10. if you are asked to give the intersection of any plane and any line not in that
plane, your answer will always be be a
11. if you are asked to give the intersection of two non-parallel planes, your
answer will always be a
12. if you are asked to give the intersection of two lines, your answer will always
be a

10. If you are asked to give the intersection of any plane and any line not in that  plane, your answer will always be be a point

11. If you are asked to give the intersection of two non-parallel planes, your answer will always be a line

12. If you are asked to give the intersection of two lines, your answer will always  be a point

Step-by-step explanation:

10. In space, a plane and a line that is not lying on this plane or parallel to it will always meet at point.

For instance, line PN intersects plane A at point N

11. The intersection of two planes in space is a line.

For instance, the intersection of planes QRST and PQRN is line RQ.

12. Two lines that are not parallel in the same plane will always intersect at a point.

For instance, lines TQ and PQ intersects at point Q

However, if two parallel lines are identical and coincide with each other, they intersect at infinitely many points.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!