Require Export C12_Angles_Opposes.
Section SUPPLEMENTARY_ANGLES.
Definition Supplementary (alpha beta : AS) :=
exists A  : Point,  exists B  : Point,  exists C : Point,  exists D : Point,
A <> C /\ Between B A D /\ Angle B A C = alpha /\ Angle C A D = beta.

(* Specifications: *)
Lemma SupplementaryCommut : forall alpha beta : AS,
Supplementary alpha beta ->
Supplementary beta alpha.
Lemma HalfLineBetweenBetween : forall A B C D : Point,
Between A C D ->
HalfLine C A B ->
Between B C D.
Lemma SupplementBC : forall A B C D A' B' C' D' : Point,
A <> C ->
A' <> C' ->
B <> C ->
Between B A D ->
Between B' A' D' ->
Angle B A C = Angle B' A' C' ->
Angle C A D = Angle C' A' D'.
Lemma Supplement : forall A B C D A' B' C' D' : Point,
A <> C ->
A' <> C' ->
Between B A D ->
Between B' A' D' ->
Angle B A C = Angle B' A' C' ->
Angle C A D = Angle C' A' D'.
Lemma SupplementaryEq : forall alpha beta gamma : AS,
Supplementary alpha beta ->
Supplementary alpha gamma ->
beta = gamma.
Lemma OpposedAtVertex : forall A B C D E : Point,
Between B A D ->
Between C A E ->
Angle B A C = Angle D A E.
Lemma CongruentOpposedTriangles : forall A B C D I : Point,
Between A I C ->
Between B I D ->
Distance I A = Distance I C ->
Distance I B = Distance I D ->
CongruentTriangles A I B C I D.
Lemma SupplementaryRec : forall A B C D : Point,
Clockwise B A C ->
Clockwise C A D ->
Supplementary (Angle B A C) (Angle C A D) ->
Between B A D.