Hinge Theorem

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.

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

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

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

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

$QT\ <\ ST$

$QT\ >\ ST$

$QT\ =\ ST$

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

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

$x<6$

$24<x$

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

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

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

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

$27<x<37$

$2\ <x<6$

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

$6<x<22$

$2<x<22$

$-20<x<21$

$8<x<37$

$34<x<37$

$8<x<24$

$x>7$

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

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

$x<7$

$m\angle FBG\ <m\angle BFA$

$m\angle FBG\ >m\angle BFA$

$m\angle FBG\ \ge m\angle BFA$

$m\angle FBG\ \le m\angle BFA$

$m\angle GBC\ <m\angle FBA$

$m\angle GBC\ >\ m\angle FBA$

$m\angle GBC\ \le\ m\angle FBA$

$m\angle GBC\ \ge\ m\angle FBA$