Question

In the given figure, AB is the diameter of the circle with center O. If ∠BOD = 15° & ∠EOA = 85°, then find the value of ∠ECA.​

Answer 1

Answer:

∠EOA = 85°,     ∠BOD = 15°

∠EOD = 180° - (85° + 15°) = 80°

In Δ OED, OE = OD (radii)

∠OED = ∠ODE = 50°

In Δ OEC, ∠EOC = 80°+15°

= 95°

⇒ ∠ECA = 180°- (95 + 50°) = 35°

Step-by-step explanation:

Answer 2

Answer:

∠$ECA=35$°

Step-by-step explanation:

We have

∠$BOD=15$° and ∠$EOA=85$°

$EO=OD$  (Radius of circle)

Hence,

∠$OED=$ ∠$ODE=x$

∠$ADE+$∠$EOD+$∠$BOD=180$° (Linear pair sum)

$85$°$+$∠$EOD+15$°$=180$°

∠$EOD=180$°$-100$

∠$EOD=80$°

In Δ $EOD,$

$x+$∠$EOD+15$°$=180$°

$2x+80$°$=180$°

$2x=180$°$-80$°

$x=\frac{100}{2}$

$x=50$°$,$ Hence ∠$OEC=50$

In Δ $EOC,$

∠$EOC+$∠$ECA+$∠$OEC=180$°

$($∠$EOD+$∠$BOD)+$∠$ECA+50$°$=180$°

$80$°$+15$°$+$∠$ECA+50$°$=180$°

∠$ECA=180$°$-145$°

∠$ECA=35$°