Isosceles Triangle and Equilateral Triangle

A triangle is a polygon with 3 vertices and 3 sides which makes 3 angles. The total sum of the three angles of the triangle is 180 degrees. Three types of triangles are differentiated based on the length of their vertex.

1. Equilateral Triangle

2. Isosceles triangle

3. Scalene Triangle

Here, we will learn about Isosceles and the Equilateral triangle and their theorem, and based on which we will solve some examples.

Isosceles Triangles

An isosceles triangle is a triangle that has at least two congruent sides. These congruent sides are called the legs of the triangle. The point at which these legs join is called the vertex of the isosceles triangle, and the angle opposite to the hypotenuse is called the vertex angle and the other two angles are called base angles.

Properties of Isosceles Triangle:

1. An isosceles triangle has two equal sides.

2. It has  two equal base angles

3. An isosceles triangle that has 90 degrees is called a right isosceles triangle.

From the properties of the Isosceles triangle, the Isosceles triangle theorem is derived.

Isosceles Triangle Theorem:

1. If two sides of a triangle are congruent, then the corresponding angles are congruent.

2. (Converse) If two angles of a triangle are congruent, then the sides corresponding to those angles are congruent.

Proving of Theorem

Theorem 1: If two sides of a triangle are congruent, then the corresponding angles are congruent

Proof: Assume an isosceles triangle ABC where AC = BC. We need to prove that the angles corresponding to the sides AC and BC are equal, that is, ∠CAB = ∠CBA.

First, we draw a bisector of angle ∠ACB and name it as a CD.

Now in ∆ACD and ∆BCD we have,

AC = BC                                                    (Given)

∠ACD=∠BCD                                                   (By construction)

CD = CD                                                          (Common in both)

Thus, ∆ACD ≅∆BCD                                     (By congruence)

So, ∠CAB = ∠CBA                                          (By congruence)

Theorem 2: (Converse) If two angles of a triangle are congruent, then the sides corresponding to those angles are congruent.

Proof: Assume an Isosceles triangle ABC. We have to prove that AC = BC and ∆ABC are isosceles.

Construct a bisector CD that meets the side AB at right angles.

Now in ∆ACD and ∆BCD we have,

∠ACD=∠BCD                                               (By construction)

CD = CD                                                       (Common in both)

∠ADC = ∠BDC = 90°                                   (By construction)

Thus, ∆ACD ≅ ∆BCD                                   (By ASA congruence)

So, AB = AC                                              (By Congruence) or ∆ABC is isosceles.

Example

Question: Find angle X Solution:

Let triangle be ABC

In ∆ABC

(Image will be uploaded soon)

AB=BC (Given)

So,

∠A=∠C (angle corresponding to congruent sides are equal)

45 degree =∠C

∠A+∠B+∠C=180 degree (Angle sum property)

45 + x +45 =180

X= 180-90 X= 90 degrees.

Facts About Isosceles Triangle

The name of the isosceles triangle was found in the mid-18th century. It was named after the Greek word Isosceles which is made up of two words Isos meaning equal and scales meaning legs. The two words join together and make the definition which means having two sides and equal lengths. This is known as the defining property of an isosceles triangle.

1. Based on sides- Any triangle which has two or more sides equal can be considered as an isosceles triangle. The third side of an isosceles triangle can be shorter or longer than the other two sides, unlike an equilateral triangle which has all sides equal. Generally, when we look at an isosceles triangle the base side of the triangle is the different one and the rest two of the standing sides are similar in size.

2. Based on angles- As we learned above that an isosceles triangle has two equal sides and the base side can be shorter or longer therefore the base angles are always equal. There is another way to find out the equal angle is by looking at the two equal sites which can take place through looking at eyes. All three angles can be the same only in the special case of an equilateral triangle.

3. Symmetry- Unless we are looking at an equilateral triangle, isosceles triangles have one plane symmetry. In case a triangle is not arranged where the shortest or the longest side is placed on the base there is one more way to find out the symmetry by drawing a line from the top of the triangle to the midpoint of the base. There is one more property of an isosceles triangle and that is that the midpoint of the triangle is also the bisector of the angle between the two equal sides and perpendicular bisector of the base.

Uses of Triangle

1. Triangle in architecture- Triangles have a very important role in the field of architecture. It is said that geometry and architecture are two disciplines that are linked fundamentally. The sides and the angles of triangles help the architects to build the blueprint of any building that they are going to construct.

