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.
For the above triangle:
a + b > c
b + c > a
a + c > b
The Converse of the Triangle Inequality theorem states that
It is not possible to construct a triangle from three line segments if any of them is longer than the sum of the other two.
Step1: Using the triangle inequality theorem for the above triangle gives us three statements:
s + 4 > 7 ⇒ s > 3
s + 7 > 4 ⇒ s > –3 (not valid because lengths of sides must be positive)
7 + 4 > s ⇒ s < 11
Step 2: Combining the two valid statements:
3 < s < 11
Answer: The length of s is greater than 3 and less than 11
The following video describes triangle inequality by trying to construct triangles with different length segments.
The Angle-Side Relationship states that
In a triangle, the side opposite the larger angle is the longer side.
In a triangle, the angle opposite the longer side is the larger angle.
Example 1: Compare the lengths of the sides of the following triangle.
Step1: We need to find the size of the third angle. The sum of all the angles in any triangle is 180º.
∠A + ∠B + ∠C = 180°
⇒ ∠A + 30° + 65° = 180°
⇒ ∠A = 180° - 95°
⇒ ∠A = 85°
Step 2: Looking at the relative sizes of the angles.
∠B < ∠C < ∠A
Step 3: Following the angle-side relationship we can order the sides accordingly. Remember it is the side opposite the angle.
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