corresponding angles in transversal cutting


The following theorem is valid in Euclidean geometryMathworldPlanetmath:

Theorem 1.

If two lines ( and m) are cut by a third line, called a transversal (t), and one pair of corresponding angles (e.g. ( α and β) are congruent, then the cut lines are parallelMathworldPlanetmathPlanetmath.

Its converse theorem is also valid in Euclidean geometry:

Theorem 2.

If two parallel lines ( and m) are cut by a transversal (t), then each pair of corresponding angles (e.g. α and β) are congruent.


The angle β in both theorems may be replaced with its vertical angle β1.  The angles α and β1 are called alternate interior angles of each other.

Corollary 1.

Two lines that are perpendicularPlanetmathPlanetmath 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.


  • 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