Writing Proofs Lailani Bacuyag-Riopirio How to Write a Proof? Proof – logical argument in which each statement is supported /justified by given information, definitions, axioms, postulates, theorems and previously proven statements. Postulate -a statement which is accepted without proof. Theorem – statement accepted after it is proven deductively. How to Write a Proof? Paragraph Form. It is one way of proof where you write a paragraph to explain why a given conjecture for a given situation is true. Two Column Proof. It is one way of proof where you use two columns to prove. Flow Chart Form - It is another way of writing a proof where a series of statements are organized in logical order using boxes and arrows. How to Write a Proof? Indirect Proof -method of reasoning usually written in a paragraph form. The opposite of the statement to be proven is assumed true until the assumption leads to a contradiction 1. Axioms of Equality 1.1Reflexive Property of Equality For all real numbers p, p=p E E E οBAE π΅πΈ, πΈπ΄ and π΅π΄ οRAE πΈπ΄, π΄π and πΈπ A B A A B R R 1. Axioms of Equality 1.2 Symmetric Property of Equality For all real numbers p and q, if p=q, then q=p 1. Axioms of Equality 1.2 Symmetric Property of Equality For all real numbers p and q, if p=q, then q=p 1.3 Transitive Property of Equality For all real numbers p, q and r, if p=q and q=r, then p=r. 1.4 Substitution Property of Equality For all real numbers p and 1, if p=q, then q can be substituted for p in any expression. 2. Properties of Equality 2.1. Addition Property of Equality For all real numbers p, q and r, if p=q, then p+r=q+r 2.2 Multiplication property of Equality For all real numbers p, q and r, if p=q, the pr=qr 3. Definitions, Postulates and Theorems on Points, Lines, Angles And Angle Pairs 3.1 Definition of Midpoint If points P, Q and R are collinear (P-Q-R) and Q is the midpoint of PR, then PQ = QR. P Q R 3. Definitions, Postulates and Theorems on Points, Lines, Angles And Angle Pairs 3.2 Segment Addition Postulate If points P, Q and R are collinear (P-Q-R) and Q is between points P and R, then PQ + QR=PR. P If PQ= 10 cm, what is a. QR? b. PR? Q R 3. Definitions, Postulates and Theorems on Points, Lines, Angles And Angle Pairs 3.3 Definition of Angle Bisector If ππbisects οPQR, then οPQS = οSQR. 0 If οPQS = 25 , π€βππ‘ ππ P a. mοSQR? Q S b. mοPQR? R 3. Definitions, Postulates and Theorems on Points, Lines, Angles And Angle Pairs 3.4 Angle Addition Postulate If point S lies in the interior of οPQR, then mοπππ + P ποπππ = ποπππ . Q S R 3. Definitions, Postulates and Theorems on Points, Lines, Angles And Angle Pairs 3.5 Definition of Supplementary Angles. Two angles are supplementary if the sum of their measures is 1800 . R If mοPQR+mοπ΄π΅πΆ=1800 , π‘βππ οπππ πππ οπ΄π΅πΆ are supplementary A 1200 P Q B 600 C 3. Definitions, Postulates and Theorems on Points, Lines, Angles And Angle Pairs 3.6 Definition of Complementary Angles Two angles are complementary if the sum of their measures is 900 . N D Y 600 300 S U A πΌπ πππ + οπ·π΄π = 900 , π‘βππ οπππ πππ οπ·π΄π πππ ππππππππππ‘πππ¦ angles 3. Definitions, Postulates and Theorems on Points, Lines, Angles And Angle Pairs 3.7 Definition of Linear Pair Linear pair is a pair of adjacent angles formed by two intersecting lines. 3.8 Linear Pair Theorem If two angles form a linear pair, then they are supplementary. 3. Definitions, Postulates and Theorems on Points, Lines, Angles And Angle Pairs Y 1 S Common Side 2 A Non Adjacent Sides D Note: mο1 +m ο2= 1800 Common side : π΄π Non Adjacent sides: ππ΄ πππ π΄π· 3. Definitions, Postulates and Theorems on Points, Lines, Angles And Angle Pairs 3.9 Definition of Vertical Angles Vertical angles refers o two non-adjacent angles formed by two intersecting lines. 3.10 Vertical Angles Theorem Vertical angles are congruent 3. Definitions, Postulates and Theorems on Points, Lines, Angles And Angle Pairs 1 2 4 3 1. Vertical Angles: ο1 and ο3 ο2 and ο3 2. If mο1=1100 , ππππ mο2, mο3, mο4. mο2=700 mο3=1100 mο4=700 Exercises: Lailani B. Riopirio What property/definition/theorem/postulate best describes each of the following statements? 1. If A-B-C, then AB+BC=AC. Segment Addition Postulate (SAP) 2. If A-B-C, and B is the midpoint, then AB=BC. Definition of Midpoint 3. If PQ bisects οAPB, then mοAPQ=mοBPQ. Definition of Angle Bisector What property/definition/theorem/postulate best describes each of the following statements? 4. If A-B-C-D then BC=BC. Reflexive Property 5. The sum of mοABC and mοDEF is 90o. Definition of Complementary Angles 6. The sum of mοHOT and mοDIE is 180o Definition of Supplementary Angles.
