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Geometric Inequalities

Geometric Inequalities

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

What are Geometric Inequalities?

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!

Basic Geometrical Inequality Postulates

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.

Comparison Postulate

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.

Geometrical Inequalities Comparison Postulate StudySmarterLine segments used to illustrate Comparison Postulate - StudySmarter Originals

We can say

Transitive Property

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.

Geometrical Inequalities Transitive Property StudySmarterTriangle used to illustrate Transitive Property - StudySmarter Originals

Using the image above, the transitive property states that:

if and , then

Substitution Postulate

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.

Geometrical Inequalities Substitution Property StudySmarterTriangle used to illustrate Substitution Postulate - StudySmarter Originals

Using the figure above, the substitution postulate states that:

if and , then

Addition Postulate

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

Geometrical Inequalities Addition Postulate Study SmarterLine Segment - StudySmarter Originals

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.

Geometrical Inequalities Addition Postulate StudySmarterTwo line segments - StudySmarter Originals

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.

Geometrical Inequalities Addition Postulate StudySmarterTwo line segments - StudySmarter Originals

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

If and , then or

Subtraction Postulate

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.

Geometrical Inequalities Subtraction Postulate StudySmarterTriangle used to illustrate Subtraction Postulate - StudySmarter Originals

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

If and , then or

Multiplication Postulate

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

Geometrical Inequalities Multiplication Postulate StudySmarterLine segments - StudySmarter Originals

If , , and , then

Division Postulate

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

Geometrical Inequalities Division Postulate StudySmarterTriangle used to illustrate Division Postulate - StudySmarter Originals

If , then

Geometric Inequality Theorems

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

  1. Triangle Inequality Theorem.

  2. Pythagorean Inequality Theorem.

  3. Exterior Angle Inequality Theorem.

  4. Greater Angle Theorem.

  5. Longer Side Theorem.

Let's take a deeper look using some examples.

Triangle Inequality Theorem

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

Geometrical Inequalities Triangle Theorem StudySmarterA Triangle - StudySmarter Originals

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.

Geometrical Inequalities Number line inequality StudySmarter

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,

Pythagorean Inequality Theorem

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.

Geometrical Inequalities Pythagorean inequality theorem StudySmarterA Triangle - StudySmarter Originals

If , then the triangle is a right triangle.

If , then the triangle is acute.

If , then the triangle is obtuse.

Exterior Angle Inequality Theorem

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:

Geometrical Inequalities Exterior angle inequality theorem StudySmarterA Triangle with an exterior angle - StudySmarter Originals

Greater Angle Theorem

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

Geometrical Inequalities Greater angle theorem StudySmarterA Triangle - StudySmarter Originals

Let's see an example.

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

Geometrical Inequalities Greater Angle Theorem Examples StudySmarter

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,

Longer Side Theorem

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,

If , then

Geometrical Inequalities Longer side theorem StudySmarterA Triangle - StudySmarter Originals

Let's take a look at a quick example.

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

Geometrical Inequalities Longer side theorem example StudySmarter

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 - Key takeaways

  • 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

Frequently Asked Questions about Geometric Inequalities

Geometric inequality refer to the relationship between geometric quantities and measurements that are unequal.

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

The triangle theorem states that the length of one side of a triangle is smaller than the sum of the other two sides. 

Final Geometric Inequalities Quiz

Question

What are geometric inequalities?

Show answer

Answer

Geometrical inequalities refer to the relationship between geometrical quantities and measurements that are unequal.

Show question

Question

List some of the basic inequality postulates.

Show answer

Answer

  • Comparison Postulate.
  • Transitive Property.
  • Substitution Postulate.
  • Addition Postulate.
  • Subtraction Postulate.
  • Multiplication Postulate.
  • Division Postulate.

Show question

Question

The basic inequality postulates are the same as the properties of geometric inequalities.

Show answer

Answer

True

Show question

Question

List some of the geometric inequality theorems.

Show answer

Answer

  • Triangle Inequality Theorem.
  • Exterior Angle Inequality Theorem.
  • Greater Angle Theorem.
  • Longer Side Theorem.

Show question

Question

What is the Triangle Inequality Theorem in geometry?

Show answer

Answer

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

Show question

Question

What is the Exterior Angle Inequality Theorem?

Show answer

Answer

The Exterior Angle Inequality Theorem states that the measure of the exterior angle of a triangle is greater than the non-adjacent interior angles of the triangle.

Show question

Question

What is the Pythagorean Inequality Theorem?

Show answer

Answer

The theorem states that a triangle is a right triangle if the square of the longest side is equal to 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.

Show question

Question

What is the Greater Angle Theorem?

Show answer

Answer

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

Show question

Question

Which of the following theorems tell you if a triangle is right, obtuse or acute?

Show answer

Answer

Pythagorean Inequality Theorem.

Show question

Question

What is the Exterior Angle Inequality Theorem?

Show answer

Answer

The Exterior Angle Inequality states that the measure of the exterior angle of a triangle is greater than the non-adjacent interior angles of the triangle. 

Show question

Question

The comparison postulate states that a number or quantity can be substituted for its equal in an inequality. 

Show answer

Answer

False

Show question

Question

What are the theorems used in solving two triangle inequalities?

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Answer

  • Side-Angle-Side (SAS) Inequality Theorem or Hinge Theorem.
  • Side-Side-Side (SSS) Inequality Theorem.

Show question

Question

State the SAS Inequality Theorem.

Show answer

Answer

The SAS Inequality Theorem states that if two sides of a triangle are congruent to two sides of another triangle, and the measure of the included angle in the first triangle is larger than the measure of the included angle in the second triangle, then the third side of the first triangle is longer than the third side of the second triangle. 

Show question

Question

State the SSS Inequality Theorem.

Show answer

Answer

 The SSS Inequality Theorem states that if two sides of a triangle are congruent to two sides of another triangle and the third side of the first triangle is longer than the other, then the measure of the included angle in the first triangle is larger than the measure of the included angle in the second triangle. 

Show question

Question

What is another name for the SAS Inequality Theorem?

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Answer

The Hinge Theorem.

Show question

Question

What is does the SSS acronym stand for?

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Answer

Side-Side-Side.

Show question

Question

What does the SAS acronym stand for?

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Answer

Side-Angle-Side.

Show question

Question

If the measure of the included angle in one triangle is larger than the measure of the included angle in another triangle, then the third side of the first triangle is longer than the third side of the second triangle. 

True or False.

Show answer

Answer

True

Show question

Question

If the third side of one triangle is longer than the third side of another triangle, then the measure of the included angle in the first triangle is larger than the measure of the included angle in the second triangle.

True or False.

Show answer

Answer

True.

Show question

Question

How many possible ways are there to prove the SAS inequality theorem?

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Answer

2

Show question

Question

What does the triangle inequality theorem say?

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Answer

The triangle inequality theorem says that the sum of two sides of a triangle is greater than the third side of the triangle. 

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Question

Using algebra, how can the triangle inequality theorem be written?

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Answer

\[ \begin{align} & |AC| + |BC| > |AB| \\ & |AB| + |AC| > |BC| \\ & |AB| + |BC| > |AC|. \end{align}\]

Show question

Question

Describe the interior angles of the triangle, and name one thing that is true about them.

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Answer

The interior angles are the ones inside the triangle, and their measures add up to \(180^\circ \).

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Question

What does the angle side theorem tell you?

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Answer

The Angle Side theorem says that if one side is longer than another, then their angle opposite the longest side is bigger than the angle opposite the shorter side.

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Question

What are three ways to write the measure of an angle?

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Answer

  • \(m (\angle A)\) 

  • \(\text{meas }\angle A\), or 

  • \(\measuredangle A\)


Show question

Question

What kind of proof do you use to prove the triangle inequality theorem?

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Answer

A geometric proof.

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Question

What does it mean if two triangle sides are congruent? 

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Answer

If two triangle sides are congruent it means that they are equal in length, this means that the angles opposite will also be congruent and equal in size. 

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Question

A triangle has the side lengths \(2\) and \(5\), find the possible lengths of the third side of the triangle.

Show answer

Answer

Using the triangle inequality theorem, the third side is between \(3\) and \(7\) units long.

Show question

Question

What is a triangle inequality?

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Answer

A triangle inequality is an inequality that is true about any kind of triangle.

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Question

What is the triangle inequality theorem?

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Answer

For any triangle, if you add up the length of any two sides, it will be larger than the length of the remaining side.

Show question

Question

If you were told that a triangle has the side lengths \(18\) and \(9\), what would you use to find the possible lengths of the third side of the triangle?

Show answer

Answer

The triangle sides and angles inequality theorem.

Show question

Question

What is the exterior angle triangle inequality?

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Answer

The exterior angle triangle inequality says that the measure of an exterior angle is bigger than the measure of either of its two remote interior angles.

Show question

Question

What theorem tells you that for any triangle, if you add up the length of any two sides, it will be larger than the length of the remaining side?

Show answer

Answer

This is the triangle inequality theorem.

Show question

Question

What theorem tells you that for any triangle, if one side is longer than another, then their angle opposite the longest side is bigger than the angle opposite the shorter side? 

Show answer

Answer

This is the angle side triangle theorem.

Show question

Question

What theorem tells you that the measure of an exterior angle is bigger than the measure of either of its two remote interior angles?

Show answer

Answer

This is the exterior angle triangle inequality.

Show question

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