corresponding angles in transversal cutting

The following theorem is valid in Euclidean geometry:

Theorem 1.

If two lines ($\mathrm{\ell }$ and $m$) are cut by a third line, called a transversal ($t$), and one pair of corresponding angles (e.g. (http://planetmath.org/Eg) $\alpha$ and $\beta$) are congruent, then the cut lines are parallel.

Its converse theorem is also valid in Euclidean geometry:

Theorem 2.

If two parallel lines ($\mathrm{\ell }$ and $m$) are cut by a transversal ($t$), then each pair of corresponding angles (e.g. $\alpha$ and $\beta$) are congruent.

Remark.

The angle $\beta$ in both theorems may be replaced with its vertical angle ${\beta }_1$.  The angles $\alpha$ and ${\beta }_1$ are called alternate interior angles of each other.

Corollary 1.

Two lines that are perpendicular to the same line are parallel to each other.

Corollary 2.

If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.

Corollary 3.

If the left sides of two convex angles are parallel (or alternatively perpendicular) as well as their right sides, then the angles are congruent.

Title	corresponding angles in transversal cutting
Canonical name	CorrespondingAnglesInTransversalCutting
Date of creation	2013-03-22 17:15:12
Last modified on	2013-03-22 17:15:12
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	12
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 51M04
Classification	msc 51-01
Related topic	EuclideanAxiomByHilbert
Related topic	HarmonicMeanInTrapezoid
Defines	transversal
Defines	vertical angle
Defines	alternate interior angle