# 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.

## 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 |