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. 46.
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. 47):
PQ + QR = PR
PR − PQ = QR
PR − QR = 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. 48. Then the following equations hold concerning the angular measures:
∠PQS + ∠SQR = ∠PQR
∠PQR  ∠PQS = ∠SQR
∠PQR  ∠SQR = ∠PQS
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. 49, are known as vertical angles and always 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. 410, 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 .
Fig. 410 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. 411, 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 .
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. 412, 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.
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. 413). This
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. 414).
Practice problems for these concepts can be found at: Basics Of Geometry for Physics Practice Test
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