First, if a transversal intersects two parallel lines, then the alternate interior angles are congruent.
2.
As the proof only requires the use of Proposition 27 ( the Alternate Interior Angle Theorem ), it is a valid construction in absolute geometry.
3.
Euclid's Proposition 27 states that if a transversal intersects two lines so that alternate interior angles are congruent, then the lines are parallel.
4.
To prove proposition 29 assuming Playfair's axiom, let a transversal cross two parallel lines and suppose that the alternate interior angles are not equal.
5.
In the figure at the right, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because vertical angles are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent.
6.
The Euclidean proof of the HSEAT ( and simultaneously the result on the sum of the angles of a triangle ) starts by constructing the line parallel to side " AB " passing through point " C " and then using the properties of corresponding angles and alternate interior angles of parallel lines to get the conclusion as in the illustration.
7.
By the alternate interior angle theorem, " l " is parallel to " n " . ( The alternate interior angle theorem states that if lines a and b are cut by a transversal t such that there is a pair of congruent alternate interior angles, then a and b are parallel . ) The foregoing construction, and the alternate interior angle theorem, do not depend on the parallel postulate and are therefore valid in absolute geometry.
8.
By the alternate interior angle theorem, " l " is parallel to " n " . ( The alternate interior angle theorem states that if lines a and b are cut by a transversal t such that there is a pair of congruent alternate interior angles, then a and b are parallel . ) The foregoing construction, and the alternate interior angle theorem, do not depend on the parallel postulate and are therefore valid in absolute geometry.
9.
By the alternate interior angle theorem, " l " is parallel to " n " . ( The alternate interior angle theorem states that if lines a and b are cut by a transversal t such that there is a pair of congruent alternate interior angles, then a and b are parallel . ) The foregoing construction, and the alternate interior angle theorem, do not depend on the parallel postulate and are therefore valid in absolute geometry.
10.
By the alternate interior angle theorem, " l " is parallel to " n " . ( The alternate interior angle theorem states that if lines a and b are cut by a transversal t such that there is a pair of congruent alternate interior angles, then a and b are parallel . ) The foregoing construction, and the alternate interior angle theorem, do not depend on the parallel postulate and are therefore valid in absolute geometry.
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