You want to prove a theroum in a two coulum proof. You start with your given statement and list deductions in the left hand column. What are the tree main types of reasoning you will use for reasons in the right hand column?

In a two-column proof, you typically use three main types of reasoning to justify the deductions in the right-hand column. These types of reasoning are:

1. Definitions: The basic meanings of terms, which are agreed upon in geometry. For example, when you state that a figure is a rectangle, you're using the definition of a rectangle which is a quadrilateral with four right angles.

2. Postulates (or axioms): These are statements that are accepted as true without proof. They serve as the starting points for reasoning within a mathematical system. An example of a postulate is the Parallel Postulate in Euclidean geometry, which states that through a point not on a given line, there is exactly one line parallel to the given line.

3. Theorems: These are statements that have been proven based on definitions, postulates, and previously proven theorems. They are the logical conclusions that one can draw from the groundwork laid by the definitions and postulates. An example of a theorem is the Pythagorean Theorem, which relates the lengths of the sides of a right triangle.

When proving a theorem in a two-column proof, you'll make a statement in the first column and then provide a reason in the second column, which will be based on definitions, postulates, or theorems.

Extra: Understanding how two-column proofs work is fundamental in geometry. The process is much like a logical argument in which each step is justified by reasoning. When constructing your proof, you always begin with the given information and what you are trying to prove (the "Prove" statement). Each step towards this "Prove" statement must follow logically from the previous one with clear reasons:

- Definitions are the fundamental building blocks. For example, when you are dealing with a circle, you use the definition of a circle (the set of points equidistant from a center point) to make further deductions.

- Postulates or axioms are self-evident truths. For example, the Ruler Postulate allows us to measure distances using a ruler, and it is accepted without requiring proof.

- Theorems are richer and more complex than postulates. Once proven, they become powerful tools for deducing new truths. Theorems often require a proof that relies on previously established theorems, postulates, and definitions. An example is the Angle Addition Postulate, which allows you to infer that if point B is in the interior of angle AOC, then the measure of angle AOB plus the measure of angle BOC equals the measure of angle AOC.

Performing a two-column proof is an exercise in critical thinking and requires a logical, step-by-step approach to problem-solving, making it an invaluable skill in mathematics education.