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 states and investigates the triangle inequality theorem
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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