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Fundamental Rules of Geometry for Physics Help (page 2)

By — McGraw-Hill Professional
Updated on Sep 17, 2011

Perpendicular Bisector

Suppose that L is a line segment connecting two points P and R . Then there is one and only one perpendicular line M that intersects line segment L in a point Q such that the distance from P to Q is equal to the distance from Q to R . That is, every line segment has exactly one perpendicular bisector. This is illustrated in Fig. 4-6.

Basics of Geometry Fundamental Rules Perpendicular Bisector

Fig. 4-6 . Perpendicular-bisector principle.

Distance Addition And Subtraction

Let P, Q , and R be points on a line L such that Q is between P and R . Then the following equations hold concerning distances as measured along L (Fig. 4-7):

 

Basics of Geometry Fundamental Rules Distance Addition And Subtraction

Fig. 4-7 . Distance addition and subtraction.

PQ + QR = PR

PRPQ = QR

PRQR = PQ

Angle Addition And Subtraction

Let P, Q, R , and S be four points that lie in a common plane. Let Q be the vertex of three angles ∠PQR, ∠PQS, and ∠SQR, as shown in Fig. 4-8. Then the following equations hold concerning the angular measures:

∠PQS + ∠SQR = ∠PQR

∠PQR - ∠PQS = ∠SQR

∠PQR - ∠SQR = ∠PQS

Basics of Geometry Fundamental Rules Angle Addition And Subtraction

Fig. 4-8 . Angular addition and subtraction.

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. 4-9, are known as vertical angles and always have equal measure.

Basics of Geometry Fundamental Rules Vertical Angles

Fig. 4-9 . Vertical angles have equal measure.

Alternate Interior 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. 4-10, angles labeled x are alternate interior angles; the same holds true for angles labeled y . Alternate interior angles always have equal measure. Line N is perpendicular to lines L and M if and only if x = y .

Basics of Geometry Fundamental Rules Alternate Exterior Angles

Fig. 4-10 Alternate interior angles have equal measure.

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. 4-11, angles labeled x are alternate exterior angles; the same holds true for angles labeled y . Alternate exterior angles always have equal measure. The line N is perpendicular to lines L and M if and only if x = y .

Basics of Geometry Fundamental Rules Alternate Exterior Angles

Fig. 4-11 Alternate exterior angles have equal measure.

Corresponding Angles

Let L and M be parallel lines. Let N be a transversal line that intersects L and M at points P and Q , respectively. In Fig. 4-12, angles labeled w are corresponding angles; the same holds true for angles labeled x, y , and z . Corresponding angles always have equal measure. The line N is perpendicular to lines L and M if and only if w = x = y = z = 90° = π/2 radians; that is, if and only if all four angles are right angles.

Basics of Geometry Fundamental Rules Corresponding Angles

Fig. 4-12 Corresponding angles have equal measure.

Parallel Principle

Suppose that L is a line and P is a point not on L . There exists one and only one line M through P such that line M is parallel to line L (Fig. 4-13). This

Basics of Geometry Fundamental Rules Parallel Principle

Fig. 4-13 The parallel principle.

is one of the most important postulates in euclidean geometry. Its negation can take two forms: Either there is no such line M , or there exists more than one such line M 1 , M 2 , M 3 ,.... Either form of the negation of this principle constitutes a cornerstone of noneuclidean geometry that is important to physicists and astronomers interested in the theories of general relativity and cosmology.

Mutual Perpendicularity

Let L and M be lines that lie in the same plane. Suppose that both L and M intersect a third line N and that both L and M are perpendicular to N . Then lines L and M are parallel to each other (Fig. 4-14).

Basics of Geometry Fundamental Rules Mutual Perpendicularity

Fig. 4-14 Mutual perpendicularity.

 

Practice problems for these concepts can be found at:  Basics Of Geometry for Physics Practice Test

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