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Introduction to Proofs Lesson 8 Question of the Day: Match the definition with a term from the box. Place the letter in the blank 1.) _______: two angles that add to be 90 2.) _______: equidistant from the endpoints of a segment; gives two congruent pieces 3.) _______: two or more adjacent angles that form a line 4.) _______: three or more angles with the same vertex 5.) _______: bisects a segment and forms right angles with the segment 6.) _______: angles inside the parallel lines on opposite sides of the transversal 7.) _______: add up pieces of a segment to get the entire segment 8.) _______: angles opposite each other when two lines cross 9.) _______: the angles of a triangle add to 180 10.) ______: angles outside the parallel lines on opposite sides of the transversal 11.) ______: lines that intersect at right angles 12.) ______: angles inside the parallel lines on the same side of the transversal 13.) ______: base angles are congruent; two sides are congruent 14.) ______: add up the small pieces of an angle to get the whole angle 15.) ______: two angles that add to be 180 16.) ______: Angles on the same location of the parallel lines 17.) ______: two or more lines that never intersect 18.) ______: to be equal a. line b. angles on a line c. midpoint d. ray e. vertical angles i. vertex j. segment addition f. complementary angles g. angle addition h. angle k. perpendicular bisector l. angles at a point m. congruent o. alternate interior p. alternate exterior q. triangle sum r. same side interior s. corresponding angles t. isosceles triangle u. perpendicular lines v. parallel lines n. supplementary angles 16 ~Deductive Reasoning: the ability to identify the steps used to reach a conclusion based on known facts ~Geometric Proof: A step-by-step explanation that identifies the steps used to reach a conclusion about a geometric statement. - can be written in two columns, as a paragraph, or flow chart -statements: geometric concepts (definitions and theorems) applied to the picture -goes in the left column - reasons: in the right column; what concept (definitions and theorems) was used -ex: angles on a line, midpoint -Every new concept (step) of the proof is a row ~Hints to writing proofs: 1.) Mark the figure with the information given. This is how you find out where to start. The conclusion to be proved is the end of the proof. 2.) Always keep in mind what is trying to be proved and what needs to be known to get there. 3.) Don't skip any steps, even simple ones. ~Algebraic Properties of Equality: PROPERTY ALGEBRA EXAMPLE GEOMETRY EXAMPLE Reflexive: An object is equal to itself. Transitive: If two things are equal to the same thing, they are equal to each other. Substitution: Things that are equal may be plugged in for each other. Bringing the concrete (specific) to the abstract (general): 36 102 Use the picture to illustrate the exterior angle of a triangle theorem. w Given a triangle with line BC, prove <x +<y = <z . 17 EXAMPLES: 1.) GIVEN: AB = CD Prove: AC = BD Statements Reasons Given Segment Addition Segment Addition Substitution Property Transitive Property 2.) Given: m<1 + m<2 = 66°, m<1 + m<2 + m<3 = 99°, m<3 = m<1, m<1 = m<4 Prove: m<4 = 33° 1 2 3 Statements m<1 + m<2 = 66°, m<1 + m<2 + m<3 = 99°, m<3 = m<1, m<1 = m<4 4 Reasons Substitution m<3 = 33 m<3 = m<4 Substitution ̅̅̅̅ Prove: PQ = 3.) Given: Q is the midpoint of 𝑃𝑅 1 𝑃𝑅 2 Statements Reasons PQ + QR = PR Substitution 2(PQ) = PR 18