Big Trouble in Little Geometry. Chapter 5.1: The Indirect Proof By Steve Sorokanich. The Indirect Proof?!. The indirect proof, also known as Modus Tollens, or “Proof by Contradiction” in Latin, uses negation of a fact in order to prove another fact to
Chapter 5.1: The Indirect Proof
By Steve Sorokanich
in Latin, uses negation
of a fact in order to
prove another fact to
also be a negative.
Well, let’s put it this way, a regular proof
Goes through the logic “If p, then q” or
“p => q”. An example of this would be
Proving two triangles congruent by SSS.
It follows that “If three sides of one triangle are
Congruent to the three sides of another triangle,
Then the two triangles are congruent to
“if not q, then not p” or “~q => ~p”. When q and p rely on each other in order to be true, then when q is negated, p is also negated. In our example, if it is proven that three sides of a triangle are not congruent, then it follows that the triangles are not congruent to each other because if they were congruent, all three pairs of sides would be congruent.
Indirect Proofs follow several steps in
Order to create an organized, logical proof.
(a) given information or
(b) a theorem, definition, or other known fact.
It’s as simple as 1, 2, 3a, 3b, 4!
Either ray RS bisects angle PRQ or RS does not bisect Angle PRQ
Assume ray RS bisects angle PRQ
Then angle PRS is congruet to angle QRS (bisector divides an angle into 2 congruent angles)
It is given that seg RS is perpendicular to PQ, so that angle PSR an angle QSR are right angles (perpendicular angles form right angles) and angle PSR is congruent to angle QSR (right angles are congruent)
Since segments RS and RS are congruent through reflexive property, then triangle PSR is congruent to triangle QSR (ASA). Therefore, segments PR and QR are congruent (CPCTC).
But this is impossible because it contradicts the given fact that segments PR and QR are not congruent.
Therefore the assumption is false and it follows that ray RS does not bisect angle PRQ because that is the only other possibility.
Chapter 5 Packet 1, page 1, problem 1
It is given that
But it is impossible Because it contradicts
Therefore the assumption is false and it follows that
Because that is the only other possibility
Chapter 5, Packet 1, Page 1 Problem 2
Either angle AOB is congruent to angle BOC or angle AOB is not congruent to BOC
Assume that angle AOB is congruent to angle BOC
It is Given That Circle O, so segments AO and OC are congruent (all radii of a circle are congruent). Since segment BO is congruent to BO (Reflexive), then triangle BAO is congruent to triangle BCO (SAS). Therefore segements AB and BC are congruent by CPCTC.
But this is impossible because it contradicts the given fact that segment AB is not congruent to segment BC.
Therefore the assumption is false and it follows that angle AOB is not congruent to angle BOC because that is the only other possibility
Geometry for Enjoyment and Challenge. Chapter
5, section 1, The Indirect Proof.