What theorem explains that the sum of lengths of any two sides of a triangle should always be greater than the other third side?

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Geometry is the branch of mathematics that deals with the study of different types of shapes and figures and sizes. The branch of geometry deals with different angles, transformations, and similarities in the figures seen. 

Triangle

A triangle is a closed two-dimensional shape associated with three angles, three sides, and three vertices. A triangle associated with three vertices says A, B, and C is represented as △ABC. It can also be termed as a three-sided polygon or trigon. Some of the common examples of triangles are signboards and sandwiches. 

To prove: The sum of two sides of a triangle is greater than the third side, BA + AC > BC

Assume: Let us assume ABC to be a triangle.

Proof:

Extend the line segment BA to D,

Such that, AD = AC

⇒ ∠ ADC = ∠ ACD

Observing by the diagram, we obtain, 

∠ DCB > ∠ ACD

⇒ ∠ DCB > ∠ ADC

⇒ BD > AB (Since the sides opposite to the larger angle is larger and the sides opposite to smaller angle is smaller)

⇒ BA + AD > BC

⇒ BA + AC > BC.

Hence proved. 

Note: Similarly it can be also proved that, BA + BC > AC or AC + BC > BA

Hence, The sum of two sides of a triangle is greater than the third side.

Question 1. Prove that the above property holds for the lowest positive integral value. 

Solution: 

Let us assume ABC to be a triangle. 

Each of the sides is 1 unit. 

Now, 

It is an equilateral triangle where all the sides are 1 each. 

Taking sum of two sides, 

AB + BC ,

1 + 1 > BC

1+1 > 1 

2 > 1

Question 2. Illustrate this property for a right-angled triangle

Solution: 

Let us assume the sides of the right angles triangle to be 5,12 and 13.

Now, 

Taking the smaller two sides, we obtain, 

5 + 12 > 13

17 > 13

Hence, the property holds. 

Question 3. Does this property hold for isosceles triangles?

Solution: 

Let us assume a triangle with sides 2x, 2x, and x.

Now, 

Taking the sum of equal two sides, we obtain, 

2x + 2x = 4x 

which is greater than the third side, equivalent to x. 

Can a triangle be formed with any three side lengths?  Try it yourself and see.  Grab 3 pencils or pens with different lengths.  Can you put them end to end to form a triangle? In the picture above, a triangle can be formed with the three pencils.  Will this always work?  What if you have two really short pencils and one long pencil? Will the two short ones always be able to reach to form a triangle? In the picture above, you can see that the two short pencils aren't long enough to form a triangle.  There are times when a triangle can't be formed with three given side lengths.  Sometimes the two shortest sides won't be long enough to touch each other to form a triangle.  ​So how do we know if three given side lengths can form a triangle (without physically testing it out)? When you're given three side lengths, imagine the two shortest ones put end to end and the longest side placed directly under them. If the two short ones put together are longer than the longest side, they'll be able to angle up to form a triangle. If you put the two shortest sides end to end and they're not as long as the longest side, they won't be able to reach when you try to form a triangle.

Welcome to Kate's Math Lessons!   Teachers: make sure to check out the study guides and activities.

What if you put the two short ones end to end and they're exactly the length of the longest side? In order to "bump out" to form a triangle, there has to be a little extra room.  If they're just barely touching the ends of the longest side when they're parallel to the longest side, there's no room for them to angle out and form a triangle.  The two shortest sides put end to end must be longer than the longest side.  If you tried to put the two shortest ones at an angle, they would be very close to forming a triangle, but not quite long enough.

The Triangle Inequality Theorem is just a more formal way to describe what we just discovered. We found that when you put the two short sides end to end (that's the sum of the two shortest sides), they must be longer than the longest side (that's why there's a greater than sign in the theorem).

This is just one way to state the Triangle Inequality Theorem.  Another way to state it is to say that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.  

Here's the important thing to remember:  

Determine if the given side lengths can form a triangle:  4, 6, and 8.

First, identify the two shortest sides: 4 and 6.  If you find the sum of the two shortest sides, is it greater than the longest side? 

There's a visual of this below.  If you put the two shortest sides end to end, they will be longer than the longest side.  This means they're long enough to reach when you angle them out to form a triangle.

Determine if the given side lengths can form a triangle:  7, 3, and 2.

​First, identify the two shortest sides: 3 and 2.  If you find the sum of the two shortest sides, is it greater than the longest side?

There's a visual of this below.  If you place the two shortest sides end to end, they aren't long enough to be placed at an angle to form a triangle.

Determine if the given side lengths can form a triangle:  6, 6, and 12.

First, identify the shortest two sides: 6 and 6.  Is the sum of the two shortest sides greater than the longest side?  In this case 6 + 6 is exactly 12.  Does it work when the sum is exactly the same as the longest side?

No, it does not work when the two shortest sides add up to the exact same length as the longest side.  The sum must be larger (not equal to) the longest side.  If it's exactly the same, there's no room to for the shorter two sides to angle up to form a triangle.

Ready to try a few problems on your own? Click the START button below to try a practice quiz.

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