In the diagram below, AC ICB, AB I CD, and mZA = 31°. Find mZB. С
D
B
You may assume lines that appear straight are straight, but the figure is not otherwise drawn to scale.

m∠B = 59°

Step-by-step explanation:

From ΔADC,

m∠DAC = 31°

By triangle sum theorem,

m∠ADC + m∠DAC + m∠DCA = 180°

90° + 31° + m∠DCA = 180°

m∠DCA = 180° - 121°

             = 59°

m∠ACB = m∠ACD + m∠BCD

90° = m∠ACD + 59°

m∠ACD = 90° - 59°

              = 31°

In ΔCDB,

m∠CDB + m∠DBC + m∠DCB = 180°

90° + m∠DBC + 31° = 180°

m∠DBC = 180° - 121°

             = 59°

Therefore, measure of angle B = 59°.

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!

step-by-step explanation: