In this figure, and are vertical angles (and therefore congruent), as are and . Supplementary and Complementary Angles Supplementary angles are two angles that together add up to 180o. Complementary angles are two angles that add up to 90o. Whenever you have vertical angles, you also have supplementary angles. In the diagram of vertical angles above, and , and , and , and and are all pairs of supplementary angles. Parallel Lines Cut by a Transversal Lines that will never intersect are called parallel lines, which are given by the symbol ||. The intersection of one line with two parallel lines creates many interesting angle relationships. This situation is often referred to as “parallel lines cut by a transversal,” where the transversal is the nonparallel line. As you can see in the diagram below of parallel lines AB and CD and transversal EF, two parallel lines cut by a transversal will form eight angles.
Among the eight angles formed, three special angle relationships exist:Alternate exterior angles are pairs of congruent angles on opposite sides of the transversal, outside of the space between the parallel lines. In the figure above, there are two pairs of alternate exterior angles: and , and and . Alternate interior angles are pairs of congruent angles on opposite sides of the transversal in the region between the parallel lines. In the figure above, there are two pairs of alternate interior angles: and , and and . Corresponding angles are congruent angles on the same side of the transversal. Of two corresponding angles, one will always be between the parallel lines, while the other will be outside the parallel lines. In the figure above, there are four pairs of corresponding angles: and , and , and , and and . In addition to these special relationships between angles, all adjacent angles formed when two parallel lines are cut by a transversal are supplementary. In the previous figure, for example, and are supplementary. Math IC questions covering parallel lines cut by a transversal are usually straightforward. For example:
In the figure below, if lines m and n are parallel and = 110 o, then f – g =
If you know the relationships of the angles formed by two parallel lines cut by a transversal, answering this question is easy. Are alternate exterior angles, so . Is adjacent to , so it must be equal to 180o – 110o = 70o. From here, it’s easy to calculate that f – g = 110o – 70o = 40o. Perpendicular Lines Two lines that intersect to form a right (90o) angle are called perpendicular lines. Line segments AB and CD are perpendicular.
A line or line segment is called a perpendicular bisector when it intersects a line segment at the midpoint, forming vertical angles of 90o in the process. For example, in the figure above, since AD = DB, CD is the perpendicular bisector of AB. Keep in mind that if a single line or line segment is perpendicular to two different lines or line segments, then those two lines or line segments are parallel. This is actually just another example of parallel lines being cut by a transversal (in this case, the transversal is perpendicular to the parallel lines), but it is a common situation when dealing with polygons. We’ll examine this type of case later.英语作文
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