Inequalities in a Triangle

Inequalities in a Triangle

So far, you have been mainly studying the equality of sides and angles of a triangle or triangles. Sometimes, we do come across unequal objects, we need to compare them. For example, line-segment AB is greater in length as compared to line segment CD (i) and ∠ A is greater than ∠ B

 Let us now examine whether there is any relation between unequal sides and unequal angles of a triangle. For this, let us perform the following activity:
Activity: Fix two pins on a drawing board say at B and C and tie a thread to mark a side BC of a triangle.
Fix one end of another thread at C and tie a pencil at  the other (free) end . Mark a point A with the pencil and draw Δ ABC. Now, shift the pencil and mark another point A′ on CA beyond A (new position of it)

So, A′C > AC (Comparing the lengths) Join A′ to B and complete the triangle A′BC.
What can you say about ∠ A′BC and ∠ ABC? Compare them. What do you observe?
Clearly, ∠ A′BC > ∠ ABC
Continue to mark more points on CA (extended) and draw the triangles with the side BC and the points marked.
You will observe that as the length of the side AC is increased (by taking different positions of A), the angle opposite to it, that is, ∠ B also increases.
Let us now perform another activity:
Activity: Construct a scalene triangle (that is a triangle in which all sides are of different lengths). Measure the lengths of the sides.
Now, measure the angles.  What do you observe?

 In Δ ABC of BC is the longest side and AC is the shortest side.
Also, ∠ A is the largest and ∠ B is the smallest.
Repeat this activity with some other triangles.
We arrive at a very important result of inequalities in a triangle. It is stated in the form of a theorem as shown below: