#### Need Help?

Get in touch with us

# Hinges Theorem

Sep 12, 2022

## Key Concepts

• Hinge theorem.
• Converse of hinge theorem.
• Indirect Proof.

## Introduction to Hinges theorem

Imagine a gate with two doors, the doors can be opened wider.

The doors form a triangle on opening from the hinges side, and the swing side forms different triangles by moving the door wider.

The wider the door moves; the angle and length also increases.

### Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle and the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.

### Converse of hinge theorem

If two sides of one triangle are congruent to two sides of another triangle, the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.

Example 1:

Solution:

Given: AB = 5 units, CD = 3 units

And we know that AC ≅ AC, by reflexive property.

Also, AB > CD

Two sides of ∆ABC are congruent to two sides of ∆ADC and the third side of ∆ABC is longer than the third side of ∆ADC.

Therefore, by the converse of the hinge theorem, m ∠ACB > m ∠CAD

### Indirect Reasoning

Suppose, a student looks around the gate of the school and concludes that ice creams were not sold.

At first, he assumed that ice creams were served that day as that day is Saturday.

So, he looked to the right side of the gate when ice creams are sold there will be a counter.

As there was no counter so the thing he assumed, that ice creams were sold was false.

This is called indirect reasoning.

### Indirect Proof

In indirect proof, we assume a temporary contradictory statement.

This leads to the logical impossibility that our assumption is false.

This states that the given statement is true.

### Procedure to write indirect proof

#### How to Write an Indirect Proof?

STEP 1: Identify the statement that we need to prove. Assume that this statement is false temporarily, by assuming that its opposite is true.

STEP 2: Reason logically until you reach a contradiction.

STEP 3: Conclude that the given statement must be true because the contradiction proves the temporary assumption false.

Example 1:

Write an indirect proof that a prime number is not divisible by 4.

(Given that x is a prime number.

We need to prove x is not divisible by 4.)

Solution:

STEP 1

Assume temporarily that x is divisible by 4. This means that x/4=n for some whole number n. So, multiplying both sides by 4 gives x = 4n.

STEP 2

If x is prime, then, by definition, x cannot be divided evenly by 2.

However, x = 4n.

We know prime numbers are: 2, 3, 5, 7 … (except 2 all are odd numbers, only 2 is even prime) but 2 is not divisible by 4 as 2< 4

This contradicts the given statement that x is prime.

STEP 3 Therefore, the assumption that x is divisible by 4 must be false, which proves that x is not divisible by 4.

Example 2:

Write an indirect proof of the converse of the hinge theorem.

Proof:

Given GJ =KM, HJ = LM, GH > KL

We need to prove that m ∠J > m ∠M.

Assume temporarily m ∠J ≯ m ∠M, then it follows either m ∠J = m ∠M, m ∠J < m ∠M.

Case 1: m ∠J = m ∠M, then ∆ GHJ ≅ ∆KLM by SAS congruence postulate and GH = KL.

Case 2:m ∠J < m ∠M, then by hinge theorem GH < KL.

Both the cases contradict the given statement GH > KL.

So, our temporary assumption is false.

This proves that m ∠J > m ∠M.

### Real-Life Examples

The hinge theorem is used in real-life situations such as trap door openings, surveying, transportation, and urban planning. The hinge theorem is also called the alligator hinge. As the jaws of alligators are fixed, the angle is increased when the prey size is increased.

Example:

Two people Jack, and Jim came to a park and opened the two gates, Jack pushed around the gate by

40° and Jim pulled around the gate by 55°. Both the gates are of the same width, who opened the gate farther from the starting point?

Solution:

Given, two people Jack and Jim came to a park and opened the two gates, Jack pushed around the gate by

40° and Jim pulled around the gate by 55°. Both the gates are of the same width.

We draw a diagram to find the angles, then

We use the linear pair of angles theorem to find the angles to find the included angles.

By the hinge theorem, we can conclude that the distance from O to Jack is more than the distance from point O to Jim.

So, Jack opened farther from the starting point O.

## Exercise

• Indirect proof is also called ____

Questions (2 and 3)

In the given figure AB = CD

• .If AD > BC then,
• If ∠CAB > ∠DCA then,
• What is the temporary assumption used to prove that “If a + b ≠ 18, and a = 16, then b ≠ 2”?

Questions (5 – 10)

Use the Hinge theorem and complete the relationships in the given questions.

### What have we learned

#### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

#### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

#### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division  A → Addition S → Subtraction         Some examples […]