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By — McGraw-Hill Professional
Updated on Oct 3, 2011

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.

Parallel Lines

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).

Adjacent Angles

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.

Vertical Angles

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.)

 

Some Basic Rules More about Lines and Angles Vertical Angles

Fig. 1-9 . Vertical angles have equal measure.

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.

 

Some Basic Rules More about Lines and Angles Alternate Interior Angles

Fig. 1-10 . Alternate interior angles 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 Some Basic Rules More about Lines and Angles Alternate Interior Angles . So in shorthand, we can write ( N Some Basic Rules More about Lines and Angles Alternate Interior Angles L and N Some Basic Rules More about Lines and Angles Alternate Interior Angles 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 Some Basic Rules More about Lines and Angles Alternate Exterior Angles L and N Some Basic Rules More about Lines and Angles Alternate Exterior Angles M ) iff x = y .

Some Basic Rules More about Lines and Angles Corresponding Angles

Fig. 1-11 . Alternate exterior angles have equal measure.

Corresponding 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. 1-12, 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:

Some Basic Rules More about Lines and Angles Corresponding Angles

Fig. 1-12 . Corresponding angles have equal measure.

w = x

y = z

w = y

x = z

In shorthand, this statement is written as follows:

( N Some Basic Rules More about Lines and Angles Corresponding Angles L and N Some Basic Rules More about Lines and Angles Corresponding Angles 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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