What theorem states that if 2 sides of a triangle are not congruent then the larger angle is opposite the larger side?

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The following diagrams show the Triangle Inequality Theorem and Angle-Side Relationship Theorem. Scroll down the page for examples and solutions.

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.

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.

Example 1: Find the range of values for s for the given triangle.

Solution:
Step 1: 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

What is the Triangle Inequality Theorem? The following video states and investigates the triangle inequality theorem. The sum of lengths of any two sides of a triangle must be greater than the length of the third.

We really only need to make sure the sum of the lengths of the two shorter sides is greater than the length of the longest side.

Examples:
Which lengths can form a triangle?

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Description of the Triangle Inequality
The following video describes triangle inequality by trying to construct triangles with different length segments.

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What are the conditions required to draw a triangle to illustrate triangle inequality?

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Intuition behind the triangle inequality theorem

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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.

Solution:
Step 1: 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.

Answer:

Examples of the angle-side relationships in triangles

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Angle side relationships in Triangles If 2 sides of a triangle are not congruent, then the larger angle is opposite the larger side. If 2 angles of a triangle are not congruent, then the larger side is opposite the larger angle.

The measure of Angle A is greater than the measure of Angle B, and the measure of Angle B is greater than the measure of Angle C. Find the possible values for the length of side AC.

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By theorem, we need to prove that angle C > angle B.Now, take a point P on line AB such that AP = AC. Join the two points C and P to set CP. Let us assume angle APC is \[{{x}^{o}}\] and angle BCP is \[{{y}^{o}}.\]

We know that AP = AC. If two sides are equal in a triangle, then the angles opposite to the sides are equal. Applying this condition to the triangle APC, we get the angle APC and angle ACP, all equal. As angle APC is \[{{x}^{o}},\] we get the angle ACP also \[{{x}^{o}}.\] Mathematically, we write these steps as follows:\[\angle ACP=\angle APC\]\[\angle ACP={{x}^{o}}\]On line AB, we can say that it is a straight line which is equal to \[{{180}^{o}}.\] So, we get, \[\angle APC+\angle CPB={{180}^{o}}\]By substituting the value of angle APC, we get the equation as, \[x+\angle CPB={{180}^{o}}\]By subtracting x on both the sides of the equation, we get it as, \[\angle CPB={{180}^{o}}-x\]We know the theorem that sum of all the angles in a triangle is \[{{180}^{o}}.\] By applying this to the triangle CPB, we get it as, \[\angle CBP+\angle BPC+\angle PCB={{180}^{o}}\]By substituting the value of BPC from the above equation, we get, \[\angle CBP+{{180}^{o}}-x+\angle PCB={{180}^{o}}\]By substituting the value of PCB as per our assumption, we get, \[\angle CBP+{{180}^{o}}-x+y={{180}^{o}}\]By canceling out the common terms, we can write it as, \[\angle CBP-x+y=0\]By adding x and subtracting y on both the sides, we get,\[\angle CBP=x-y\]From the diagram, we can say that, \[\angle CBP=\angle CBA\]\[\angle CBA=x-y.....\left( i \right)\]From the diagram, we can say that,\[\angle BCA=\angle BCP+\angle PCA\]By substituting the values, we get it in the form of \[\angle BCA=x+y....\left( ii \right)\]From (i) and (ii), we can say the inequality in the form of \[\angle BCA>\angle CBA\]\[\Rightarrow \angle C>\angle B\]Hence, we have proved that the angle opposite to the greater side is larger.Note: Be careful while marking the angles in the diagram. As when you miss any angle, you may not get the idea to proceed. The idea of making the angles as sum, the difference of angles is very important. This is derived from a small idea that the sum is greater than the difference if the numbers are positive. Here, the angles are positive.