WGU C992 Task 1 |Latest Update with Complete
Solution
Kelsi Offerman
College Geometry – C992
April 4, 2025
Task 1: Axiomatic Systems
Consider the axiomatic system and theorem below:
• Axiom 1: If there is a pair of points, then they are on a line together.
• Axiom 2: If there is a line, then there must be at least two points on it.
• Axiom 3: There exists at least two distinct points.
• Axiom 4: If there is a line, then not all the points can be on it.
Theorem 1: Each point is on at least two distinct lines.
A. List all undefined terms involved in the given axiomatic system,
including all elements and relations.
Undefined Terms
• Point
• Line
• On
B. Explain how the axioms require that the system has three distinct points.
Axiom 1: “If there is a pair of points, then they are on a line together.” This axiom is the basic
idea that lines are defined by the points on them.
Axiom 2: “If there is a line, then there must be at least two points on it.” This axiom is the basic
idea that a line does not exist without at least two points.
Axiom 3: “There exists at least two distinct points.” This axiom is the basic idea that a pair of
points exists in the system. This system will not exist with zero or one point in it.
Axiom 4: “If there is a line, then not all the points can be on it.” This axiom is the basic idea that
no line can go through all points.
Suppose there are two distinct points K and L by axiom 3. According to Axiom 1, Points K and L
will be on one line. Axiom 4 says that if there is a line, then not all points can be on it. There
must be another point, point O, since only two points can be on the line. Therefore, we would
need at least three distinct points to satisfy all four axioms.
C. Prove Theorem 1 for three points, using only the provided axioms.
Suppose there are the same three distinct points from above, K, L, and O. We can show that
these points lie on two distinct lines and cannot lie on only one line.
Points K and L are on line KL. Points L and O are on line LO. Points K and O are on line KO (axiom
1). According to axiom 3, there exist at least two distinct points, so there are two or more points.
This shows that there are at least points K and L. Axiom 1 says that if there are a pair of points,
they are on a line together, so points K and L are on the same line, line KL. Axiom 4 says, if there
Solution
Kelsi Offerman
College Geometry – C992
April 4, 2025
Task 1: Axiomatic Systems
Consider the axiomatic system and theorem below:
• Axiom 1: If there is a pair of points, then they are on a line together.
• Axiom 2: If there is a line, then there must be at least two points on it.
• Axiom 3: There exists at least two distinct points.
• Axiom 4: If there is a line, then not all the points can be on it.
Theorem 1: Each point is on at least two distinct lines.
A. List all undefined terms involved in the given axiomatic system,
including all elements and relations.
Undefined Terms
• Point
• Line
• On
B. Explain how the axioms require that the system has three distinct points.
Axiom 1: “If there is a pair of points, then they are on a line together.” This axiom is the basic
idea that lines are defined by the points on them.
Axiom 2: “If there is a line, then there must be at least two points on it.” This axiom is the basic
idea that a line does not exist without at least two points.
Axiom 3: “There exists at least two distinct points.” This axiom is the basic idea that a pair of
points exists in the system. This system will not exist with zero or one point in it.
Axiom 4: “If there is a line, then not all the points can be on it.” This axiom is the basic idea that
no line can go through all points.
Suppose there are two distinct points K and L by axiom 3. According to Axiom 1, Points K and L
will be on one line. Axiom 4 says that if there is a line, then not all points can be on it. There
must be another point, point O, since only two points can be on the line. Therefore, we would
need at least three distinct points to satisfy all four axioms.
C. Prove Theorem 1 for three points, using only the provided axioms.
Suppose there are the same three distinct points from above, K, L, and O. We can show that
these points lie on two distinct lines and cannot lie on only one line.
Points K and L are on line KL. Points L and O are on line LO. Points K and O are on line KO (axiom
1). According to axiom 3, there exist at least two distinct points, so there are two or more points.
This shows that there are at least points K and L. Axiom 1 says that if there are a pair of points,
they are on a line together, so points K and L are on the same line, line KL. Axiom 4 says, if there
