Maths Formulas

Triangle Definition & Properties of a Triangle

Definition of a Triangle

A triangle is defined as a basic polygon with three edges and three vertices. It is a very basic yet significant shape in geometry. Students who wonder what is a triangle can find the answer here. The layman definition of a triangle is a flat geometric figure that comprises 3 sides and 3 angles. Triangles are also divided into different types based on the measurement of sides and angles.

Studying triangles is one of the important parts of geometry. Major concepts, such as Pythagoras theorem and trigonometry, depend heavily on properties of triangle. Learners score the most marks on this topic as it is not only easy to comprehend but also has high weightage. In this article, apart from learning what is a triangle, we will also cover other important information on shape of a triangle, angles of triangle, properties of triangle, types of triangle, shape of a triangle, perimeter of triangle, area of triangle and more. Please refer to the table of contents added below to learn more.

Table of content:

  1. Parts of Triangle
  2. Types of Triangle
  3. Angles of Triangle
  4. Properties of Triangle
  5. Perimeter of Triangle
  6. Area of Triangle
  7. Similarity of Triangles
  8. Solved Questions
  9. FAQs on Triangles

1) Parts of a Triangle

A triangle consists of a number of parts that primarily comprise 3 angles, 3 vertices and 3 sides. Each part of a triangle allows you to draw various derivations and calculate the true measurements of the geometric figure. Once a learner understands what is a triangle along or the true definition of a triangle, he/she can begin grasping all the concepts in-depth. Students need to direct their focus on the basics and grasp parts of a triangle from the ground up.

Lets understand the concept in-depth and grasp what are the parts of a triangle. Refer to the below triangle diagram DEF

In the above image:

The three angles are, ∠DEF, ∠EFD, and ∠FDE.

The three sides are side DE, side EF, and side FD.

The three vertices are D, E, and F

2) Types of Triangle

A triangle has a predetermined set of angles and sides. If any of these set values changes, the shape of a triangle changes too. For example, if all sides of a triangle are equal, then it’s termed as an equilateral triangle. Thus, confirming the change in the shape of a triangle. Understanding different types of triangles will make you more familiar with the true definition of a triangle.

Here are different shape of a triangle based on different angle and side measurement:

Equilateral Triangles – Shape of a Triangle #1:

The above added diagram is an Equilateral Triangle – a prominent part of the types of triangle. Its all three sides are equal. So, we can conclude that they have identical angles and identical sides. No matter what the length of the sides is, angles in equilateral triangles measure 60° each.

The equilateral Triangle has sides of equal length, thus, we only need the length of 1 side to calculate the perimeter of triangle. The angle bisector, altitude, median, and the perpendicular bisector of a given side are all on the same line and is one of the three lines of symmetry of the triangle. This will quench your curiosity to learn more about what is a triangle in detail.

Isosceles Triangles – Shape of a Triangle #2:

The above-mentioned diagram represents an isosceles triangle that is an extended part of the true definition of a triangle. The 2 sides of an isosceles triangle are equal, which means two of the angles of triangle are also equal. Irrespective of where the apex or the peak points are, it’s still going to be called an isosceles triangle. The hash mark in the figure denotes the congruency. Any sides or angles with the same number of hashes through them are congruent.

The base of the isosceles triangles is shorter in length than the sides. If we know the measurement of any of the two angles, we can find the value of the third angle and solve your doubt of what is a triangle? Since the angles of triangle add up to 180°, then the third angle is 180 – (2 times the base angle) – ( 180 – 2b ). Hence, the formula is 180 – 2b

If the isosceles triangle has a right angle, then that triangle will be called the right isosceles triangle. A right isosceles triangle has one angle at 90° angle and other angles at 45° each. Right angles are shown by a square at a line intersection rather than a curve.

Scalene Triangles – Shape of a Triangle #3:

When it comes to a scalene triangle, all the side lengths of triangle have different measures. None of the sides are equal in length to any of the other sides in this case. In a scalene triangle, even the interior angles are also different.

Acute triangles:

Acute triangles have all acute angles i.e. angles less than 90°. Thus, helping you understand what is a triangle in actuality.

Right Triangles:

The Right Triangles have one angle that is 90°. Thus, playing a significant role in determining the shape of a triangle.

Obtuse triangles:

Obtuse triangles have one angle which is greater than 90°. This defines the true shape of a triangle.

3) Angles of Triangle

Every triangle comprises sides, angles, and vertices respectively. In this section, we’ll cover the angles of triangle and understand the true definition of a triangle.

Angles of triangle, an extensive part of the answer to what is a triangle, can be further divided into:

  • Interior angles of triangle
  • Exterior angles of triangle

A.   Interior Angles of Triangle

Interior angles are situated within the triangle. Take the above diagram as reference, it has 3 interior angles of triangle respectively. The ∠ x, ∠y and ∠z are denote interior angles. The sum of these angles is always 180° implies, ∠ x + ∠y + ∠z = 180°. Let us see the proof of this statement.

