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Tips for Writing proofs!

Tips for Writing proofs!. Step-by-ste p. Draw the diagram. WHY? The diagram helps you see what you have and where to go with it . . . The diagram helps you process what the “givens” mean in or lend to a situation . . Sometimes you can make some ALLOWABLE assumptions from the diagram:

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Tips for Writing proofs!

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  1. Tips for Writing proofs! Step-by-step

  2. Draw the diagram • WHY? • The diagram helps you see what you have and where to go with it . . . • The diagram helps you process what the “givens” mean in or lend to a situation . . • Sometimes you can make some ALLOWABLE assumptions from the diagram: • Straight angles • Vertical angles • Supplementary Angles • Or maybe, • ⇘ Complementary Angles ⇙ • (but ONLY if you have two angles whose sum is equal to a MARKED right angle!) • Etc. . .

  3. Usecoloredhighlighters or pencils • Why? • When you mark the diagram using color along with your tick marks, it is easier to see and remember what you have left to do in order to COMPLETE your proof!

  4. Write the first “given” as Statement 1 • Why? • A “GIVEN” statement often leads to other statements . . . • If a ray bisects and angle, then it divides the angle into 2 congruent angles • If a point is a midpoint, then it divides a segment into 2 congruent segments • If two lines (segments) are perpendicular, then they form right angles • If two angles are right angles, then they are congruent • If two angles are complementary or supplementary to congruent angles or the same angle, then the angles are congruent (by substitution) • So, be sure to list these types of “other statements” that are related to the first given before writing the next “given” . . . and so on. . .

  5. REASONS should be written in “If ___, then ___.” form • Why? • Writing your reasons in “If – then” form helps you to be absolutely certain that you have provided all the necessary links in your logical argument BEFORE . . . • you make your final “Prove” statement!

  6. When proving two triangles congruent, use one of the following Postulates: • SSS (side – side – side) • SAS (side – included angle – side) • ASA (angle – included side – angle) • HL (RIGHT triangleS with . . . • pairs of corresponding ≅ hypotenuse and ≅ leg) • Remember! SSA and AAA do NOT work • for proving ∆ congruence!

  7. If you have two pairs of corresponding parts and you need a third part, look for: • Segments or angles that are part of BOTH triangles . . . Then use Reflexive Property! • Vertical angles . . . These can be assumed, so use the theorem “vertical angles are congruent.” • Segments or angles that can be proven congruent by segment or angle Addition or Subtraction Properties. • If one or more circles are involved and you need congruent segments, do not forget that • “All radii of a given circle are congruent!”

  8. C P C T C • Before using C P C T C, • You MUST have proven two triangles to be congruent FIRST!!! • Also . . .

  9. Beyond C P C T C . . . • Many proofs involve steps beyond C P C T C! • AFTER using CPCTC, you may be able to identify: • Medians • Altitudes • Angle Bisectors • Midpoints • Etc . . .

  10. Proving a median • If you are trying to prove that a segment is a median, • YOU NEED: • A MIDPOINT • . . . and • TWO congruent segments

  11. To prove that a segment is an ALTITUDE: • YOU NEED: • Perpendicular Segments • ~ Or ~ • Right Angles

  12. Auxiliary lines • Sometimes you may find it necessary to introduce a line in a given diagram thus CREATING the two triangles you are then able to prove as congruent ! • When that is the case YOU MUST write: • STATEMENT: “Draw ____(name of segment)____” • REASON: “Two points determine a LINE” (… segment or ray) • Be sure to follow immediately with: • STATEMENT: __(name of segment)__≅ __(name of segment)__ • REASON: Reflexive Property (any segment is always congruent to itself!)

  13. DPT Cards • Be sure to keep adding to your DPT cards. • Include illustrations of Definitions, Postulates, and Theorems! • Review your DPT cards AT LEAST once per week!

  14. Don’t forget your Textbook! • Remember to USE your textbook as a study RESOURCE. • Be sure to READ each section very carefully – and read them more than once. • Make your own NOTES from what you read and observe in each section. • Study each SAMPLE PROBLEM very closely so that you know how a proof should look from beginning to end. • Notice how every Sample Proof has all FIVE parts of a two-column proof: √ Givens √ Prove (or conclusion) √ DIAGRAM You DO always include and use the DIAGRAM, right? √ Statements √ Reasons

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