2. Making a rectangle with a triangle- We have studied that a rectangle has four sides and four right angles therefore if we take three triangles and place them together in such a manner that it creates 90 degrees at every corner then we will figure out that it creates a perfect rectangle with bisectors.

3. Identification of triangle- We have studied that there are three types of a triangle and that are isosceles triangle, equilateral triangle, and scalene triangle. Different triangles have different uses but to find the used first we need to figure out which triangle we are looking for and which triangle has the longest sides, equal sides, or all unequal sides.

Equilateral Triangles

In an equilateral triangle, all three sides of the triangle are equal which makes all the three internal angles of the triangle to be equal. An equilateral triangle is also known as an equiangular triangle. Equilateral triangles have unique characteristics. The following characteristics of equilateral triangles are known as corollaries.

Properties of Equilateral Triangle

1. The Equilateral Triangle has 3 equal sides.

2. The Equilateral Triangle has 3 equal angles.

3. The total sum of the interior angles of a triangle is 180 degrees, therefore, every angle of an equilateral triangle is 60 degrees.

4. It is a 3 sided regular polygon.

The following corollaries of equilateral triangles are derived from the properties of an equilateral triangle and the Isosceles triangle theorem.

Isosceles Triangle Theorem:

1.   A triangle is said to be equilateral if and only if it is equiangular.

2.  Each angle of an equilateral triangle is the same and measures 60 degrees each. 

Theorem1: Each angle of an equilateral triangle is the same and measures 60 degrees each.

Proof: Let an equilateral triangle be ABC

AB=AC=>∠C=∠B. --- (1) since angles opposite to equal sides are equal. (Isosceles triangle theorem)

Also, AC=BC=>∠B=∠A   --- (2) since angles opposite to equal sides are equal. . (Isosceles triangle theorem)

From (1) and (2) we have

∠A=∠B=∠C --- (3)

In △ABC,

∠A+∠B+∠C=180 degree (Angle sum property)

=>∠A+∠A+∠A=180 degree

=>∠A=180/3 =60 degree

Therefore, ∠A=∠B=∠C=60 degree

Therefore the angles of the equilateral triangle are 60 degrees each.

Hence Proved

Theorem 2: A triangle is said to be equilateral if and only if it is equiangular.

Proof: Let an equilateral triangle be ABC

AB=AC=>∠C=∠B. --- (1) since angles opposite to equal sides are equal. (Isosceles triangle theorem)

Also, AC=BC=>∠B=∠A   --- (2) since angles opposite to equal sides are equal. . (Isosceles triangle theorem)

From (1) and (2) we have

Therefore, ∠A=∠B=∠C --- (3)

Therefore, an equilateral triangle is an equiangular triangle.

Hence Proved

Solved Example-

Question: show that angles of an equilateral triangle are 60 degrees each

Solution: Let an equilateral triangle be ABC AB=AC=>∠C=∠B. --- (1) since angles opposite to equal sides are equal. (Isosceles triangle theorem)

Also, AC=BC=>∠B=∠A   --- (2) since angles opposite to equal sides are equal. . (Isosceles triangle theorem)

From (1) and (2) we have

∠A=∠B=∠C --- (3)

In △ABC,

∠A+∠B+∠C=180 degree (Angle sum property)

=>∠A+∠A+∠A=180 degree

=>∠A=180/3 =60 degree

Therefore, ∠A=∠B=∠C=60 degree

Difference between Isosceles Triangle, Equilateral Triangle, and Scalene Triangle

1. Equilateral triangles have all equal sides and angles whereas isosceles triangles have only two sides equal and one side can be shorter or longer and a scalene Triangle has all unequal sides.

2. In an equilateral triangle, all the angles are 60 degrees whereas in an isosceles triangle two angles are 60 degrees and in the case of a scalene Triangle there is no certainty of having a 60-degree angle.

3. An equilateral triangle and isosceles triangle can make a rectangle whereas a scalene triangle cannot make a perfect rectangle.

4. Equilateral triangle breaks into 2 words equal and lateral (lateral means legs) whereas isosceles breaks into two words that means two equal sides and one odd size and in the case of a scalene triangle, the word skellen means uneven therefore all sides of this triangle are unequal.

5. Triangles are divided into three types based on their angles and they are acute angle, obtuse angle, and right angle.