Section 2 - Euclid's Axiomatic Approach Flashcards
what is Euclid’s parallel axiom?
If a straight line crossing two straight lines makes the interior angles on one side together less than two right angles, then the two straight lines will meet on that side.
what is Playfair’s axiom?
given a line l and pt P (not on l), there is exactly one line on P // to l.
(this is equivalent to Euclid’s parallel axiom)
what does 𝙸.32 state?
the sum of the angles in a △ is 2 right angles
how do we prove the sum of the angles in a △ is 2 right angles?
- let ABC be a △
- draw a line DE through C // to AB
- ∠BAC=∠DCA & ∠ABC=ECB
- since ∠DCA+∠ECB+∠ACB = 180 degrees, ∠ACB+∠CBA+∠BAC also = 180 degrees
what is the SAS axiom?
if △s ABC & A’B’C’ have |AB|=|A’B’|, ∠ABC=∠A’B’C’ and, |BC|=|B’C’|, then |AC|=|A’C’|, ∠BCA=∠B’C’A’ and, ∠CAB=∠C’A’B’
we can use this to prove SSS and ASA
what does 𝙸.5 state?
In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another
how do we prove the base angles in an isosceles△ are equal?
- suppose △ABC has |AB|=|BC|
- △ABC & △CBA are congruent by SAS
- the base angles are equal
what is a parallelogram?
quadrilateral in which opposite sides are //
how do we prove that opposite sides of a //-gram are equal?
- let ABCD be a //-gram
- draw AC
- ∠DAC=∠ACB & ∠CAB=∠DCA
- by ASA △ACD=△CAB
- then |AD|=|CB| & |CD|=|AB|
what does 𝙸.15 state?
vertically opposite ∠s are equal
what does 𝙸𝙸.4 state?
If a straight line is cut at random, then the sum of the rectangles contained by the whole and each of the segments equals the square on the whole.
e. g. (a+b)² = a² + 2ab + b²
e. g. a(b+c) = ab +ac
AAA gives …
△s are congruent
AAS gives …
△s are same
ASA gives …
△s are same
SAS gives …
△s are same
SSA gives …
△s are same only if A is right angle
SSS gives …
△s are same
area of a //-gram is same as ….
area of rectangle with same base
how do you convert a rectangle into a //-gram with same base & height?
- given □OPQR & ST on extension of PQ
- |ST|=|OR|=|PQ|
- area of //-gram OSTR= area OPTR - area △OPS
- by SSS △OPS=△RQT
- so area □OPQR = area □OPQR + area △RQT - area △OPS
- so area □OPQR = area //-gram OSTR
this argument is reversible
//-grams with same …….. have same area
base & height
since area of //-gram is bh, what it area of△?
1/2 bh
what is the written version of pythagoras?
in any right angle △ the sum of the squares on the legs equals the square on the hypotenuse
what is Thales’ thm? how can we restate it?
//s cut any lines they cross into proportional segments
or
a line drawn // to one side of a △ cuts the other two sides proportionally
how do we prove that a line drawn // to one side of a △ cuts the other two sides proportionally?
- given △ABC
- let P be on AB & Q be on AC s.t. PQ//BC
- draw BQ & PC
- then area △PQC = area △QPB (same base & height)
- so area △ABQ = area △ACP
- now area △ABQ = area △APQ + area △PBQ
- △APQ & △PBQ have same height
- so |AP|/|PB| = area△APQ/area△PBQ
- similarly |AQ|/|QC| = area△APQ/area△PQC = area△APQ/area△QPB
- then |AP|/|PB| = |AQ|/|QC|
how do we prove that if A&B are two pets on a circle then for all pts C on one of the arcs joining them, ∠ABC is constant?
- let 0 be the centre of the circle, with pts A,C,B
- draw OA, OB, OC, AC, & BC
- know |OA|=|OB|=|OC| (radii)
- then ∠OAC=∠OCA & ∠OCB=∠OBC (isosceles △s)
- so ∠AOC = 180 - 2∠OCA & ∠BOC = 180 - 2∠OCB
- then ∠AOB= 360 - 2(∠OCA+∠OCB)
- but ∠AOB is a constant independent of C so ∠OCA+∠OCB=∠ACB is a constant
how do we find √x for any constructible length x
use the corollary that if AB is a diameter of a circle, and C is a pt on the circle, then √ACB is a right angle
- take a pt D on AB s.t. |AD|=x & |DB|=1
- then bisect AB at D and extend this line to the circle
- call this pt C
- then |DC|=√x by similar △s
how can we find the centre of a circle?
- take any 4 pts A,B,C,D on the circle
- draw lines AB & CD and bisect them
- the perpendicular bisectors will intersect at the centre of the circle
