Writing two column proofs in math 601 (Mr. Kadish) Here is a rundown on what you can and can’t use as definitions, theorems and postulates in two-­‐column proofs. Remember: you may always write a complete sentence in “if… then” format describing your justification for a step in a proof. Symbols: You may always use the following to represent words in 2 column proofs: ∠ angle Δ = ≠ ≅ m∠ triangle is equal to is not equal to is congruent to the measure of angle… 
AB line AB AB segment AB or line segment AB AB
 = the measure of segment AB AB ray AB ⊥ perpendicular || parallel rt. right str. straight comp complement supp supplement ~ is similar to In your mathematical writing, there are many other symbols you may use {≥,∴,⇔,⇒,⊃,⊇,∈∉,∞,Σ,∀,π…} and we’ll use more this year. Using Definitions: Write as an “if…then” sentence (using symbols listed above) There are too many to list! As an example, if you are given a midpoint and want to state that this creates ≅ segments, you would write the definition in “If…then” form, making sure to start your conditional statement with the hypothesis (a midpoint) and ending with the necessary conclusion (≅ segements): “If a point is the midpt. of a segment, then it divides the segment into two ≅ segments.” Exeption: The definition of congruent triangles: Δs in which all pairs of corresponding parts are ≅. Use the abbreviation CPCTC, standing for “Corresponding Parts of Congruent Triangles are Congruent,” when showing that (corresponding)parts within ≅ Δs are ≅. Theorems (in the most abbreviated form you may use) If two ∠s are rt. ∠s, they are ≅ If two ∠s are str. ∠s, they are ≅ If ∠s are supp to the same ∠, then they are ≅ If ∠s are supp to ≅ ∠s, then they are ≅ If ∠s are comp to the same ∠, then they are ≅ If ∠s are comp to ≅ ∠s, then they are ≅ Addition prop. for segment (or ∠) ≅ Subtraction prop. for segment (or ∠) ≅ Multiplication property for segment (or ∠) ≅ Division property for segment (or ∠) ≅ *Through chapter 3 only. Transitive Property for segment (or ∠) ≅ Reflexive Property Vertical ∠s are ≅ All radii of a  are ≅ (these are the angle-­‐side theorems used in isosceles Δs-­‐ to If , then . show sides are ≅ based on the ∠s opposite them and vice versa. Maybe a picture is worth a thousand words-­‐ but If , then . here they’ll save you about a dozen) Equidistance Thm. (≅ segmets ⇒ ⊥ bis.) Equidistance Thm (⊥ bis ⇒ ≅ segments) Postulates: SSS (list ≅ parts) SAS (list ≅ parts) ASA (list ≅ parts) AAS (list ≅ parts) HL (list ≅ parts) Two points determine a line. (use whenever you add an auxiliary line) *** You do not need to use the reflexive property when using the addition or subtraction property when the same segment or ∠ is being added or subtracted from ≅ segments or ∠s. You do need to use the reflexive property to show that a segment or ∠ is equal to itself when you are proving that overlapping objects are ≅. This will come up frequently in establishing triangle congruence. Some general notes on writing proofs: Statements Column Reasons Column In this column, list each step in your logical chain. You must start with something given or properly assumed from the diagram. Each step must follow directly from some previous step(s). This is a list of simple statements you are making. How you know each statements is true belongs in the reasons column. In this column write Theorems, postulates and definitions which justify each logical step in the statements column. * note: in our deductive structure of proof, the reasons are general ideas of geometry, not specific to this problem only. In this column, Don’t write: if ∠DEF is a rt .∠, then… Do write: If an ∠ is a rt. ∠, then… A simple example of a proof Statements  
1) AB ⊥ BC 2) m∠ABC = 90° 3) m∠QRT = 90° 4) ∠ABC ≅ ∠QRT Common errors: Converse error:  
Given: AB ⊥ BC m∠QRT = 90° Prove: ∠ABC ≅ ∠QRT Hypothesis (If we know a and b are true) Conclusion (show that d is true) Reasons (the logical chain, symbolically) 1) given 1) we know a 2) If lines are ⊥ , then they 2) b follows logically from a intersect at a 90° ∠ 3) given 3) we know c 4) d follows logically from the 5) If 2 ∠s have the same combination of b and c measure, then they are ≅ }
D Given: and trisect Prove: S R 2. ∠DAB ≅ ∠BAC ≅ ∠CAE 1. given 2. If two rays divide an ∠ into three ≅ ∠ s, then they trisect the ∠ . 
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1. AB and AC trisect ∠DAE C A CONVERSE ERROR Notice the difference between ≅ and =. B E S 
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1. AB and AC trisect ∠DAE R 1. given 2. If two rays trisect 2. ∠DAB ≅ ∠BAC ≅ ∠CAE an ∠ , then they divide the ∠ into three ≅ ∠ s. CORRECT REASON The definition of ≅ ∠s is ∠s which have the same measure, but ∠A ≅ ∠B is a different statement than m∠A = m∠B. Similarly, a rt. ∠ is defined as an ∠ measuring 90°, but pay attention to when you write ∠A is a rt. ∠ or that m∠A=90°. If you want to prove two rt. ∠s are ≅, you must have a statement for each establishing that they are rt. ∠s, or a statement for each establishing that they are ∠s measuring 90°-­‐ don’t have one of each! Often, as with ⊥ lines, you can use the definition to conclude that there is a rt. ∠, or that there is 90° ∠. Make sure you match the language with any conclusion you draw later (we have a definition stating that ∠s with the same measure are ≅. We have a theorem stating that ∠s which are right ∠s are ≅. We do not have any established definition, theorem or postulate stating that an ∠s measuring 90° and a rt. ∠ are ≅) Language use must be precise!