Triangle Inequality Theorem
The
triangle inequality theorem states that the sum of the lengths of any two sides of a triangle
is greater than the length
of the third side. E.g. in the triangle shown in Fig 1: (c + b) > a, (b + a)
> c, (a + c) > b.
For example, Given: with the following measures, can a triangle be formed?
4, 11, and 9 units
6, 19, and 13 units
Notice that in the first case: 4 + 11> 9, 11 + 9> 4 and 9 +4> 13. Therefore,
triangle can be formed. In second case; if you note carefully 9 + 8 = 17 i.e.
measure of third side. Hence the triangle cant be formed.
(More text below video...)
(Continued from above)angleside relationship: In a triangle, the side opposite the larger angle is the longer side.
For example, Given: In
ABC (Figure below),
A
= 35
and B =
60,
find out the length of the sides.
First find out the measure of the third angle. You know that sum of three angles in any triangle is
180
i.e.
A +
B +
C =
180.
By substituting values for A and B, we get
C =
85.
Now find out the length: From the angle measures,
A
< B
< C.
Considering angle side relationship, BC < AC <
AC.
This is the final answer.
Inequalities in two triangles:
The theorems on inequalities in two triangles are stated below:
If two triangles have two congruent sides, the triangle with the larger third side will have a larger included angle.
If two triangles have two congruent sides, the triangle with the larger included angle will have a larger third side. (Figure below):
Remember:
if two sides of a triangle are unequal, then the greater angle is opposite to the greater side. If two angles of a triangle are unequal, the greater side is opposite to the greater angle.
of all the segments that can be drawn to a given straight line from a point outside it, the perpendicular is the shortest
in a right triangle, hypotenuse is the longest side.
sum of any two sides of a triangle is greater than the third side
the difference between any two sides of a triangle is less than the third side.
The video above will explain more details about Triangle Inequality
and relationship of the involved properties, with the help of several examples and practice problems.
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