YOU are better than YOU think. Show yourself  how:

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 Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and writing

Do not leave here without it -  Logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

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After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Explore collaborative whiteboards from groupboard,  twiddla  or scriblink.

Parallel Lines 

In other words, the postulate assumes the sum of interior angles for that transversal between two lines sum to 180 degrees (two right angles) when and only when the lines are parallel. 

Theorem: Corresponding Angles for a transversal between two lines when and only when the sum of interior angles for that transversal sum to 180 degrees (two right angles)

Proof:

Here = 180 degrees. Now (corresponding angles equal) implies gamma + beta = 180 degrees. The latter in turn implies the two lines are parallel.  

Conversely, if the two lines  are parallel, then  = 180 degrees. The latter along with = 180 degrees (due to their being components of a straight angle) gives So corresponding angles are equal. 

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The above theorem immediately implies the foregoing.

Theorem: Alternative angles for a transversal between two lines when and only when the lines are parallel.

Proof: If alternating angles and are equal then + = 180 degrees implies +   = 180 degrees. So the sum of interior angles on one side of the transversal sum to 180 degrees.  When the lines are parallel. 

Conversely, If the two lines are parallel then  + + 180 degrees combined with +=180 degrees gives = . So alternate angles are equal.  

Parallel Postulate (Alternative View)

The angle side angle method fails to construct a triangle on both sides of the segment if the sum of the angles is a straight angle.

The parallel postulate may be stated as follows: Two lines will not intersect if the sum of interior angles on any side of a transversal sum to two right angles. 

Equivalent ways of implying the sum of interior angles for a transversal equal a straight angle are follows.

• alternate angles equals
• corresponding angles equal
• sum of exterior angles equal a straight angle, two right angles or 180 degrees)

• sum of exterior angles exceed a straight angle
• alternate angles are not equal - does not specify the side
• corresponding angles are not equal - does not specify the side