Question

In the figure, if O is the center of the circle and two chords of the circle EF and GH are parallel to each other. Show that ∠EOG ≅ ∠FOH.

Answer 1

Construction: Draw seg EO, seg FO, seg GO, seg HO, seg GF.

Proof:

Chord EF || chord GH, GF is the transversals.

∠EFG ≅ ∠HGF   ...(i) [Alternate angles]

`{:(∠EFG = 1/2 m(arc  GE)),(∠HGF = 1/2 m(arc  HF)):}}`   `{:(...("ii")),(...["Inscribed angle theorem"]),(...("iii")):}`

∴ m(arc GE) = m(arc HF)   ...(iv) [From (i), (ii) and (iii)]

`{:("Also"","  ∠EOG = m(arc  GE)),(∠FOH = m(arc  HF)):}}`   `{:(...("v") ["Definition of measure of arc"]),(...("vi")):}`

∴ ∠EOG = ∠FOH   ...[From (iv), (v) and (vi)]