<em>Statement/Reason</em> is a method of presenting your logical thought process as you go from the "givens" in a problem statement to the desired conclusion. Each <em>statement</em> expresses the next step in the solution process. It is accompanied by the <em>reason</em> why it is true or applicable.
For example, if you have an equation that says ...
... x + 3 = 5
Your next "statement" might be
... x + 3 - 3 = 5 - 3
The "reason" you can make that statement is that the <em>addition property of equality</em> allows you to add the same quantity to both sides of an equation without violating the truth of the equality. You know this because you have studied the properties of equality and how they relate to the solution of equations.
In geometry (where you're more likely to encounter statement/reason questions), you know the statements you're allowed to make because you have studied the appropriate postulates and theorems. The "reason" is generally just the name of the applicable postulate or theorem. The "statement" is the result of applying it to your particular problem.
For example, if you have ∠ABC and ∠CBD, you might want to say (as part of some problem solution) ...
... m∠ABC + m∠CBD = m∠ABD
The reason you can say this is the <em>angle addition postulate</em>, which you have studied. It will tell you that the measures of non-overlapping angles with a common side and vertex can be added to give the measure of the angle that includes them both. (Many such postulates seem obvious, as this one does.)
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<em>Side comment on geometric proofs</em>
As you go along in geometry, you study and develop more and more theorems that you can use to find solutions to problems. Sometimes, you're required to use a restricted subset of the ones you know in order to prove others.
As an example, in some problems, you may be able to use the fact that the midline of a triangle is parallel to the base; in other problems, you may be required to prove that fact.
I sometimes found it difficult to tell which theorems I was allowed to use for any given problem. It may help to keep a list that you can refer to from time to time. Your list would tell you the name of the theorem, axiom, or postulate, and what the meaning of it is, and where it might be applied.
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<em>Which reason fits which statement?</em>
The "reason" is telling how you know you can make the statement you made. It is anwering the question, "what allows you to make that statement?"
<em>How do I form true statements?</em>
The sequence of statements you want to make comes from your understanding of the problem-solving process and the strategy for solution you develop when you analyze the problem.
Your selection of statements is informed by your knowedge of the properties of numbers, order of operations, equality, inequality, powers/roots, functions, and geometric relationships. You study these things in order to become familiar with the applicable rules and properties and relationships.
A "true" statement will be one that a) gets you closer to a solution, and b) is informed by and respects the appropriate properties of algebraic and geometric relations.
In short, you're expected to remember and be able to use all of what you have studied in math—from the earliest grades to the present. Sometimes, this can be aided by remembering a general rule that can be applied different ways in specific cases. (For me, in Algebra, such a rule is "Keep the equal sign sacred. Whatever you do to one side of an equation, you must also do to the other side.")
