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Home >> Triangle >> Properties >> Angles opposite to equal sides of triangle are equal >>

Angles opposite to equal sides of a triangle are equal

Sum of Two Sides Angle Sum Property Angles opposite to equal sides of triangle are equal Angle opposite to longer side is greater Pythagoras Theorem
Exterior Angle Property of a Triangle Mid point property of Triangle Triangles on same base & between same parallel lines

Before you study this property of a triangle, you are advised to read:

What are Congruent Triangles ?

In the below diagram, we have a Triangle ABC which have two equal sides i.e. AC and BC (as highlighted by pink line)



Now, Angle opposite to side AC is Angle B (as shown below):



And Angle opposite to side BC is Angle A (as shown below):



Side AC = Side BC (given)

And as explained in the above diagrams:
Angle opposite to side AC is Angle B and Angle opposite to side BC is Angle A

So, as per the given property, which says that Angles opposite to equal sides of a triangle are equal , we can conclude that:
Angle B = Angle A



How this property is obtained:

In the below diagram, we have a Triangle ABC which have two equal sides i.e. AC and BC (as highlighted by pink line)



Now, draw the bisector of Angle C, which meets side AB at D (as shown below):



Now, let's prove △ ADC ≅ △ BDC

In △ ADC and △ BDC:
AC = BC (given)
Angle ACD = Angle BCD (by construction of bisector)
CD = CD (common sides)

So, by SAS rule of congruency, it proved that
△ ADC ≅ △ BDC

And since corresponding angles of congruent triangles are also equal, so we get:
Angle DAC = Angle DBC

Or we can also write it as:
Angle A (Angle opposite to side BC) = Angle B (angle opposite to side AC)

Hence, this proves the property of triangle : Angles opposite to equal sides of a triangle are equal



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