Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Noether's theorem wikipedia , lookup
History of geometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euler angles wikipedia , lookup
Cartesian coordinate system wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
History of trigonometry wikipedia , lookup
Four color theorem wikipedia , lookup
Line (geometry) wikipedia , lookup
3-6-09 Chapter 3-Logical Reasoning and Methods of Proof 3-1Implications and Proofs Synthetic proof- a proof built using system of postulates and theorems in which the properties of figures are studied (not the actual measurements) Postulates and theorems are listed on pages 668-671 Implication- if-then statement. If a, then b Ex. If a piece of fruit is and orange then it is round. pq “p implies q” (if p is true then q is true) Converse- if q, then p Ex. If a piece of fruit is round, then it is an orange. Not necessarily true. Inverse- if not p then not q Ex. If a piece of fruit is not an orange, then it is not round. Not necessarily true. Contrapositive- if not q, then not p Ex. If a piece of fruit is not round, then it is not an orange. ****The contrapositive is the logical equivalent to an implication. Ex. Given quadrilateral ABCD with sides AD and BC parallel and congruent. To prove that ABCD is a parallelogram. Properties of Quadrilaterals A quadrilateral is simply a foiur-sided polygon. a) Sun of the interior angles is 360⁰ A Trapazoid is a quadrilateral with at least (we’ll talk about it more in 3-3) one pair of parallel sides. a) Isosceles if the legs are congruent A Kite is a quadrilateral with two pairs of consecutive, congruent sides. a) Diagonals are perpendicular. A Parallelogram is a quadrilateral with two pairs of parallel sides. a) Opposite sides congruent. b) Opposite angles are congruent c) Diagonals bisect each other. (bisect- divides into two congruent sections.) d) Consecutive angels are supplementary. A Rhombus is a parallelogram with four congruent sides. a) Diagonals are perpendicular bisectors of each other. (perpendicular bisectors—90Degree angles and bisect). A Rectangle is a parallelogram with four congruent, right angles. a) Diagonals are congruent. A Square is a regular parallelogram ( four congruent sides: four congruent angles.) A square is…ALL OF THE ABOVE!! (parallelograms on…) All squares are rectangles, rhombuses, and parallelograms. As an implication…if a quadrilateral is a square then it is a (rectangle/rhombus/parallelogram). In this case the converse is not necessarily true! If a quadrilateral is a (rectangle/rhombus/parallelogram) then it is a square. A Rhombus is a kite. A kite is not necessarily a rhombus **SSS, ASA, AAS, SAS ** Given: Quadrilateral ABCD with segment AD parallel (||)segment BC and AD =BC Prove: ABCD is a parallelogram. Statements 1. AD=BC and segment AD || segment BC 2. Construct segment AC Justifications 1. Given! 2. Through any two points there is exactly one line. 3. m<ACB=m<CAD 3.Alt. Int. < Theorem, 4. AC=AC 4. Reflexive P.o.E. 5. ∆ABC=∆CDA 5. SAS!! (Side angle side) 6. m<BAC =m<DCA 6. CPCTE* 7. Segment AB ||CD Converse of Alt. Int. < Theorem… 8. ABCD is a parallelogram 8. steps 1-7 AND definition of a Parallelogram… *CPCTE- Corresponding Parts of Congruent Triangles are Equal… Given: A quadrilateral PQRS with PQ=RS and QR=PS Prove: PQRS is a parallelogram Statements Justifications 1. PQ=RS ; QR=PS 1. Given Construct segment QS Through any two points, there is exactly one line QS=QS Reflexive PoE ∆QRS=∆SPQ SSS M<SQR=m<QSP CPCTE M<RSQ-m<SQP Segment PQ||RS and Segment Converse of Alt. Int. < Theorem…!! QR||PS PQRS is a parallelogram Steps 1-6 and definition of a parallelogram. 3-17-09 3-2 Coordinate Proof… Coordinate proof- a proof based on a coordinate system in which all points are represented by ordered pairs; most coordinate proofs will not use specific coordinates but rather variables so that the proofs may be used for all situations. Important Formulas Used in Coordinate Proofs Distance… Midpoint… d ( x2 x1 ) ( y2 y1 ) 2 x x y y2 M 1 2 , 1 2 2 Slope… m y2 y1 x2 x1 Median of a Triangle- a segment joining a vertex to the midpoint of the opposite side. Altitude of a Triangle- segment from a vertex that is perpendicular to the opposite sides. Other important notes for coordinate proofs: *slopes of parallel lines must be equal *slopes of perpendicular lines must be negative reciprocals (perpendicular slopes have a product of -1) Theorem: opposite sides of a parallelogram are congruent. Given : Parallelogram ABCD with diagonal AC. Prove: AB=CD and AD=BC. Statements 1. Parallelogram ABCD with Diagonal AC 2. Segment AB|| CD; Segment AD||BC 3. m<1 = M<2: m<3=m<4 4. AC=AC 5. ∆ABC=∆CDA 6. AB=CD; AD=BC Justification. Given Definition of a Parallelogram! Alternate Interior Angle Theorem! Reflexive Property… ASA (Angle Side Angle!) CPCTE! Given: Drawing- trapezoid OPQR. Prove: The line connecting the midpoints of the legs of a trapezoid is parallel to the bases… (Segment OR and PQ are the legs!!) x x y y2 M 1 2 , 1 2 2 M of OR=(b,c) M of PQ=(a+d, c) m y2 y1 x2 x1 M of QR=(2c-2c)/(2d-2d)=0 m of OP=0 Slope of the line connecting the midpoints: m=(c-c)/(a+d)-b)=0 In order to determine that the line connecting the midpoints is parallel to the bases, I need to prove their slopes are equal. To do so, I used the midpoint formula to find the coordinates of the midpoints of the legs. They were M of OR=(b,c) M of PQ=(a+d, c) M of QR=(2c-2c)/(2d-2d)=0 m of OP=0. I found the slope of this segment and the slopes of both bases to be equal to 0. Therefore, I can state the line joining the midpoints of the legs is parallel to the bases OP and QR. Page…150-151 #’s 2-4,6-9,17-18….