PROOF HELP - Quia



PROOF HELP

1. List all the Given info as the 1st step (reason is “Given”).

2. If one step has congruent segments/angles and the next step has = segment/ angle measures, then the reason will probably be “Definition of Congruent Segments/Angles” (although it’s really a pointless step). You’ll see these occasionally in the book, but a lot of teachers aren’t that picky about it.

3. If one step has = segment/angle measures and the next step has congruent segments/angles, then the reason will probably be “Definition of Congruent Segments/Angles” (again, pretty pointless).

4. If you have a large angle and it is made up of smaller angles then you will probably need to have a statement with “Angle Addition Postulate,” especially if the Prove statement deals with them.

5. If you have a large segment and it is made up of smaller segments then you will probably need to have a statement with “Segment Addition Postulate,” esp. if the Prove statement talks about them.

6. Look for vertical angles. Make a statement that they are vertical angles with the reason as “Definition of Vertical Angles.” (Not that necessary, though.)

7. After you define vertical angles, make a statement that they are congruent with the reason as “Vertical angles are congruent.”

8. Look for linear pairs. Make a statement that they are linear pair with the reason as “Definition of Linear Pair.” (Not that necessary, though.)

9. After you define linear pairs make a statement that they are supplementary with the reason “Linear pairs are supplementary” or “Linear Pair Postulate.” You may also say that the angles add to = 180 by Linear Pair Postulate instead of stating that they are supplementary 1st.

10. It must be stated (or given) that angles are complementary or supplementary BEFORE you add them to = 90 or 180 degrees. In reverse, you can state that angles are complementary/supplementary by Definition of Compl/Suppl Angles if they’ve already been added to = 90 or 180 degrees.

11. Look for perpendicular lines. State that they are perpendicular (if not given) with the reason as “Definition of Perpendicular Lines.” Then you can mention that angles are right angles (reason: “Perpendicular lines form right angles”). Then you can state that an angle = 90 degrees by “Definition of Right Angle.”

12. Look for right angles. If not already given, state that they are right angles by “Definition of Right Angle.” Then you can mention that they are 90 degrees (also by “Definition of Right Angle”), or you can state that lines are perpendicular by “Definition of Perpendicular Lines.” Remember that if U have right angles already mentioned, U can state that they R congruent (“Rt. Angles R Congruent”).

13. You won’t need to use Symmetric Property in a proof. The book might, but you won’t – it’s unnecessary. Reflexive Property will be used whenever you a common/shared segment or angle “piece” & need other segments/angles (usually used with Segment/Angle Addition Postulates & possibly with Addition/Subtraction Properties). Transitive Property & Substitution Property are practically interchangeable. Anytime you’re doing a replacement, you can use Substitution Property.

14. Addition/Subtraction/Multiplication/Division Properties are unnecessary. You must be doing an operation with the same (or equal) thing(s) on both sides in order to use them. If major math is happening, but only on 1 side, then probably the reason will be Distributive Property.

15. Reflexive Property (1 object congruent/= to itself) ; Symmetric Property (2 objects congruent/= to each other said 1 way, then reversed) ; Transitive Property (3 objects congruent/= to each other, but in a chain). Remember which is which this way : “R S T –> 1 2 3.”

16. You must state the Prove statement as your last step (but “Prove” or “Proof” is NOT the reason).

Reason options : given, math facts, definitions, postulates, axioms, properties, & previously proven theorems.

You need a definition as the reason when a statement says what something is or when a statement happens because of what something does (provided that the info isn’t already from the given info).

Example : If it is given that a point is a midpoint of a segment, then it will bisect the segment into 2 congruent pieces. Stating that those pieces are congruent is because of “Definition of Midpoint.” This also works in reverse. If 2 segments are given as congruent & those segments are attached in a collinear fashion, then you can state that the shared/common point is a midpoint by “Definition of Midpoint.”

Remember that ALL definitions are biconditional & can work “both ways.”

When proving a conditional statement (including a theorem), the “if” stuff is the Given, and the “then” stuff is what you are trying to Prove.

