Jeff can win the game if he rolls 11 on the two number cubes if he rolls an even sum, however he will lose his next turn

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?

Write the expression in simplified radical form. show all steps and work including any foil method work. (hint: use the conjugate. no decimal answers.) (9-2√3)/(12+√3)

In the given figure ar(adf) =ar(cda) and ar(cdf) = ar(cdf). show that abdc and cdfe are trapeziums.