mathematics

Hinge theorem

Hinge theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hinge theorem rather than just read about it. In short: In geometry, the hinge theorem (sometimes called the open mouth theorem) states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. This theorem is given as Proposition 24 in Book I of Euclid's Elements.

Key takeaways

Hinge theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hinge theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hinge theorem from memory before moving on to harder problems.

Reference excerpt

In geometry, the hinge theorem (sometimes called the open mouth theorem) states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. This theorem is given as Proposition 24 in Book I of Euclid's Elements.

Proof from the law of cosines The theorem is an immediate corollary of the law of cosines. For two triangles with sides { a , b , c } {\displaystyle \{a,b,c\}} and { a , b , c ^ } {\displaystyle \{a,b,{\hat {c}}\}} with angles γ {\displaystyle \gamma } and γ ^ {\displaystyle {\hat {\gamma }}} opposite the respective sides c {\displaystyle c} and c ^ {\displaystyle {\hat {c}}} , the law of cosines states:

c 2 = a 2 + b 2 − 2 a b cos ⁡ γ , c ^ 2 = a 2 + b 2 − 2 a b cos ⁡ γ ^ . {\displaystyle {\begin{aligned}c^{2}&=a^{2}+b^{2}-2ab\cos \gamma ,\\{\hat {c}}^{2}&=a^{2}+b^{2}-2ab\cos {\hat {\gamma }}.\end{aligned}}}

The cosine function is monotonically decreasing for angles between 0 {\displaystyle 0} and π {\displaystyle \pi } radians, so γ ^ > γ {\displaystyle {\hat {\gamma }}>\gamma } implies c ^ > c {\displaystyle {\hat {c}}>c} (and the converse as well).

Scope and generalizations The hinge theorem holds in Euclidean spaces and more generally in simply connected non-positively curved space forms. It can be also extended from plane Euclidean geometry to higher dimension Euclidean spaces (e.g., to tetrahedra and more generally to simplices), as has been done for orthocentric tetrahedra (i.e., tetrahedra in which altitudes are concurrent) and more generally for orthocentric simplices (i.e., simplices in which altitudes are concurrent).

Converse The converse of the hinge theorem is also true: If the two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is greater 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. In some textbooks, the theorem and its converse are written as the SAS Inequality Theorem and the SSS Inequality Theorem respectively.

References

Illustrations

Worked examples

Example 1 — a first encounter with Hinge theorem

Start with the simplest possible case. Write down what Hinge theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hinge theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hinge theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hinge theorem

In research: Hinge theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hinge theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hinge theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elementary geometry, Theorems about triangles, so understanding it makes those chapters shorter.
In everyday life: Look for Hinge theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Hinge theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hinge theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hinge theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hinge theorem in simple terms?

Why does Hinge theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hinge theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hinge theorem.