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Hinge Theorem — Definition, Formula & Examples

The Hinge Theorem is a rule for two triangles that share two pairs of equal sides. The triangle whose angle between those sides is wider has the longer third side, the same way a door's gap grows as its hinge opens.

If two sides of △ABC\triangle ABC are congruent to two sides of △DEF\triangle DEF, with AB‾≅DE‾\overline{AB} \cong \overline{DE} and AC‾≅DF‾\overline{AC} \cong \overline{DF}, and m∠A>m∠Dm\angle A > m\angle D, then BC>EFBC > EF. The converse (the SSS Inequality) also holds: if the two pairs of sides are congruent and BC>EFBC > EF, then m∠A>m∠Dm\angle A > m\angle D.

Worked Example

Problem: In △ABC\triangle ABC and △DEF\triangle DEF, AB=DE=3AB = DE = 3 and AC=DF=4AC = DF = 4. The included angles are m∠A=90∘m\angle A = 90^\circ and m∠D=60∘m\angle D = 60^\circ. Which is longer, BCBC or EFEF?
Step 1: Check the conditions. Two pairs of sides are congruent, and ∠A\angle A and ∠D\angle D are the included angles, because each one lies between the two known sides.
AB=DE=3,AC=DF=4AB = DE = 3, \quad AC = DF = 4
Step 2: Compare the included angles and apply the Hinge Theorem. The larger angle is opposite the longer third side.
90∘>60∘  ⟹  BC>EF90^\circ > 60^\circ \implies BC > EF
Step 3: Confirm the result with the Law of Cosines. The theorem itself does not need this step.
BC=32+42−2(3)(4)cos⁡90∘=5,EF=32+42−2(3)(4)cos⁡60∘=13≈3.61BC = \sqrt{3^2 + 4^2 - 2(3)(4)\cos 90^\circ} = 5, \qquad EF = \sqrt{3^2 + 4^2 - 2(3)(4)\cos 60^\circ} = \sqrt{13} \approx 3.61
Answer: BCBC is longer. BC=5BC = 5 and EF=13≈3.61EF = \sqrt{13} \approx 3.61.

Why It Matters

Geometry courses use the Hinge Theorem in indirect proofs and in triangle-inequality problems. A typical task asks you to find the range of values for xx when an angle or side is written as an expression. The same idea explains why a compass, a pair of scissors, or a robotic arm reaches farther as its joint opens wider.

Common Mistakes

Mistake: Comparing angles that are not between the two congruent sides, or applying the theorem when only one pair of sides is congruent.
Correction: The theorem requires both pairs of sides to be congruent, and the angles you compare must be the included angles. Before you compare anything, check that each angle sits between the two known sides.

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