You will explore angles formed by parallel and nonparallel lines intersected by a transversal and determine how to use these angles to identify parallel lines.
After completing this tutorial, you will be able to complete the following:
Over two thousand years ago in ancient Greece, Euclid wrote the Elements, which gathered and improved upon the geometric work done by his predecessors Pythagoras, Eudoxus, and others. This book became the standard for geometry in the classical world and, though few schools today follow the text of the Elements, most high school geometry textbooks use the same concepts Euclid taught.
Euclid included several postulates, or facts, that are taken for granted and used as the starting point for logical deductions of other theorems. His fifth postulate states, "If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on the side on which the angles sum to less than two right angles." This gives a criterion for two lines to meet under certain conditions (and therefore to be nonparallel). As a result, Euclid's fifth postulate is often called the "parallel postulate."
The parallel postulate was of a much more sophisticated nature than Euclid's other postulates and axioms. Many critics of Euclid's work believed that the parallel postulate should not be a postulate, but a theorem to be derived from other axioms. However, it was later determined that the axioms these critics used to prove the parallel postulate were, in fact, equivalent to the parallel postulate.
In this Activity Object, we use the following definitions to discuss the angles formed when two lines are intersected by a transversal, which is a third line that intersects the lines at different points.
· Corresponding angles-two angles that have the same position at either intersection.
· Exterior angles-angles that lie outside the region between two lines.
o Alternate exterior angles-nonadjacent angles outside the region between two lines, but on opposite sides of the transversal.
o Same side exterior angles-exterior angles on the same side of a transversal.
· Interior angles-angles that lie inside the region between two lines.
o Alternate interior angles-nonadjacent angles inside the region between two lines, but on opposite sides of the transversal.
o Same side interior angles-interior angles on the same side of a transversal.
The following theorems about the angles formed when two lines are intersected by a transversal are fundamental for Euclidean geometry. In fact, each one is equivalent to Euclid's parallel postulate.
· Two lines cut by a transversal are parallel if and only if corresponding angles are congruent.
· Two lines cut by a transversal are parallel if and only if alternate interior angles are congruent.
· Two lines cut by a transversal are parallel if and only if alternate exterior angles are congruent.
· Two lines cut by a transversal are parallel if and only if same side interior angles are supplementary.
· Two lines cut by a transversal are parallel if and only if same side exterior angles are supplementary.
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||Students should know the definitions of parallel lines, supplementary angles, and adjacent angles.|
|Type of Tutorial||Skills Application|
|Key Vocabulary||angle, exterior angles, interior angles|