5 different proof styles : 2-column, paragraph, flow (use statements & reasons, although flow proofs do show more connection between/among the steps), coordinate (uses the xy-graphing grid, general ordered-pairs like (x, y), & is more mathematical in process (slope, midpoint formula, & distance formula), and indirect (using contradiction as a process of elimination so that the only option remaining is what you are actually trying to prove).

Follow your book examples and other proof problems (from teacher or otherwise). Really look at the structure & format used in them. Look at the odd-numbered problem proofs in the back of your book & use them as a guide for the even problem proofs. You must write/copy everything down every time (including the Given, the Prove, & any picture) – it’s the only way that the info (& the proof process) gets in your brain properly & more quickly. If there isn’t a picture, then draw one that represents the Given & Prove information. Make the connections. Only you can do this, not your teacher. There is no way that your teacher can do the thinking for you. All we can do is guide your thinking in the proper fashion. If there was an “EZ” way to do proofs, we would have told you from the beginning.

You do “proofs” everyday. Anytime you have to explain or show why or how something happened (in math or otherwise), you are giving a proof. Anytime you reason through a situation to verify, justify, or prove something (or your side of the story), you are doing a proof. What you haven’t really done before is write it down in a proper, organized, logical format. Now you will. Think of it like you do a persuasive essay in English class – you take a position & provide logical, supporting statements to justify your position. That’s exactly what a proof is except your “target audience” is usually your Geometry teacher Don’t worry – ALL geometry students before you, with you, & after you have & will do proofs. You are not alone. Fighting against them only makes the process more painful, because they will never go away completely. Work for the proofs, work with the proofs – don’t work against them. Practice is the only way to improve your proving skills, both in class & in life.

Parallel Line Proofs

If lines are Given to you as parallel, then they will cause angle stuff to happen (either congruent or supplementary). What matters most is recognizing the positioning of the angles :

Corresponding, Alternate Interior, Alternate Exterior, Same-Side Interior, Same-Side Exterior, Linear Pairs, & Vertical

Those angle positions will give away the name of the reason that you need. Corresponding & Linear Pairs are the only ones that are postulates – the rest are theorems. Same-Side Interior/Exterior & Linear Pairs are supplementary – the rest are congruent. Don’t forget about “Perpendicular Transversal Theorem” – “If a transv. is perpend. to 1 of 2 || lines, then it’s perpend. to the other.”

If you need to show that lines BECOME parallel, then let the angle positions & whether they are supplementary or congruent give away the name of the reason. If the angles amounts/positions CAUSE the lines to become parallel, then say that the lines R || by the “Converse” version of the previous theorems/postulates. Don’t be surprised if you have to use both “regular” and “converse” versions in the same proof.

Also, remember the 2 other ways to prove lines become parallel (7 ways total):

a) 2 lines || to same line R || b) In a plane, 2 lines perpendicular to same line R ||.

Congruent Shapes (especially triangles)

Make/Draw/Copy pictures. Mark on those pictures. Let those markings guide you towards 3 things: consistency, correspondence, & congruency (the 3 C’s). For correspondence, think about the pairs of sides & angles that would match up if the shapes were positioned the same way. Be consistent in how you name the shapes – again, let the picture markings help you.

Definition of Congruence : if 2 shapes R congruent, then all of their corresponding part pairs R congruent (CPCTC – corresp. parts of congruent triangles R congruent)

if 2 shapes have all of their corresponding part pairs congruent, then the shapes themselves R congruentto help U get corresponding part pairs congruent, remember to “Look 4”:

1) Given (& what it leads to) 2) Parallel Lines (usually Alt. Int. Angle Th.)3) Vertical Angles (Vert. Angles R Congruent) 4) Shared Sides/Angles (Reflex. Prop)

After saying that triangles are congruent, then you can mention any of the other 3 corresponding part pairs as congruent by CPCTC. Those CPCTC things may also help you prove other stuff, such as: other congruent triangles; midpoints; || lines; angle/segment bisectors; etc.

Isosceles Triangles

Triangle Base Angle Theorem (Isos. Triangle Th.) : 2 sides of a triangle R congruent, so U already know that it’s isos., therefore U mention about the base angles congruent.

Triangle Base Angle Converse Th. (Isos. Triangle Conv. Th.) : 2 angles of a triangle R congruent which makes 2 sides congruent which causes the triangle to become isosceles.

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