Answer:
11) ∠LPN = 57°
12) x = 11
Explanation:
11) see the below figure
∠LOP = 102° and ∠NOP = 144°
As OL, OP, ON are radius of the circle
∠OLP = ∠OPL = x
so in ΔOLP sum of angles is 180
102 + ∠OLP + ∠OPL = 180
102 + 2x = 180
2x = 180 - 102 = 78
x = 39
so ∠OPL = 39°
as ON = OP, ∠OPN = ∠ONP = y
so in ΔOPN sum of angles is 180
144 + ∠OPN + ∠ONP = 180
144 + 2y = 180
2y = 180 - 144
2y = 36
y = 18
so ∠OPN = 18°
now ∠LPN = ∠OPL + ∠OPN
= 39 + 18
∠LPN = 57°
12) angle of arc VYX = 282
The angle between a tangent and a radius is 90°.
∠UVX = 4X - 5
see the second figure below
∠OVU = 90°
∠XOV = 360 - ∠arc VYX
= 360 - 282
= 78°
in ΔOVX
∠OXV = ∠OVX = x
and ∠OXV + ∠OVX + ∠VOX = 180
2x + 78 = 180
2x = 180 - 78 = 102
x = 51
now ∠UVX + ∠XVO = 90
4x - 5 + 51 = 90
4x + 46 = 90
4x = 44
x = 11