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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 † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 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). 2