GEOM QZ #5.2: Hinge Theorem(5.6)

the vertex of the angle acts like a hinge to open the triangle.

it's named after Joseph Hinge, the man who discovered it.

"hinge" is the Spanish word for angle.

the triangle makes a hinge shape, like a tent.

$$m\angle ACB\ >\ m\angle GDE$$

$$m\angle ACD\ <\ m\angle GDE$$

$$m\angle ACB\ =\ m\angle GDE$$

$$\angle ACB\ \cong\ \angle GDE$$

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$$QT\ <\ ST$$

$$QT\ >\ ST$$

$$QT\ =\ ST$$

$$\overline{QT}\ \cong\ \overline{ST}$$

$$\frac{7}{2}<x<24$$

$$x<6$$

$$24<x$$

$$\frac{11}{2}<x<6$$

$$\frac{5}{3}<x<8$$

$$-\frac{3}{2}<x>8$$

$$\frac{32}{3}<x<27$$

$$27<x<37$$

$$2\ <x<6$$

$$6<x<15\ \frac{1}{3}$$

$$6<x<22$$

$$2<x<22$$

$$x>7$$

$$-\frac{17}{3}<x>7$$

$$-\frac{17}{3}<x<7$$

$$x<7$$