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The Hinge Theorem

The Hinge Theorem. Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is opposite the larger included angle.

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The Hinge Theorem

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  1. The Hinge Theorem

  2. Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is opposite the larger included angle.

  3. Hinge Theorem Converse: If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is opposite the longer third side.

  4. Consider ∆ABC and ∆XYZ. If , and mY> mB, then XZ > AC. This is the Hinge Theorem (SAS Inequality Theorem).

  5. Examples Write an inequality relating the given side lengths. If there is not enough information to reach a conclusion, write no conclusion. 1. JL and MO 2. ST and BT

  6. Write an inequality relating the given angle measures. 3. mM and mR 4. mU and mX

  7. TR > ZX. What is the range of possible values for x? The triangles have two pairs of congruent sides, because RS =XY and TS =ZY . So, by the Converse of the Hinge Theorem, mS > mY. • Write an inequality: 72 > 5x + 2 Converse of the Hinge Theorem 70 > 5x Subtract 2 from each side. 14 > x Divide each side by 5.

  8. Write another inequality: mY > 0 The measure of an angle of a triangle is greater than 0. 5x + 2 > 0 Substitute. 5x > −2 Subtract 2 from each side. Divide each side by 5. So,

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