How many angles with measures less than $180^{\circ}$ are formed by the $4$ intersecting lines shown?[asy] import olympiad; size(150); pair A=(0,0), B = 4*dir(0), C=4*dir(40),D=4*dir(100),E=4*dir(150),F=4*dir(180),G=4*dir(220),H=4*dir(280),I=4*dir(330); dot(A,red); draw(A--B,EndArrow); draw(anglemark(B,A,C,20),black); label("$40^\circ$",anglemark(B,A,C),4*dir(20)); draw(A--C,EndArrow); draw(A--D,EndArrow); draw(A--E,EndArrow); draw(anglemark(D,A,E,20),black); label("$50^\circ$",anglemark(D,A,E),4*dir(120)); draw(A--F,EndArrow); draw(A--G,EndArrow); draw(A--H,EndArrow); draw(anglemark(G,A,H,20),black); label("$60^\circ$",anglemark(G,A,H),4*dir(250)); draw(A--I,EndArrow); [/asy]
