Given the symbols were duplicated when written, I will re-write the statement:
Given: ∠F is
supplementary to ∠G and ∠G is supplementary to ∠H .
Conjecture: ∠F is supplementary to ∠H .
It is False.
Definition: two angles are supplementary if, and only if, they add up 180°.
=>
∠F is supplementary to ∠G => ∠F = 180° - ∠G => ∠G = 180° - ∠F
∠G is supplementary to ∠H => ∠G = 180° - ∠H
=> 180° - ∠F = 180° - ∠H
=> ∠F = ∠H,
Then, you have gotten the two angles are equal, and they will be supplementary only if ∠F = ∠ G = 90°, because 90° + 90° = 180°. In any other case, they are not supplementary.
This is a counter example:
∠G = 60°
∠F is supplementary to ∠G => ∠F = 180° - G = 180° - 120°
∠H = 30°
∠G is supplementary to ∠H => ∠G = 180° - H = ∠G = 180° - 30° = 150°
Then, ∠H + ∠F = 150° + 120° = 270° ≠ 180° => they are not supplementary.
