Big Trouble in Little Geometry

1. Big Trouble in Little Geometry Chapter 5.1: The Indirect Proof By Steve Sorokanich

2. 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 also be a negative.

3. Pshhh. Yeah Right! 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 Each other.”

4. What’s So Great about Indirect Proofs? • Indirect proofs are the contrapositive of direct proofs. They follow the logic of “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.

5. The Golden Form of Indirect Proofs Indirect Proofs follow several steps in Order to create an organized, logical proof. • List the possiblilities for the conclusion • Assume that the negation of the desired conclusion is correct. • Write a chain of reasons until you reach an impossibility. This will be a contradiction of either (a) given information or (b) a theorem, definition, or other known fact. • State the remaining possibility as the desired conclusion It’s as simple as 1, 2, 3a, 3b, 4!

6. Le Examples 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

7. Practice Problem Either Assume 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

8. Practice Problem…Solution 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

9. References • Calkins,Keith J. “A Review of Basic Geometry, Lesson 11: Direct and Indirect Proofs. 30 May,2008. Andrews University. <http://www.andrews.edu/~calkins/m ath/webtexts/geom11.htm>. Geometry for Enjoyment and Challenge. Chapter 5, section 1, The Indirect Proof.