Agnes has 23 collectible stones, all of which are labradorite crystals or galena
crystals. Labradorite crystals are worth
$20 each, while galena crystals are
worth $13 each. Agnes earns $439 by
selling her entire collection. How many
stones of each type did she sell?
hi

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?

Given the diagram below, michael writes, "segment ac is congruent to segment ac." which of the following reasons allow him to write this statement?

The data set represents the ages of players in a chess club. 27, 34, 38, 16, 22, 45, 54, 60. what is the mean absolute deviation of the data set?