal, then its two ___________ are also equal.
- The measure of each angle of an equilateral triangle is ___________.
- A triangle has ___________ sides.
- An equilateral triangle is ___________ also.
- A triangle whose all angles are less than 90°, the triangle is known as ___________ angled triangle.
- A triangle with one angle greater than 90°, the triangle is known as ___________ angled triangle.
- If the measure of one angle of an isosceles triangle is equal to 90° then the other two angles are less than ___________.
Write True or False Worksheet
| Type: | True False |
| Count: | 1 |
| S.N. | Statement | ✓ or ✕ |
|---|---|---|
| 1) | A triangle has 3 vertices. | |
| 2) | If an isosceles triangle has one angle of 90° then the other two angles must be obtuse angles. | |
| 3) | A triangle has four angles. | |
| 4) | Each acute angled triangle is equilateral. | |
| 5) | A triangle with 90°, 60° and 30° angles is a right angled triangle. | |
| 6) | The sum of all angles of a triangle is 180°. | |
| 7) | A triangle with a measure of each angle as 60°, is an equilateral triangle. | |
| 8) | An obtuse angle triangle has a right angle. | |
| 9) | A triangle has three collinear points. | |
| 10) | An equilateral triangle is an acute angled triangle. |
Match Columns Worksheet
| Type: | Matching |
| Count: | 1 |
| 1) | A triangle with all equal angles | a) | Scalene triangle |
| 2) | A triangle with all sides unequal | b) | Right angled triangle |
| 3) | A triangle with one right angle | c) | Obtuse angled triangle |
| 4) | A triangle with all acute angles | d) | Isosceles right angled triangle |
| 5) | A triangle with 2 equal sides and one right angle | e) | Equilateral triangle |
| 6) | A triangle with one obtuse angle | f) | Isosceles triangle |
| 7) | A triangle with 2 equal sides/td> | g) | Acute angled triangle |
Multiple Choice Questions Worksheet
| Type: | MCQ |
| Count: | 1 |
- x
- y
- z
- ∠A
- 1800
- 3600
- 7200
- 14400
- 500
- 400
- 600
- 700
- 300
- 400
- 900
- 600
- scalene triangle
- acute triangle
- equilateral triangle
- isosceles triangle
- 900
- 1000
- 1100
- 1200
- true
- false
- maybe
- none of these
- scalene triangle
- equilateral triangle
- isosceles triangle
- right angled triangle
- scalene triangle
- obtuse triangle
- equilateral triangle
- right angled triangle
- obtuse triangle
- acute triangle
- right angled triangle
- All of the above
