What is the area of the trapezoid shown? Express your answer in simplest exact form. [asy]

unitsize(1.5cm);

pair a=(0,0); pair b=(1,sqrt(3)); pair c=(3,sqrt(3)); pair d=(4,0);

draw(a--b--c--d--a);

label("2",(a+b)/2,NW);

label("2",(b+c)/2,N);

label("2",(c+d)/2,NE);

label("~$60^\circ$",a, NE);

label("$60^\circ$~",d, NW);

[/asy]

Here is their argument. given the obtuse angle x, we make a quadrilateral abcd with ∠dab = x, and ∠abc = 90◦, and ad = bc. say the perpendicular bisector to dc meets the perpendicular bisector to ab at p. then pa = pb and pc = pd. so the triangles pad and pbc have equal sides and are congruent. thus ∠pad = ∠pbc. but pab is isosceles, hence ∠pab = ∠pba. subtracting, gives x = ∠pad−∠pab = ∠pbc −∠pba = 90◦. this is a preposterous conclusion – just where is the mistake in the "proof" and why does the argument break down there?

∠aed is formed inside a circle by two intersecting chords. if minor arc bd = 70 and minor arc ac = 180, what is the measure of ∠ aed? a) 45° b) 50° c) 55° d) 60°

Asequence is defined recursively by the formula  f(n  +  1)  =  f(n)  +  3 . the first term of the sequence is  –4. what is the next term in the sequence? –7–117