Honors Geometry Companion Book, Volume 1

1.1.1 Understanding Points, Lines, and Planes (continued)

To draw opposite rays with a common endpoint W , begin by drawing a point and labeling it W , the endpoint for both rays. Opposite rays extend in opposite directions from a common endpoint to form a line. So, if the first ray extends from W to the left, then in order for the two rays to form a line, the second ray must extend from W to the right. (Note that opposite rays do not have to form a horizontal line; any line can be formed.)

Example 3 Identifying Points and Lines in a Plane Many facts regarding geometric figures will be presented as a postulate. A postulate, or axiom, is a statement that is accepted as true without proof. Three postulates regarding the relationship between points, lines, and planes are listed below. • Through any two points there is exactly one line. • Through any three noncollinear points there is exactly one plane. • If two points lie in a plane, then the line containing those points lies in the plane.

The given figure shows one plane, F ; three points, C , D , and E ; and one line. The green line passes through two points, C and D . This line can be named  or it can be named with a reference to the two points it passes through, C and D .

Example 4 Representing Intersections When two lines or planes cross each other, they are said to intersect. Consider two intersecting lines. The point at which the two lines intersect is the point that the two lines have in common. This common point is called the intersection of the lines. Generally, the intersection of two figures is the set of all points that two figures have in common. An intersection may contain one point, a limited number of points, or an infinite number of points, depending on the figures. The following postulates describe the intersections of lines and planes.

• If two unique lines intersect, then they intersect at exactly one point. • If two unique planes intersect, then they intersect at exactly one line.


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