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corresponding angles in transversal
cutting∗
Wkbj79†
2013-03-21 23:12:58
`
m
t
α
β1
β
.
The following theorem is valid in Euclidean geometry:
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 parallel.
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.
Remark. 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.
∗ hCorrespondingAnglesInTransversalCuttingi created: h2013-03-21i by: hWkbj79i version: h39588i Privacy setting: h1i hTheoremi h51M04i h51-01i
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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).
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