The sum of any two sides of a triangle is ... than the third side

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We can extend BA past A into a straight line.Then there is a point D such that DA = CATherefore, from isosceles triangle has two equal angles,$\angle ADC = \angle ACD$Thus, in $\Delta DCB,$$\angle BCD > \angle BDC{\text{ }}\left[ {{\text{side opposite greater angle is larger}}} \right]$Thus, $ BD > CD \\ BA + AD > CD{\text{ }}\left[ {{\text{since, AC = AD}}} \right] \\ BA + AC > CD \\ $The sum of any two sides of the triangle is greater than the third side and the correct option is “A”.Note- In order to solve these types of questions, remember the theorems and properties of triangles. Also the concept of parallel line and angles. To prove the above theorem we use the isosceles triangle. We can also take an equilateral triangle to prove as we know that all sides of the equilateral triangle are equal therefore the sum of two sides is twice the third side. Hence, the sum of two sides of the triangle is greater than the third side.

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Here we will prove that the sum of any two sides of a triangle is greater than the third side.

Given: XYZ is a triangle.

To Prove: (XY + XZ) > YZ, (YZ + XZ) > XY and (XY + YZ) > XZ

Construction: Produce YX to P such that XP = XZ. Join P and Z.

Statement

1. ∠XZP = ∠XPZ.

2. ∠YZP > ∠XZP.

3. Therefore, ∠YZP > ∠XPZ.

4. ∠YZP > ∠YPZ.

5. In ∆YZP, YP > YZ.

6. (YX + XP) > YZ.

7. (YX + XZ) > YZ. (Proved)

Reason

1. XP = XZ.

2. ∠YZP = ∠YZX + ∠XZP.

3. From 1 and 2.

4. From 3.

5. Greater angle has greater side opposite to it.

6. YP = YX + XP

7. XP = XZ

Similarly, it can be shown that (YZ + XZ) >XY and (XY + YZ) > XZ.

Corollary: In a triangle, the difference of the lengths of any two sides is less than the third side.

Proof: In a ∆XYZ, according to the above theorem (XY + XZ) > YZ and (XY + YZ) > XZ.

Therefore, XY > (YZ - XZ) and XY > (XZ - YZ).

Therefore, XY > difference of XZ and YZ.

Note: Three given lengths can be sides of a triangle if the sum of two smaller lengths greater than the greatest length.

For example: 2 cm, 5 cm and 4 cm can be the lengths of three sides of a triangle (since, 2 + 4 = 6 > 5). But 2 cm, 6.5 cm and 4 cm cannot be the lengths of three sides of a triangle (since, 2 + 4 ≯ 6.5).

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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.

Sample Questions

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.

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Sum of length of any two sides of a triangle is always equal to the third side.

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