s. Only one circle can be drawn with a given point as center and a given distance as radius in a
plane.
t. All radii of the same circle or of equal circles are equal.
u. All diameters of the same circle or of equal circles are equal.
v. A straight line can intersect a circle at only two points. If the two points coincide, the
straight line is tangent to the circle.
w. A circle can intersect another circle at only two points.
x. A diameter bisects a circle.
y. Only one line parallel to a given line passes through a given point.
other, are equal. This method of establishing equality is called the method of superposition. In practice
you will not actually move the geometric magnitudes but merely compare them mentally. If you
conceive that one straight line can be placed upon another straight line so that the ends of both coincide,
the lines are equal. If you determine that one angle can be placed over another angle so that their
vertices coincide and their sides go in the same directions, the angles are equal. If you determine that
one figure can be placed upon another figure so that they coincide at all places, the figures are equal.
Figures that coincide exactly when superposed are congruent.
PART E - THEOREMS FOR LINES, ANGLES, AND TRIANGLES
3-8. Basic Concepts of Theorems and Corollaries. A geometric rule that can be proved by using
postulates, illustrations, and logical reasoning is called a theorem. A secondary rule whose truth can be
easily deducted from a theorem is called a corollary. You can use theorems and corollaries to solve
geometric problems without having to use the basic relationships that are established by the postulates.
3-9. Theorems for Lines. The following theorems show the relationship between straight lines in the
same plane.
a. Theorem 1. Only one perpendicular line can be drawn from a given line to a given point
outside that line. A corollary for this theorem is that a perpendicular line is the shortest line that can be
drawn from a given line to a given point.
b. Theorem 2. Two lines in the same plane and perpendicular to the same line are parallel.
Therefore, if a straight line is perpendicular to one of two parallel lines, it is also perpendicular to the
other. This relationship is shown in Figure 3-12.
EN0591
3-8
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