Overview: Lines and Angles
Lines in geometry exist in relationships to one another by the angles they form when they intersect. Those lines that do not intersect at all are parallel to each other. Those that intersect at one point or another form angles where they intersect. Perpendicular lines intersect at right angles, and have special relationships to one another.
What Are Parallel Lines?
Parallel lines do not intersect at any point. If they are in the same plane, parallel lines have the same tilt. When they are graphed, both lines have the same slope. If a transversal line crosses both parallel lines, the corresponding angles have the same measurement. This property of parallel lines allows students to deduce the measurement of one angle formed by one of the parallel lines and the transversal, if the measure of a corresponding angle is known.
Parallel Lines Postulate
As an example, in the figure below, lines a and b are parallel, and line t is a transversal cutting them. Suppose angle β measures 1100 where the transverse line t cuts through line a. Its corresponding angle where the transverse line t cuts through line b will also measure 1100. The measure of angle β and angle θ is 1800, so angle θ (theta) measures 180-110 or 700.
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What Are Perpendicular Lines?
Perpendicular lines exist in a special relationship to one another, as they intersect at a 900 angle. They do not have to be horizontal or vertical, but just intersect at right angles. By extension of the Parallel Lines Theorem, if two lines are perpendicular to the same line, they are parallel to each other. If one angle measures 900, all the other angles (adjacent, vertical, and corresponding) that are formed by a transverse line cutting another line will also measure 900. Suppose angle β in the figure above measures 900. Angle θ also measures 900. This application is also called the Perpendicular to Parallels Theorem.
Construction of Parallel and Perpendicular Lines
Geometric properties of normal two-dimensional space ensure that parallel and perpendicular lines are readily constructed using nothing but an unmarked straightedge and compass. By using the endpoints of a line segment, students can construct another line perpendicular to that segment, and then construct a parallel line with the same slope as the first line. Students can use the process to demonstrate how the theorems work, by creating the transverse perpendicular to the first line, then constructing another line parallel to it.
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