Question

E
Draw two different triangles and measure their sides. Is the sum of any
shorter sides greater than the third side?​

Answer 1

Answer:

Given a triangle ABC, the sum of the lengths of any two sides of the triangle is greater than the length of the third side.

In the words of Euclid:

In any triangle two sides taken together in any manner are greater than the remaining one.

(The Elements: Book I: Proposition 20)

Proof

Triangle Inequality.png

Let ABC be a triangle

We can extend BA past A into a straight line.

There exists a point D such that DA=CA.

Therefore, from Isosceles Triangle has Two Equal Angles:

∠ADC=∠ACD

Thus by Euclid's fifth common notion:

∠BCD>∠BDC

Since △DCB is a triangle having ∠BCD greater than ∠BDC, this means that BD>BC.

But:

BD=BA+AD

and:

AD=AC

Thus:

BA+AC>BC

A similar argument shows that AC+BC>BA and BA+BC>AC.

Answer 2

Answer:

no mate it's not possible......