Given:

kg

∥

ev

, m

gv

=50°

m∠ekg=96°

find: m∠enk, m∠keg

m∠ENK = 50°

m∠KEG = 59°

Step-by-step explanation:

* Lets revise some facts on the circle

- The measure of the circle is 360°

- Two parallel secants intercept equal arcs

- If two chords intersect each other inside the circle, then the

 measure of the angle between them equals half the sum of the two

 intercepted arcs

- The measure of any inscribed angle equals half the measure

 of the intercepted arc

* Now lets solve the problem

∵ KG // EV ⇒ given

∴ The measure of arc VG = the measure of arc EK

∵ The measure of arc VG = 50° ⇒ given

∴ The measure of arc EK = 50°

- EG and VK are chords in circle O and intersect each other in point N

∴ m∠ENK = 1/2 (measure of arc EK + measure of arc VG)

∵ measure of arcs EK and VG is 50°

∴ m∠ENK = 1/2(50 + 50) = 1/2(100) = 50°

- ∠EKG is an inscribed angle in circle O and subtended by the arc EVG

∴ m∠EKG = 1/2 measure of arc EVG

∵ m∠EKG = 96° ⇒ given

∴ 96 = 1/2 measure of arc EVG ⇒ Multiple both sides by 2

∴ The measure of arc EVG = 192°

- The measure of the circle = 360°

∴ Measure of arc EVG + measure of arc KG + measure of arc KE = 360°

∵ Measure of arc EK = 50° ⇒ proved

∵ Measure of arc EVG = 192° ⇒ proved

∴ Measure of arc KG = 360 - (192 + 50) = 360 - 242 = 118²

- Angle KEG is an inscribed angle subtended by arc KG

∴ m∠KEG = 1/2 measure of arc KG

∵ Measure of arc KG = 118°

∴ m∠KEG = 59°