Question

Prove the following theorem. Using the ruler function axiom. List all axioms and definitions used.

Let P and Q be two points, then the line segment AB=BA (AB and BA have lines over them to show line segments)

Answer 1

the definitions used are following:

Definition 1. A collection of three or more points is collinear if there is some line containing all those points.

Definition 2. Two lines are parallel if they never meet.

Definition 3. When two lines meet in such a way that the adjacent angles are equal, the equal angles are called right angles, and the lines are called perpendicular to each other.

Definition 4. A circle is the set of all points equally distant from a given point. That point is called the center of the circle.

What about the term “line segment”? We all know what that is—it’s the portion of a line between two points. But what does “between” mean?

With a little thought, we can define “between” using two concepts we already have: the undefined term distance and the definition of collinear . Once we’ve done that, we can define what a line segment is. It’s important to get these two definitions in the proper order.

Definition 5. Given three distinct collinear points A, B, C, we say that B is between A and C if AC > AB and AC > BC.

Definition 6. The line segment AB between two points A and B consists of A and B themselves, together with the set of all points between them.

list of aioms are:

Axiom 1. If A, B are distinct points, then there is exactly one line containing both A and B, which we denote ←→AB (or ←→BA).

This is Euclid’s first axiom. Notice that it includes a definition of notation. The next group of axioms concern distance.

Axiom 2. AB = BA.

Axiom 3. AB = 0 iff A = B.

The word “iff” is mathematician’s jargon for “if and only if”. That is, the axiom says that two different things are true. First, if A = B, then AB = 0. Second, if AB = 0, then A = B. Logically, these are two separate statements.

Axiom 4. If point C is between points A and B, then AC + BC = AB.

Axiom 5. (The triangle inequality) If C is not between A and B, then AC + BC > AB.

Now, some axioms about angle measure.

Axiom 6. (a.) m(∠BAC) = 0◦ iff B, A, C are collinear and A is not between B and C. (b.) m(∠BAC) = 180◦ iff B, A, C are collinear and A is between B and C

Axiom 7. Whenever two lines meet to make four angles, the measures of those four angles add up to 360◦ .

Axiom 8. Suppose that A, B, C are collinear points, with B between A and C, and that X is not collinear with A, B and C. Then m(∠AXB) + m(∠BXC) = m(∠AXC). Moreover, m(∠ABX) + m(∠XBC) = m(∠ABC). (We know that ∠ABC = 180◦ by Axiom 6.)

Axiom 9. Equals can be substituted for equals. Two axioms about parallel lines:

Axiom 10. Given a point P and a line `, there is exactly one line through P parallel to `L.

Axiom 11. If ` and ` 0 are parallel lines and m is a line that meets them both, then alternate interior angles have equal measure, as do corresponding angles.

Now for two axioms that connect number and geometry:

Axiom 12. For any positive whole number n, and distinct points A, B, there is some C between A, B such that n · AC = AB.

Axiom 13. For any positive whole number n and angle ∠ABC, there is a point D between A and C such that n · m(∠ABD) = m(∠ABC)

Assume A and B are distinct points.

Note that AB = {A, B} ∪ {P | A ? P ? B} and BA = {B, A} ∪ {P | B ? P ? A}. Let x ∈ AB.

So x ∈ {A, B} ∪ {P | A ? P ? B}

⇒ x ∈ {A, B} or x ∈ {P | A ? P ? B}

In the case where x = A or x = B, then x ∈ BA. In the case where x 6= A and x 6= B, x is in between A and B. Therefore, Ax + xB = AB by the definition of between.

Since AB = Ax + xB, then we have that

⇒ AB = xB + Ax 4

⇒ AB = Bx + xA by Theorem 3.2.7

Since AB = Bx + xA, then x is between B and A and hence, AB ⊆ BA.

Similarly, if we take an x ∈ BA, we conclude that BA = Ax + xB and x is between A and B. Thus, BA ⊆ AB. Therefore, AB = BA.