Lecture 16

3.1 Euclid’s Axioms:
Ax1. A straight line segment can be drawn joining any two points.
Ax2. Any straight line segment can be extended indefinitely in a straight line.
Ax3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
Ax4. All right angles are congruent.
Ax5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
Remark: Ax5 does not mention the word parallel, but for a line m through A and any line n through a point B not on m, this rules out the possibility that line n is parallel to m except when two interior angles add up to a straight angle. So there is only one possible line through B parallel to m. It can be proved that this line is in fact parallel.
a
b
1
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Hawraa Abbas Almurieb Geometry
Chapter Three: Foundation of Geometry
2 | P a g e
3.2. Foundations of Geometry
3.2.1.The Five Groups Of Axioms:
I. Axioms of Incidence.
II. Axioms of order.
III. Axiom of parallels (Euclid’s axiom).
IV. Axioms of congruence.
V. Axiom of continuity (Archimedes’ axiom).
Group I: Axioms of Incidence
Ax1. For every two points A and B there exists a line a that contains them both. We write AB=a or BA=a. Ax2. For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where B ? C, then also BC = a. Ax3. There exist at least two points on a line. There exist at least three points that do not lie on the same line. Ax4. For every three points A, B, C not on the same line there exists a plane ? that contains all of them. We write ABC = ?. Ax5. For every three points A, B, C which do not lie on the same line, there exists no more than one plane that contains them all. Ax6. If two points A, B of a line a lie in a plane ?, then every point of a lies in ?. In this case we say: “The line a lies in the plane ?,”. Ax7. If two planes ?, ? have a point A in common, then they have at least a second point B in common. Ax8. There exist3.1 Euclid’s Axioms:
Ax1. A straight line segment can be drawn joining any two points.
Ax2. Any straight line segment can be extended indefinitely in a straight line.
Ax3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
Ax4. All right angles are congruent.
Ax5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
Remark: Ax5 does not mention the word parallel, but for a line m through A and any line n through a point B not on m, this rules out the possibility that line n is parallel to m except when two interior angles add up to a straight angle. So there is only one possible line through B parallel to m. It can be proved that this line is in fact parallel.
a
b
1
2
Hawraa Abbas Almurieb Geometry
Chapter Three: Foundation of Geometry
2 | P a g e
3.2. Foundations of Geometry
3.2.1.The Five Groups Of Axioms:
I. Axioms of Incidence.
II. Axioms of order.
III. Axiom of parallels (Euclid’s axiom).
IV. Axioms of congruence.
V. Axiom of continuity (Archimedes’ axiom).
Group I: Axioms of Incidence
Ax1. For every two points A and B there exists a line a that contains them both. We write AB=a or BA=a. Ax2. For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where B ? C, then also BC = a. Ax3. There exist at least two points on a line. There exist at least three points that do not lie on the same line. Ax4. For every three points A, B, C not on the same line there exists a plane ? that contains all of them. We write ABC = ?. Ax5. For every three points A, B, C which do not lie on the same line, there exists no more than one plane that contains them all. Ax6. If two points A, B of a line a lie in a plane ?, then every point of a lies in ?. In this case we say: “The line a lies in the plane ?,”. Ax7. If two planes ?, ? have a point A in common, then they have at least a second point B in common. Ax8. There exist