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Mth 97 Winter 2013 Sections 4.1 and 4.2 4.1 Reasoning and Proof in Geometry Direct reasoning or ________________________reasoning is used to draw a conclusion from a series of statements. Conditional statements, “if p, then q,” play a central role in deductive reasoning. “If p, the q” is written ______________, and can also be read “p ___________ q” or” p only if q.” p is the _____________________ and q is the___________________ What can you deduce from the following statements? If ∆ABC is an equilateral triangle, then it is also an equiangular triangle. ∆ABC has 3 congruent sides. Therefore (symbol , a 3 dot triangle), ∆ABC is an _______________________________. Not all statements are written in conditional form (if…then), but we can write most statements in this form. Rewrite the following statements as conditional statements. The only polyhedron with four vertices is a triangular pyramid. _____________________________________________________________________________________ A cylinder is not a polyhedron. _____________________________________________________________________________________ Law of Detachment Example: If p → q is a true conditional statement and p is true, then q is true. 1. If an angle is obtuse, its measure is greater than 90° and less than 180°. 2. ﮮB is obtuse. 3. __________________________________________ Law of Syllogism Example: If p → q and q → r are true conditional statements, then p → r is also true. 1. If M is the midpoint of AB , then AM = MB. 2. If the measures of two segments are equal, then they are congruent. 3. _____________________________________ Other forms of the conditional “If p, then q.” Converse of p → q q→p Inverse of p → q not p →not q Contrapositive of p → q not q →not p 1 Mth 97 Winter 2013 Sections 4.1 and 4.2 Write the converse, inverse and contrapositive of “If ABCD is a square, it has four right angles.” Converse ____________________________________________________________________________ Inverse ______________________________________________________________________________ Contrapositive _________________________________________________________________________ A biconditional statement exists only when the conditional statement and its converse are both _________ If p → q and q →p are true, then_____________, read “p if and only if q.” (Shorthand for this is p iff q.) Example: Write the converse of the statement below. If both the statement and its converse are true, write the biconditional statement. If three sides of a triangle are congruent, then the three angles of the triangle are congruent. Converse: _____________________________________________________________________________ Biconditional: ___________________________________________________________________________ Do ICA 6 Vertical Angles are formed by a pair of intersecting lines and are opposite each other. In the sketch below, the pair ﮮ1 and ﮮ3 and the pair ﮮ2 and ﮮ4 are vertical angles. Theorem 4.1 – If two angles are a pair of vertical angles, 2 1 3 4 . their measures are equal. mﮮ1 = mﮮ3 and mﮮ2 = mﮮ4 See proof on top of page 191 Proof – A proof is a convincing mathematical argument. The two kinds of proof used extensively in Geometry are paragraph proof and statement-reason proof (2 column proof). If two angles are congruent and supplementary, then they are right angles. Given: 1 2 and 1 and 2 are supplementary 1 2 Prove: 1 and 2 are right angles Statement 1. 2. 3. 4. 5. 6. 1 2 ; m 1 + m 2 = 180 m 1 = m 2 m 2 + m 2 =180; 2(m 2 ) = 180 m 2 = 90 m 1 = 90 1 and 2 are right angles Reason (Other samples of this proof are on page 191.) 1. 2. 3. 4. 5. 6. 2 Mth 97 Winter 2013 Sections 4.1 and 4.2 4.2 Triangular Congruence Conditions Definition of Congruent Triangles: ∆ABC ∆DEF, if and only if 1. All 3 pairs of corresponding angles are congruent. 2. All 3 pairs of corresponding sides are congruent. Reflexive Property – Something is congruent or equal to ________________ Examples: ∆DEF ∆DEF or 15 = 15 Symmetric Property – It doesn’t matter which side of the = or you are on. Examples: If ∆ABC ∆DEF, then ∆DEF ∆ABC. If 5 + 2 = 7, then 7 = ______________ Transitive Property – If two thing are or = to the same thing, then they are or = to each other. Examples: If ∆ABC ∆DEF and ∆DEF ∆GHI, then If ∆ABC ∆GHI. or If 4 + 4 = 8 and 8 = (2)(4), then 4 + 4 = ______________ SAS Postulate LL Theorem Postulates and Theorems used to prove Two Triangles are Congruent If two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent. If two legs of one right triangle are congruent respectively to two legs of another right triangle, then the two triangles are congruent. HL Theorem If the hypotenuse and a leg of one right triangle are congruent respectively to the hypotenuse and the leg of another right triangle, then the two triangles are congruent. ASA Postulate If two angles and the included side of one triangle are congruent respectively to two angles and the included side of another triangle, then the two triangles are congruent. SSS Postulate If three sides of one triangle are congruent respectively to three sides of another triangle, then the two triangles are congruent. 3 Mth 97 Winter 2013 Sections 4.1 and 4.2 Theorem 4.4 – Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle equals the square of the third side, then the triangle is a right triangle. For each pair of triangles decide whether they are necessarily congruent. If so, write and appropriate congruence statement and specify which congruence principle applies. F a) A B b) D c) C I H d) T H G U O M P A Y X W V N Proof using triangular congruence Given: RA TA, PA CA, RP CT R A C Prove : R T P Subgoal: T Statement RA TA, PA CA, RP CT 2. ∆RAP ________________ 3. R T 1. Reason 1. 2. 3. 4