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Maths > Triangles > Properties of Triangles
Triangles

Properties of Triangles

Properties of triangles are generally used to study triangles in detail, but we can use them to compare two or more triangles as well. With the help of these properties, we can not only determine the equality in a triangle but inequalities as well. Let’s see how!

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Properties of Triangles

Triangles are three-sided closed figures. Depending on the measurement of sides and angles triangles are of following types:

  • Equilateral Triangles: An equilateral triangle has all the sides and angles of equal measurement. This type of triangle is also called an acute triangle as all its sides measure 60° in measurement.
  • Isosceles triangle: An isosceles triangle is the one with two sides equal and two equal angles.
  • Scalene triangle: In a scalene triangle, no sides and angles are equal to each other.

Depending on angles, triangles are of following types:

  • Acute Triangle: Triangles, where all sides are acute-angled to each other, are called acute triangles. The best example of this kind of triangle is the equilateral triangle.
  • Obtuse Triangle: The obtuse angled triangle is the one with one obtuse angled side. Isosceles triangles and scalene triangles come under this category of triangles.
  • Right Angled triangle: A triangle with one angle equal to 90° is called right-angled triangle.

When we study the properties of a triangle we generally take into consideration the isosceles triangles, as this triangle is the mixture of equality and inequalities. Let’s see the figure given below before studying further about properties of triangles.

Properties of triangles

The figure given above is of an isosceles triangle PQR. What do you observe in the figure? The two sides of the triangle are equal. Now using a protractor, measure the angles as well. On measuring the angles we observe that ∠Q and ∠R are also equal. This implies that in every isosceles triangle, the angles opposite to the equal sides are also equal.

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Download Triangles Cheat Sheet PDF

Triangle Cheat sheet

The following properties of triangles shall make the concept more clear to you:

1. Angles opposite to equal sides of an isosceles triangle are also equal

In an isosceles triangle XYZ, two sides of the triangle are equal. We have XY=XZ. Here we need to prove that ∠Y =∠Z. Let’s draw the triangle first, with a point W as the bisector of ∠X.

Properties of triangles

In Δ YXW and Δ ZXW,
XY=XZ                                        (as given)
∠YXW = ∠ZXW                        (W bisects the angle ∠X)
XW=XW                                     (Common side)
So by the Side-Angle-Side (SAS) rule; Δ YXW ≅  Δ ZXW
As the corresponding angles of congruent triangles,   ∠XYW = ∠XZW
Hence ∠Y = ∠Z

Download NCERT Solutions for Class 10 Mathematics

2. The sides opposite to equal angles of a triangle are also equal

This property is the converse of the above property. For this, we need to measure the sides of the triangle with scale and angles with a protractor. On measuring the sides and angles respectively we come to the conclusion that the sides opposite to equal angles are also equal. We use the ASA congruence rule to prove the property.

Solved Question for you

Question 1: The figure below shows a triangle PQR with PQ=PR, S and T are two points on QR such that QT=RS. Show that PS=PT.

Properties of triangles

Answer : In Δ PQS and Δ PRT, PQ=PR. Since angles opposite to equal sides are equal sides ∠Q = ∠R
Also, QT= RS
So, QT-ST = RS-ST
that is, QS = TR
So, Using the SAS congruence rule we come to the conclusion thatΔ PQS ≅ Δ PRT
Hence,  PS = PT

Question 2: What is called a triangle?

Answer: An equilateral triangle property is that it has equal sides. Further, a triangle with two sides equal is called isosceles, and a triangle with all sides a different length is called scalene. A triangle can be simultaneously right and isosceles, in which case it is known as an isosceles right triangle.

Question 3: How many types of triangles are there?

Answer: Triangles are shapes with three sides. There are different names for the types of triangles. A triangle’s type depends on the length of its sides and the size of its angles (corners). There are three types of a triangle based on the length of the sides: equilateral, isosceles, and scalene.

 Question 4: What is a unique triangle?

Answer: The two angles and any side condition determines a unique triangle. Since the condition has two different arrangements, we separate it into two conditions: the two angles and included side condition and the two angles and the side opposite a given angle condition.

Question 5: Who invented triangle?

Answer: The triangle was invented by Blaise Pascal in 1653. Although it has been named after Blaise Pascal, there have been traces of the triangle, long before Blaise Pascal was born. It is believed that the Persians and the Chinese had been using it to find the square and cube root of numbers.

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