Download 3-6-09 Chapter 3-Logical Reasoning and Methods of Proof 3

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.
pq “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….