Overview:
Angles and the lines that form them are an essential part of geometry. Understanding the relationship between parallel lines, lines that are not parallel, and the different types of angles within figures is important to determining their measurement.
What Are the Relationships Between Angles on a Straight Line?
Suppose that three points A, B, and C are all on the same line. Then ∠ABC measures 180o by definition. If a ray goes between point B through point D, two new angles are formed, ∠DBC and ∠DBA. Those angles are called a linear pair because they have a common side, ray DB. The measurements of a linear pair, because the angles are on a straight line, add up to the same measurement as a straight line, or 180o. If ∠DBC measures 70o, then ∠DBA measures 180-70, or 110o. Similarly, if ∠DBC measures 135o, then ∠DBA measures 180-135, or 45°.
What Are Vertical Angles?
Suppose that ray DB were extended through point B and point E into a line. There would then be two new angles, ∠EBC and ∠EBA. There are now two new linear pairs, ∠DBA and ∠EBA and ∠DBC and ∠EBC. Suppose that ∠DBC measures 70°, then ∠EBC measures 110°, or 180-70. Since ∠DBC and ∠DBA are a linear pair, ∠DBA also measures 110°, the same as ∠EBC. The angles ∠DBA and ∠EBA are also a linear pair, and since ∠DBA measures 110°, ∠EBA measures 180-110, or 70°, the same as ∠DBC. The angles ∠DBA and ∠EBC are opposite each other, as are the angles ∠DBC and ∠EBA. They are called vertical angles, and vertical angles have the same measurement.
What Are Parallel Lines?
Parallel lines are on the same plane, but do not have any points in common. They never meet at any point on the plane. Suppose a parallel line to the line containing points A, B, and C were drawn through point D. That line contained points F and G, as well as D, and none of those points are on line ABC. The line BDE that intersects both parallel lines is called a transversal. The only point it has in common with line ABC is point B, and the only point it has in common with line FDG is point D.
What Is the Measurement of Angles Created by Parallel Lines?
Parallel lines go in the same direction, so the measurements of the new angles created by the intersection of the transversal with the new parallel line are equal to the measurements of the intersection of the transversal with the original line. Those angles are called corresponding angles.
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