A quadrilateral has angle measures of 41 52 and 54 which is the measure of the last angle

213 degrees

Step-by-step explanation:

A quadrilateral has an internal measurement of angles of 360 degrees.  Given that we have 3 measures of the angles, we simply need to find the 4th.  This can be accomplished with the following equation:

x + 52 + 54 + 41 = 360

x + 52 + 95 = 360

x + 147 = 360

x = 213

Hence, the measure of the final angle is 213 degrees.

Cheers.

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!

i belive those are the ones that are under 90 degreese.