8. There are 167 students in the fifth grade. The same number of students is in the fourth grade. How many fifth graders and fourth graders are there in all?

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?

A25- foot ladder leans against a building and reaches a point of 23.5 feet above the ground .find the angle that the ladder makes with the building

You can ride your bike around your block 6 times and the whole neighborhood 5 times in 16 minutes. you can ride your bike around your block 3 times and the whole neighborhood 10 times in 23 minutes. how long does it take you to ride around the neighborhood?