Hinges Theorem

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: 

Given BC ≅ AD, compare ∠CAD and ∠ACB. 

Solution: 

Given: AB = 5 units, CD = 3 units 

BC ≅ AD  

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.

Concept Map

What have we learned

• Relating side lengths and angle measures.
• Triangle inequality.
• Possible side lengths.