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.

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.

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.