# corresponding angles in transversal cutting

The following theorem is valid in Euclidean geometry:

###### Theorem 1.

If two lines ($\ell$ 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 ($\ell$ 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.

## References

• 1 K. Väisälä: Geometria. Kolmas painos. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1971).
 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