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corresponding angles in transversal cutting
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(Theorem)
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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. $\alpha$ 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.
- 1
- K. V¨AISÄLÄ: Geometria. Kolmas painos. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1971).
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"corresponding angles in transversal cutting" is owned by Wkbj79. [ full author list (2) | owner history (1) ]
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Cross-references: convex angles, sides, perpendicular, parallel lines, converse theorem, parallel, congruent, angles, lines, Euclidean geometry, valid, theorem
There are 8 references to this entry.
This is version 9 of corresponding angles in transversal cutting, born on 2007-06-13, modified 2008-02-20.
Object id is 9588, canonical name is CorrespondingAnglesInTransversalCutting.
Accessed 4477 times total.
Classification:
| AMS MSC: | 51-01 (Geometry :: Instructional exposition ) | | | 51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries) |
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Pending Errata and Addenda
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{.} \psline(-3,-3)(3,3) \psline(-1,-... ...put[l](3.1,3){$\ell$} \rput[l](5.2,2){$m$} \rput[r](-2.2,2){$t$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/9588/js/img1.png "\begin{pspicture}(-3,-3)(3,3) \rput[b](5,-2){.} \psline(-3,-3)(3,3) \psline(-1,-... ...put[l](3.1,3){$\ell$} \rput[l](5.2,2){$m$} \rput[r](-2.2,2){$t$} \end{pspicture}")