The definition of angles in common geometry textbooks uses the order relation. In unordered geometries, we need to use full-angles that do not involve the order relation. We will presents the definition and the basic properties of full-angles.
R1. A full-angle is defined as an ordered pair of two lines u and v denoted by ∠[u, v].
R2. ∠[u, v] = −∠[v, u].
R3. Constant full-angle ∠[u, u] is denoted by ∠[0]. We have ∠[0] + ∠[u, v] = ∠[u, v] + ∠[0] =
∠[u, v]. Also ∠[u, v] = ∠[0] iff // v.
R4. If u ⊥ v, then we define ∠[u, v] to be a constant full-angle ∠[1]. ∠[u, v] + ∠[u, v] = ∠[0], or
∠[1] + ∠[1] = ∠[0].
R5. For any line l, the operation ”addition” between two full angles is defined as∠[u, v] =
∠[u, l]+∠[l, v]. Thus the addition of two full-angles does not use the order relation and is diagramindependent.
The set of all full-angles with the addition operator “+”, is an additive (Abelian)
group.
Remark. The geometries with the definition of the addition of full-angles are Euclidean, as opposed
to non-Euclidean geometries. For example, the assertion that the sum of three angles of a triangle
ABC equals 180◦, which is equivalent to Euclidean fifth hypothesis, can be expressed as
∠[CA,AB] + ∠[AB,BC] + ∠[BC,CA] = ∠[CA,CA] = ∠[0].
R6. If l ⊥ v, we have ∠[u, v] = ∠[u, l] + ∠[l, v] = ∠[u, l] + ∠[1].
Note that this rule is the combination of Rule4 and Rule5.
R7. If AB = AC, then ∠[AB,BC] = ∠[BC,CA].
The full-angle method works particularly well for theorems involving circles and angles.
Let A,B,C, and D be four non-collinear points.5 We use ∠[ABC] to denote the full-angle
∠[AB,BC] and use ∠ABC to denote the traditional angle. In ordered geometry, ∠[ABC] =
∠[DEF] if and only if ∠ABC = ∠DEF and the two angles have the same orientation, or
∠ABC = 180 − ∠DEF and the two angles have the opposite orientations.
The Inscribed Angle Theorem
R8. (The Inscribed Angle Theorem) Points A,B,C, and D are cyclic iff ∠[ABC] = ∠[ADC]
(the left of Figure 5.)
Applying this twice we can have a combined rule, say, ∠[BAC] = ∠[BDC] iff ∠[ABC] =
∠[ADC] (the right of Figure 5), etc. Here the four points can be in any order.
These diagram-independent properties of full-angles make the proofs not only much simpler and
diagram-independent, but also rigorous.
Example 6. The same as Example 1. Let points C and D be the intersections of two circles (A)
and (B), line HF be passing through C and meeting (A) and (B) at H and F, line GE be passing
through D and meeting (A) and (B) at G and E. Show that GH is parallel to EF (Figure 6).
A Two Circle Theorem
Proof
∠[GH,EF] = ∠[GHC] + ∠[CFE] (1)
(∵ ∠[GHC] = ∠[GDC] and ∠[CFE] = ∠[CDE]) (2)
= ∠[GDC] + ∠[CDE] (3)
= ∠[GDE] = ∠[0] (4)
In this way we prove that ∠[GH,EF] = ∠[0], thus we have GH // EF from R3. In Step (2),
the two angle equality assertions use R8 with the hypotheses cyclic(HDGC) and cyclic(CFED).
This gives us a way to organize the proof in a hierarchical way: by clicking ‘∵ ∠[GHC] =
∠[GDC]’, it expands to include a substep ‘∵ cyclic(HDGC)’ on which the rule is applied; by
clicking cyclic(HDGC) it might expand a substep for the proof of the assertion cyclic(HDGC)
and in this example it is not needed to prove because it is a hypothesis.
Here we deliberately use the order HDGC instead of the clockwise order HGDC to emphasize the diagram-independent nature of the full-angle proof. This proof is a diagram-independent proof.
Also here we implicitly use hypotheses on collinear facts, e.g., in Step (1), we use the fact that
∠[GHC] is ∠[GHF] and this fact is true because H, C, and F are collinear, etc. The proof adds an auxiliary segment CD. This addition is natural for our full-angle method because
the proof requires ∠[GDC], hence requires segment CD. We will discuss the topic on
adding auxiliary geometric elements for the full-angle method.
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