Tips for Writing proofs!

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Tips for Writing proofs!

• Step-by-step

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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. . .

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Use colored highlighters 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!

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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.
  . .

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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!

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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!

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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!

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C P C T C

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

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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 . . .

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Proving a median

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

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To prove that a segment is an ALTITUDE

• YOU NEED
• Perpendicular Segments
• Or
• Right Angles

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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!)

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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!

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Dont 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
• v Givens
• v Prove (or conclusion)
• v DIAGRAM
• You DO always include and use the DIAGRAM, right?
• v Statements
• v Reasons