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Euclidean Geometry |
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Complex Numbers
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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
The above theorem immediately implies the foregoing.
| Theorem: Corresponding Angles for a transversal between two lines when and only when the lines are parallel. |
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
+
Conversely, If the two lines are parallel then
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