There are 13 angles.
We know that [tex]m\angle4=112[/tex] and [tex]m\angle12=105[/tex] (11 angles)
1.
[tex]m\angle2+m\angle3+m\angle4=180\qquad\qquad\text{linear angles}\\\\m\angle3+m\angle3+m\angle4=180\qquad\qquad m\angle2=m\angle3\\\\2m\angle3+m\angle4=180\\\\2m\angle3+112=180\\\\2m\angle3=180-112\\\\2m\angle3=68\quad|:2\\\\\boxed{m\angle3=34=m\angle2}[/tex]
(9 angles)
2.
[tex]m\angle3+m\angle12+m\angle11=180\qquad\qquad\text{triangle}\\\\34+105+m\angle11=180\\\\139+m\angle11=180\\\\m\angle11=180-139\\\\\boxed{m\angle11=41}[/tex]
(8 angles)
3.
[tex]m\angle12+m\angle13=180\qquad\qquad\text{linear angles}\\\\105+m\angle13=180\\\\m\angle13=180-105\\\\\boxed{m\angle 13=75}[/tex]
(7 angles)
4.
[tex]m\angle1+m\angle2+m\angle13=180\qquad\qquad\text{triangle}\\\\m\angle1+34+75=180\\\\m\angle1+109=180\\\\m\angle1=180-109\\\\\boxed{m\angle1=71}[/tex]
(6 angles)
5.
[tex]m\angle9=m\angle11\qquad\qquad\text{vertical angles}\\\\\boxed{m\angle9=41}[/tex]
(5 angles)
6.
[tex]m\angle10+m\angle11=180\qquad\qquad\text{linear angles}\\\\m\angle10+41=180\\\\m\angle10=180-41\\\\\boxed{m\angle10=139}[/tex]
(4 angles)
7. [tex]a||b\text{ and } c||d[/tex] so we have a parallelogram.
[tex]m\angle4+m\angle5=180\qquad\text{consecutive angles are supplementary}\\\\112+m\angle5=180\\\\m\angle5=180-112\\\\\boxed{m\angle5=68}[/tex]
(3 angles)
[tex]m\angle8=m\angle5\qquad\qquad\text{opposite angels are congruent}\\\\\boxed{m\angle8=68}[/tex]
(2 angles)
8.
[tex]m\angle7+m\angle8+m\angle9=180\qquad\qquad\text{triangle}\\\\m\angle7+68+41=180\\\\m\angle7+109=180\\\\m\angle7=180-109\\\\\boxed{m\angle7=71}[/tex]
(1 angle)
9.
[tex]m\angle1+m\angle5+m\angle6=180\qquad\qquad\text{triangle}\\\\71+68+m\angle6=180\\\\139+m\angle6=180\\\\m\angle6=180-139\\\\\boxed{m\angle6=41}[/tex]
