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Introduction to Geometry Rules for Lines and Angles
In the confines of a single geometric plane, lines and angles behave according to various rules. The following are some of the best-known principles.
Two lines are parallel if and only if they lie in the same plane and they do not intersect at any point. Two line segments or rays that lie in the same plane are parallel if and only if, when extended infinitely in both directions to form complete lines, those complete lines do not intersect at any point.
Complementary And Supplementary
Two angles that lie in the same plane are said to be complementary angles (they “complement” each other) if and only if the sum of their measures is 90° (π/2 rad). Two angles in the same plane are said to be supplementary angles (they “supplement” each other) if and only if the sum of their measures is 180° (π rad).
Suppose that L and M are two lines that intersect at a point P . Then any two adjacent angles between lines L and M are supplementary. This can be illustrated by drawing two intersecting lines, and noting that pairs of adjacent angles always form a straight angle , that is, an angle of 180° (π rad) determined by the intersection point and one of the two lines.
Suppose that L and M are two lines that intersect at a point P . Opposing pairs of angles, denoted x and y in Fig. 1-9, are known as vertical angles . Pairs of vertical angles always have equal measure. (The term “vertical” in this context is misleading; a better term would be “opposite” or “opposing.” But a long time ago, somebody decided that “vertical” was good enough, and the term stuck.)
Alternate Interior, Alternate Exterior, and Corresponding Angles
Alternate Interior Angles
Suppose that L and M are parallel lines. Let N be a line that intersects lines L and M at points P and Q , respectively. Line N is called a transversal to the parallel lines L and M . In Fig. 1-10, angles labeled x are alternate interior angles ; the same holds true for angles labeled y . Pairs of alternate interior angles always have equal measure.
If line N is perpendicular to lines L and M , then x = y . Conversely, if x = y , then N is perpendicular to lines L and M . When a logical statement works both ways like this, the expression “if and only if” (often abbreviated “iff”) is used. Here, x = y iff N is perpendicular to both L and M . The phrase “is perpendicular to” is often replaced by the symbol . So in shorthand, we can write ( N L and N M ) iff x = y .
Alternate Exterior Angles
Suppose that L and M are parallel lines. Let N be a line that intersects L and M at points P and Q , respectively. In Fig. 1-11, angles labeled x are alternate exterior angles ; the same holds true for angles labeled y . Pairs of alternate exterior angles always have equal measure. In addition, ( N L and N M ) iff x = y .
Suppose that L and M are parallel lines. Let N be a line that intersects L and M at points P and Q , respectively. In Fig. 1-12, angles labeled w are corresponding angles ; the same holds true for angles labeled x, y , and z . Pairs of corresponding angles always have equal measure. In addition, N is perpendicular to both L and M if and only if one of the following is true:
w = x
y = z
w = y
x = z
In shorthand, this statement is written as follows:
( N L and N M ) iff ( w = x or y = z or w = y or x = z )
Practice problems for these concepts can be found at: Geometry Basic Rules Practice Test.
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