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Home >> Triangle >> Properties >> Angle opposite to longer side is greater >>

Angle opposite to longer side is greater

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

Let's first understand what does this property of triangle explains ?

Observe the following diagram:



AB is the longest side of triangle ABC.

Angle opposite to side longest AB is Angle C (highlighted in pink in the below diagram)



So, as per the given property of triangle, which says, "angle opposite to longer side of a triangle is greater"; we can conclude that:
Angle C is the greatest angle in given triangle ABC (because it's opposite side AB is the longest side).



How to prove this property of triangle ?

Observe the following diagram:



Three sides of the given triangle ABC are AB, BC and CA
On measuring the length of these three sides we get:
AB > CA > BC ..... (Statement 1)

Let's find angles opposite to sides AB, BC and CA:
Angle opposite to side AB is Angle C
Angle opposite to side BC is Angle A
Angle opposite to side CA is Angle B

Now, measure all the angles and we get:
∠ C > ∠ B > ∠ A .....(statement 2)

From statement 1 and 2, we get that:
AB is the longest side and it's opposite Angle C is the greatest angle.

Hence, this proves the property, angle opposite to longer side of a triangle is greater

Also, from statement 1 and 2, we can also conclude:
BC is the smallest side and it's opposite Angle A is the least angle.
And we obtain another property, angle opposite to smaller side of a triangle is lesser

Also, with the above two properties, we can conclude a converse property:
In a triangle, side opposite to greater angle is longer and side opposite to lesser angle is smaller


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