Sum of Interior Angles of a Triangle

Statement: The sum of the interior angles is always 180°

Proof: Let us consider a ΔABC, as shown in the figure mentioned above. Draw a line PQ parallel to the side BC to prove the above property of triangles. We can observe that PQ is a straight line, so it can be concluded that:

∠PAB + ∠BAC + ∠QAC = 180……(1)

Since PQ || BC and AB and AC are transversal, therefore

∠QAC = ∠ACB (an alternate angles pair)

Also, ∠PAB = ∠CBA (an alternate angles pair)

 

Now substitute the value of ∠QAC and ∠PAB in equation (1)

∠ACB + ∠BAC + ∠CBA = 180°

Therefore, the sum of the interior angles of triangle is always 180°

B.   Exterior Angles of Triangle

Exterior angles of triangle are an extended side adjacent to one side of a triangle. Considere side CD for reference, it is adjacent to the interior side BC. Thus, making the ∠ d an exterior angle while ∠ a, ∠b and ∠c are interior angles. So, how do we find out the value of exterior angles?

Firstly, find the interior angle. Then, subtract that value from 360 to get the value of the exterior angle. If, for some reason, you can’t determine the value of interior angles of triangle, then you can deduce it by referring to either the shape of it or the number of the other angles.

4) Properties of Triangle

Each and every shape in Maths has some properties which distinguish them from each other. Similarly, properties of triangle allow learners to grasp the real definition of a triangle and help them in truly understanding their everlasting doubt of what is a triangle? Let us discuss here some of the properties of triangles.

  • A triangle comprises three angles and three sides.
  • All the exterior angles in a triangle always add up to 360 degrees.
  • The sum of the triangles’ two sides length is greater than the length of the third side. Moreover, the difference between the lengths of two sides of a triangle is less than the length of the third side.
  • The sum of consecutive exterior and interior angles is supplementary.
  • The shortest side is opposite the smallest interior angle. Similarly, the longest side is opposite the largest interior angle while defining the properties of triangle.
  • The sum of the measures of the three angles of a triangle is always 180°.
  • In a triangle, the median is a line segment that joins a vertex of the triangle to the midpoint of the opposite side.

  • In the above added triangle, AE is the median. Similarly, we can mark the midpoint of AB as D, and join CD to get CD as a median. FB median can be created using the same method to define the properties of triangle. So these are the three medians of the triangle. The three medians of the triangle are concurrent and their point of concurrence is called the centroid of the triangle.
  • The altitude has one endpoint at a vertex of the triangle and other on the line containing the opposite side. The altitudes of the triangle are concurrent and their point of concurrence is called the orthocentre of the triangle.

5) Perimeter of Triangle

A perimeter of triangle can be defined as the sum of the total length of the outer boundary of any given triangle. In simpler terms, the perimeter of triangle is equal to the sum of all the three sides. Thus, bolstering students’ understanding of what is a triangle. The unit is the same as the unit of sides of the triangle.

Perimeter of triangle = Sum of All Sides

P = a+b+c

Where, P is the perimeter of triangle

a, b, c are the sides of the triangle

6) Area of Triangle

Generally, the term “area” is defined as a region that is occupied inside the boundary of any flat object or figure. The area of triangle is measured in square units and the standard unit is signified as square meters (). There are already predefined formulas set in place by renowned scientists to calculate areas of square, circles, rectangles, triangles, and so on. Thus, helping you understand the definition of a triangle and perimeter of triangle according whenever needed.

In this section, we’ll define the area of triangle and help you grasp an integral fundamental of what is a triangle. It is the total region enclosed by the three sides of any particular triangle. The area is known by equal to half times the base and height, i.e., A = x Base x Height. The unit of area is measured in square units (, ).

Apart from the general definition of the area of triangle, the calculations are conducted in accordance with shape, i.e., area of a right angled triangle, the area of an equilateral triangle and the area of the isosceles triangle will be different from one another. So, what exactly are they defined as?

Area of Triangle – Area of a Right Angled Triangle

A right-angled triangle can be defined as a figure where one side has a 90° angle and the 2 other acute angles add up to 90° (40°+50°, or 55°+35°, and so on). Thus, the overall height of the triangle can also be referred to as the length of the perpendicular side.

I.e. Area of a Right-Angled Triangle = A = x Base x Height (Perpendicular distance)

Area of Triangle – Area of an Equilateral Triangle

An equilateral triangle is a geometrical figure where all the sides are equal. If we draw a perpendicular from the vertex of the triangle to the base, the base is automatically divided into 2 equal parts. Refer this formula when you want to find out the area of an equilateral triangle.

Area of an Equilateral Triangle = A = (√3)/4 ×

Area of Triangle – Area of an Isosceles Triangle

An triangle is an isosceles triangle if two of its sides and angles are equal as well as opposite to each other.

