A triangle is a polygon through 3 vertices and also 3 political parties which makes 3 angles .The full sum the the 3 angles the the triangle is 180 degrees. There are three types of triangle which room differentiated based on length of your vertex.

You are watching: If a triangle is equilateral then it is isosceles

Equilateral Triangle

Isosceles triangle

Scalene Triangle

In this short article we will certainly learn about Isosceles and the it is intended triangle and their theorem and based top top which we will resolve some examples.

Isosceles Triangles

An isosceles triangle is a triangle which has at least 2 congruent sides. This congruent sides are calledthe legs of the triangle. The point at which this legs join is referred to as thevertexof the isosceles triangle, and the edge opposite to the hypotenuse is dubbed thevertex angle and the various other two angles room calledbase angles.

Properties of Isosceles Triangle:

Isosceles triangle has actually two same sides.

It has actually two equal base angles

An isosceles triangle which has actually 90 levels is dubbed a best isosceles triangle.

From the nature of Isosceles triangle, Isosceles triangle organize is derived.

Isosceles Triangle Theorem:

If two sides the a triangle room congruent, then the matching angles are congruent.

(Converse) If 2 angles that a triangle space congruent, climate the sides corresponding to those angles are congruent.

Proving of Theorem

Theorem 1:If two sides of a triangle are congruent, climate the equivalent angles space congruent

Proof:Assume an isosceles triangle ABC where AC = BC. We have to prove that the angles corresponding to the political parties AC and BC space equal, that is, ∠CAB = ∠CBA.

First we attract a bisector of edge ∠ACB and also name it together CD.

Now in ∆ACD and also ∆BCD us 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 space congruent, then the sides matching those angles space congruent

Proof:Assume an Isosceles triangleABC. We need to prove that AC = BC and ∆ABC is isosceles.

Construct a bisector CD i beg your pardon meets the side abdominal at ideal angles.

Now in ∆ACD and ∆BCD us have,

∠ACD = ∠BCD (By construction)

CD = CD (Common in both)

∠ADC = ∠BDC = 90° (By construction)

Thus, ∆ACD ≅ ∆BCD (By ASA congruence)

So, abdominal = AC (By Congruence)

Or ∆ABC is isosceles.

Example

Question: uncover angle X

Solution: let triangle be ABC

In ∆ABC

AB=BC (Given)

So,

∠A=∠C (angle matching to congruent sides space equal)

45 degree =∠C

∠A+∠B+∠C=180 level (Angle amount property)

45 + x +45 =180

X= 180-90

X= 90 degrees.

Equilateral Triangles

In it is provided triangle all three sides of the triangle are equal which provides all the three internal angles the the triangle to be equal. It is provided triangle is additionally known as an equiangular triangle. Equilateral triangles have distinct characteristics. The following qualities of equilateral triangle are known as corollaries.

Properties of it is provided Triangle.

The it is provided Triangle has actually 3 equal sides.

The it is intended Triangle has 3 equal angles.

The total sum of the interior angles that a triangle is 180 degrees, therefore, every angle of an it is provided triangle is 60 degrees.

It is a 3 sided consistent polygon.

The adhering to corollaries the equilateral triangle are obtained from the properties of it is intended triangle and also Isosceles triangle theorem.

Isosceles Triangle Theorem:

A triangle is stated to be equilateral if and only if the is equiangular.

Each edge of an it is provided triangle is the same and also measures 60 degrees each.

Theorem1: every angle that an it is provided triangle is the same and also measures 60 levels each.

Proof: allow an it is intended triangle be ABC

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

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

From(1)and(2)we have

∠A=∠B=∠C--- (3)

In△ABC,

∠A+∠B+∠C=180 level (Angle amount property)

=>∠A+∠A+∠A=180 degree

=>∠A=180/3 =60 degree

Therefore, ∠A=∠B=∠C=60 degree

Therefore the angles of the it is provided triangle space 60 levels each.

Hence Proved

Theorem 2: A triangle is claimed to it is in equilateral if and only if that is equiangular.

Proof: let an it is intended triangle it is in ABC

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

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

From(1)and(2)we have

Therefore, ∠A=∠B=∠C--- (3)

Therefore, an it is intended triangle is an equiangular triangle

Hence Proved

Solved Example-

Question: display that angle of equilateral triangle room 60 degree each

Solution: allow an equilateral triangle be ABC

AB=AC=>∠C=∠B. --- (1) due to the fact that angles the contrary to same sides space equal. (Isosceles triangle theorem)

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