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# corresponding angles in transversal cutting

The following theorem is valid in Euclidean geometry:

###### Theorem 1.

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).

## Mathematics Subject Classification

51M04*no label found*51-01

*no label found*

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