Determine whether the conjecture is true or false. give a counterexample for any false conjecture. given: ∠f ∠ f is supplementary to ∠g ∠ g and ∠g ∠ g is supplementary to ∠h ∠ h . conjecture: ∠f ∠ f is supplementary to ∠h ∠ h .

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.

Yes, two vertical lines can be complementary