Answer:
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°
