Question

Is there inequality property for angles in triangles?

Answer 1

Answer:

If two sides of a triangle are unequal, the angle opposite to the longer side is greater than others.

For proving this point we need to do an activity. On a drawing board fix three pins at points PQR. A line segment is drawn at PQ. Taking P as center and some radius we draw an arc at A. Similarly with different radius we draw a few more arcs at point B, C, and D.

On joining these points with P as well as B we observe that as we move from A to D, the ∠P is becoming larger with every arc. Now, what happens to the side opposite to the angle. We observe here that the length of the side is also increasing.

Let’s take another triangle, which seems scalene in appearance. A scalene has all sides of different length. On measuring the length of the sides in a scalene triangle, we come to a conclusion that angle opposite to the longer side is the greatest while the angle opposite to the shortest side is the smallest.

Step-by-step explanation:

Solved Example for You on Triangle Inequality

X is a point on side QR of ΔPQR such that PX=PR. Show that PQ>PX

Solution: The figure shown above is of ΔPQR, with a point X on the line QR We join P with X to form a line segment PX. In ΔXPR, we see that PX = PR, which is given in the question. This gives us that ∠PXR = ∠PRX because angles opposite to equal sides are also equal.

Figure: (See attached image)

Now, ∠PXR is an exterior angle for ΔPQX. So we have that ∠PXR > ∠PQX This can also be written as

∠PRX > ∠PQX

or, ∠PRQ > ∠PQR

So, PQ>PR as the side opposite to larger angle is longer.

or, PQ>PX

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