Area of an Isosceles Triangle = A =  (base × height)

7) Similarity of Triangles

Two triangles can be called similar when they have two corresponding angles congruent and the sides proportional to each other. This concept is crucial in understanding what is a triangle and it also helps in defining the perimeter of triangle.

In the above added diagram, we can see that triangle EFG is an enlarged version of triangle ABC i.e., both of them have the same shape. The similarity in the shape of a triangle depends on the angles of triangles.

∠ABC=∠EGF,∠BAC=∠GEF,∠EFG=∠ACB

The area, altitude, and volume of Similar triangles are in the same ratio as the ratio of the length of their sides.

Properties of Similar Triangles:

(A) Reflexivity: A triangle (△) is similar to itself

(B) Symmetry: If △ ABC ∼ △ DEF, Then △ DEF ∼ △ ABC

(C) Transitivity: If△ ABC ∼△ DEF and△ DEF ∼△ XYZ, then △ ABC ∼△ XYZ

 

Tests to prove that a triangle is similar

Angle-Angle Similarity(AA)

If 2 corresponding angles of the two triangles are congruent, the triangle is called similar.

Side-Side-Side Similarity(SSS)

If the corresponding sides of the 2 triangles are proportional, then the triangles are similar.

Side Angle Side Similarity (SAS)

If 2 sides of two triangles are proportional and have one corresponding angle congruent, the two triangles are called similar.

Theorem

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides

Given: A(△ABC)~A(△PQR)

To Prove: A(△ABC)/A(△PQR)=AB²/PQ²

 

Construction: Construct seg AM perpendicular side BC and seg PN perpendicular side QR

 

Proof:

A(△ABC)/A(△PQR)=(BC)(AM)/(QR)(PN)                        …(1)

 

A(△ABC)~A(△PQR)

Therefore, ∠B=∠Q

AB/PQ=BC/QR=AC/PR                                                           …(2)

 

In triangle ABM and triangle PQN,

∠ABM=∠PQN                                                …(from 1)

∠AMB=∠PNQ                                               …(each side is a right angle)

Therefore, △ ABM ~△ PQN                                                   …(AA test of similarity)

AB/PQ=AM/PN       (c.s.s.t)                                                     …(3)

 

A(△ABC)/A(△PQR)=[(BC)/(QR)][(AB)/(PQ)]                 …(from 1, 2 and 3)

A(△ABC)/A(△PQR)=[(AB)/(PQ)][(AB)/(PQ)]                  …(from 3)

A(△ABC)/A(△PQR)=AB²/PQ²

Similarly, we can show that,

A(△ABC)/A(△PQR)=BC²/QR²=AC²/PR²

 

Hence, we have proved that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

8) Solved Questions

Q1. How many triangles can be drawn having its angles as 53° 64° and 63°?

  1. 1
  2. 2
  3. None
  4. More than 2

 

Solution: D. The three angles given are 53° 64° and 63°

So the sum of the angles is 53 + 64 + 63 =  180°

Sum of the angles of triangles is always 180°

Hence many angles can be formed with these three angles.

 

Q2. In a triangle ABC,  ∠B = ∠C = 45°  then the triangle is ……………………

  1. right-angled triangle
  2. acute-angled triangle
  3. obtuse-angled triangle
  4. equilateral triangle

 

Solution: A. In ΔABC,

∠A + ∠B + ∠C = 180° Sum of all interior angles of a triangle is 180°.

∠A + 45° + 45° = 180° [ ∠B + ∠C = 45°] given

As one of the angles is 90°

∴ It is a right-angled triangle.

9) FAQs on Triangle

Q: What is a triangle or what is the true definition of a triangle?

A: A triangle is defined as a basic polygon that has three edges and three vertices respectively.

Q: What is a triangle with one angle greater than 90° called?

A: A triangle with one angle greater than 90° is called the Obtuse Triangle.

Q: A triangle with three equal sides is called as?

A: A triangle with three equal sides is called an Equilateral triangle.

Q: What are the 7 types of triangles?                 

A: The 7 types of triangles are equilateral, right isosceles, obtuse isosceles, obtuse scalene, acute isosceles, right scalene, and acute scalene.

Q: What are the properties of triangle?

A: The properties of triangle are that firstly, the sum of all the angles of a triangle (of all types) equals 1800. Secondly, the sum of the length of the two sides of a triangle is larger than the length of the third side. Correspondingly, the distinction between the two sides of a triangle is less than the length of the third side.

Q: How to find the interior angle and understand what is a triangle?

A: To find the sum of the measure of interior angles of a triangle, use the formula (n – 2) × 180. However, to find the measure of one interior angle, we take that formula and divide it by the number of sides n: (n – 2) × 180 ÷ n.

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KUCKOO B
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KUCKOO B

I get a different answer for first example.
I got Q1 as 20.5
median 23 and
Q3 26

Yashitha
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Yashitha

Hi
Same

virat
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virat

yes

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