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How does one write robust Geometric Proofs?
One of the most challenging - and intimidating - aspect of Subtest II: Geometry, is writing Proofs for Geometric Propositions. Kids in schools HATE proofs universally (!), and I must say that teachers do their tuppence to cultivate such sentiments. The allergy of students against Proofs shouldn't be particularly surprising: many Geometry teachers themselves cower at the sight of an 'unfamiliar' Proof!
Proofs require you to reason very logically and analytically - the deductive reasoning is of a high order! - and it calls for a peculiarly methodical approach not encountered elsewhere.
With regards to the CSET, you're sure to encounter at least a couple of Proofs in the MCQ section wherein you are expected to supply the missing step of a Proof by selecting the most relevant choice. Likewise, AT LEAST one Q in the Free Response section shall be a Proof. And, obviously, this might be one you're familiar with, or something novel and challenging.
But proofs are actually quite simple once you realize that they follow a very definite 'system' or structure. Here are general Tips about approaching and writing robust Geometric Proofs:
1. There is no one RIGHT way to prove a proposition, especially if it seems rather complicated! ANY method or approach that consists of logically linked Statements - that is, you don't SKIP a step! - supported by valid Reasons shall do splendidly. The CSET examiners are very competent and professional and will accept any Proof that appears reasonable!
(This might be pushing it a bit but for one of the Geometry Free Response Qs, I used Algebra II-level Trigonometry alone since it just seemed easier for me than 'conventional' Geometry. By the way, I knew how to prove it ordinarily too...!)
2. Proofs can be written in Tabular Statement-Reason format OR in Free-flowing Paragraph form. You are expected to know both!
But relax if the latter variety is unfamiliar: simply write down your Statement (as you normally would), then say 'because', and write down your Reason. Let subsequent Statements be written together like in a paragraph. THAT'S IT!
3. Understand what you're Given and What You Need To Prove. This indicates information about where you are - the Given - and where you're going - what I call To Prove That.
After all, if you don't know ie. keep track of, where you're going, you wouln't know when you're close-by!
The To Prove That step is ALWAYS the final step (OBVIOUSLY!!) and you can blindly write that in the Statement column at the very beginning after leaving sufficient room to 'fill in' the earlier Statements!
4. To begin attacking the proof, draw, label and mark the figure as accurately and COMPLETELY as possible incorporating
a) What is given: It might clearly be stated as Given that Angle A = Angle B or PQ = QR
b) What is implicit: From the figure or what is given, you might have to draw OBVIOUS (and sometimes not so obvious!)conclusions based on your prior knowledge of Definitions, Properties, Postulates and Theorems (See Below!) like
* Two angles are Equal (because they are OBVIOUSLY Vertical) => Mark them as Congruent! * Two segments are Equal (because they are the 'shared sides' of 2 Triangles) => Mark them as Congruent! * Two segments are Equal (because a point Bisects the whole segment) => Mark them as Congruent! * Two angles are Equal (because it is Given that a Segment is the Angle Bisector) => Mark them as Congruent! * Two angles are Equal (because of the 3 segments that enclose them 2 are Parallel - forming a Z or N - thereby making the Angles Alternate Interior) => Mark them as Congruent! * Two angles are Equal (because they are the opposite angles of a parallelogram) => Mark them as Congruent!
and so on!
5. Understand that the stuff you write under the Reason column is ALWAYS one of the following:
a) A Given This is something stated or implied as 'given' in the Q itself. Or something you can 'take for granted' from the figure provided.
b) A Definition The most common ones are simple Geometric Terms like Mid-Point / Angle-Bisector / Complementary Angle / Supplementary Angle / Linear Pair / Straight Line / Perpendicular Lines / Congruent Triangles / Similar Triangles / Right Triangle / Isosceles Triangle etc.
c) A Property The most important ones are Addition Property / Subtraction Property / Transitive Property / Substitution Property / Reflexive Property / Symmetric Property.
Almost every Proof uses one or more of these Properties!
d) A Postulate This refers to statements that one takes for granted because of their axiomatic nature. Simply, they're obvious and don't need proving!
The key Postulates are: Segment Addition Postulate, Angle Addition Postulate, Linear Pair Postulate, Corresponding Angle Postulate, Corresponding Angle Converse Postulate, SSS Congruence Postulate, SAS Congruence Postulate, ASA Congruence Postulate
The following are less frequently encountered but nevertheless important: Parallel and Perpendicular Line Postulate, AA Similarity Postulate, Arc Addition Postulate
e) A Theorem This refers to hypothesis that require proving. In other words, a Theorem is something that was proved using Definitions, Properties and Postulates!
It is perfectly acceptable to use a 'simpler' Theorem in the proof of a more complex one. You can use the Vertical Angles Theorem to prove the Alternate Angle Theorem.
For a list of the most invaluable Theorems, consult the Topic: SUBTEST II: Geometry - 42 Vital Theorems on this site!
6. To actually write Proofs, first consider what is known ie. what is the Given (explicitly indicated or implied).
Next, consider what conclusions can be drawn either * from the Given OR * from some Definition, Property, Postulate or Theorem that is readily apparent and therefore, applicable.
Usually, by simply looking at the figure, a good deal of deductions can be obtained!
For the 3rd step, either * use the 2 preceding steps to draw a logical conclusion, OR again (!) * look for guidance from a Definition, Property, Postulate or Theorem
Follow the concept underlying the 3rd Step repeatedly while constantly keeping track of where you want to go - the To Prove That point!!
And then you're done! That's all there is to Proofs!!
7. It is critical, naturally, that you should have Definitions, Postulates, Properties and Theorems at the tip of your fingers!! If you lack a resolute grasp of these, it would be a futile exercise to attempt Proofs as they form the foundation: even seemingly self-evident conclusions shall then pass you by unmolested!
8. To avoid the pervasive problem of 'Where do I start?' (!) begin by writing something down, something that OBVIOUSLY follows, which could be just about anything relating to the figure or the Given or what you think could be True.
Like much of Math, simply STARING at the problem in apparent befuddlement does not help! Putting something - anything! - on paper starts your mental gears functioning.
9. Finally, writing good Proofs takes considerable Practice, a Positive Attitude and Persistence (like anything else you want to be competent at!).
Force yourself to work on 4-5 Proofs per day. Very soon - and I guarantee this! - you shall go Aha! and you'll be adept at it!
Remember: if you find something difficult to achieve, you ought to TRY HARDER, not slacken! Shying away from Proofs because you're 'not good at it' is the sorriest bloody pretext I've heard!