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Geometric Inequalities

- Calculus
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Inequality is a big problem in the world today. We hear about gender inequality, ethnicity inequality, and many more. When we talk about inequalities, we refer to the **unequal relationship** between groups or quantities.

In Math, it is the unequal relationship between two numbers or expressions. In Geometry, we look at the unequal relationships **between** **side lengths and between angles** in various shapes.

In this article, we are going to look at:

Geometric Inequalities

Geometric Inequality Postulates

Geometric Inequality Theorem

**Geometrical inequalities** refer to the unequal relationship among angles and side lengths in various shapes.

Let's recall the symbols used to represent inequalities and what they mean:

There are some general rules, or **postulates**, that can help us make comparisons about unequal side lengths and unequal angles. Let's look at them!

Various inequality postulates and properties are used when dealing with inequalities. Let's take a look at them.

Comparison Postulate

Transitive Postulate

Substitution postulate

Addition postulate

Subtraction postulate

Multiplication postulate

Division postulate

These postulates can also be applied when dealing with geometry. Let's look at the various postulates and see how they can be applied to geometry.

What is the difference between a postulate and a theorem, you may wonder? A **theorem** is a statement that can be proven to be true, but a **postulate** is a statement that is said to be true without proof.

This postulate states that a whole is greater than each of its parts and the sum of the parts is equal to the whole. So, if a, b, and c are positive numbers and , then and . Let's see how we can apply this to geometry.

Using the image below, the Comparison postulate states, assuming , and , then

We can also apply the Comparison postulate to angles. Let's look at the figure below.

We can say

The transitive property explains that if you have three real numbers, a, b, and c, and the first of the three numbers is greater than the second and the second number is greater than the third, then the first is also greater than the third. In other words, if and then, . Let's apply it to geometry using the geometric figure below.

Using the image above, the transitive property states that:

if and , then

This postulate states that in an inequality, we can swap a number of equal value in. This means if you have three real numbers a, b and c and if and , then .

Let's apply this to geometry using the figure below.

Using the figure above, the substitution postulate states that:

if and , then

Imagine you have a line segment as shown below, and somewhere on the line, you have another point . The point can only exist if .

The addition postulate has two different types.

1. The first one states that if equal quantities are added to unequal quantities, the sum is unequal in the same order.

For the figure above, the postulate is as follows when equal quantities are added to unequal quantities.

If and , then or .

2. The second one states that if unequal quantities are added to unequal quantities then the sum is unequal.

For the figure above, the postulate is as follows when unequal quantities are added to unequal quantities.

If and , then or

The subtraction postulate states that if equal quantities are subtracted from unequal quantities, then the differences are unequal in the same order. Consider the figure below.

Using the figure above, the subtraction postulate is as follows.

If and , then or

The postulate states that if unequal quantities are multiplied by equal positive quantities, the products are unequal in the same order.

If , , and , then

The postulate states that if unequal quantities are divided by equal positive quantities, the quotients are unequal in the same order.

If , then

There are various geometric inequalities theorems, and they are listed below.

Triangle Inequality Theorem.

Pythagorean Inequality Theorem.

Exterior Angle Inequality Theorem.

Greater Angle Theorem.

Longer Side Theorem.

Let's take a deeper look using some examples.

Triangle Inequality Theorem states that the length of one side of a triangle is smaller than the sum of the other two sides.

Considering the figure above, the triangle theorem states that . Let's take an example.

In , and , find the range of possible values of .

Step 1: Using the Triangle Inequality Theorem,

Step 2: Subbing in values from above:

This means that is less than 18. But we can't say is any number less than 18 because any number less than 18 may be too small to even make or complete the triangle.

Step 3: Let's write two more inequalities using the Triangle Inequality Theorem.

Lastly,

So far, we have three different inequalities.

Step 4: The actual range will be the intersection of the three inequalities, and we can get this using a number line.

From the number line, we can tell that the intersection is between 2 and 18. This means that the range of values is between 2 and 18. Therefore,

The Pythagorean Inequality Theorem tells you whether a triangle is a right triangle, an obtuse triangle, or an acute triangle.

The theorem states that you have a right triangle if the square of the longest side equals the sum of the squares of the other two sides. If the square of the longest side is less than the sum of the squares of the other two sides, the triangle is acute. If the square of the longest side is greater than the sum of the square of the other two sides, then the triangle is obtuse.

In below, C is the longest side.

If , then the triangle is a right triangle.

If , then the triangle is acute.

If , then the triangle is obtuse.

The theorem states that the measure of the exterior angle of a triangle is greater than the non-adjacent interior angles of the triangle. Take a look at the figure below.

In the figure below, the exterior angle is **, **and angle and are the non-adjacent interior angles. Using the Exterior Angle Inequality Theorem, we can conclude that:

The Greater Angle Theorem states that if the measure of one angle in a triangle is greater than the measure of another, then the side opposite the greater angle is longer than the side opposite the smaller angle. In other words

In the below,

if , then

Let's see an example.

Use the greater angle theorem to determine which side is the longest.

From the Greater Angle Theorem, we know that the longest side is the side opposite the largest angle. In the figure above, the longest side is side because it is the side opposite the largest angle,

The Longer Side Theorem says that if the length of one side of a triangle is longer than another side, the angle opposite the longer side is greater than the angle opposite the shorter side. This is the **converse** of the Greater Angle Theorem above. In the below,

Let's take a look at a quick example.

Use the longer side theorem to determine which angle is greatest.

From the theorem, we know that the angle opposite the longest side is the greatest angle. The smallest side is 6 and the opposite angle is. Next to the smallest side is 8 and the opposite angle is. The longest side is 10 and the opposite angle is.

So, according to the length of the sides, the greater angle is .- Geometrical inequalities refer to the unequal relationships between side lengths and between angles in various shapes
- The postulates of inequalities are the same as their properties. The postulate includes:
- Comparison Postulate.
- Transitive Postulate.
- Substitution postulate.
- Addition postulate.
- Subtraction postulate.
- Multiplication postulate.
- Division postulate.

- Geometric Inequality Theorems include:
- Triangle Inequality Theorem
- Pythagorean Inequality Theorem
- Exterior Angle Inequality Theorem
- Greater Angle Theorem
- Longer Side Theorem

Geometric inequalities are solved by applying the appropriate postulate and theorem.

More about Geometric Inequalities

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