Regents Examination in Geometry: Basic Relationships Among Points, Lines, and Planes

Key Ideas
Geometric building blocks can be arranged in a number of ways relative to one another. These arrangements include parallel and perpendicular for lines and planes, and collinear for points. The relationships may be definitions, postulates, or theorems.

A definition simply assigns a meaning to a word. A postulate is a statement that is accepted to be true but is not proven. A theorem is a true statement that can be proven.


Postulates and Theorems
A definition assigns a meaning to a word using previously defined words. For example, “A triangle is a polygon with three sides.” Definitions provide only the minimum amount of information needed to define the word unambiguously. Properties that can be proven using the definition are not part of the definition. For example, in the definition of a triangle, we would not mention the fact that the angles in a triangle sum to 180°. That is a theorem that can be proven.

A postulate or an axiom is a statement that is accepted to be true but cannot be proven. When proving a theorem, we cannot rely entirely on previously proven theorems because we need to start somewhere. Postulates are that starting point. Some of the postulates may seem obvious, so obvious in fact that the best one could do is to restate the postulate in different words. For example, “Exactly one straight line may be drawn through two points” is a postulate. It is obviously true but cannot be proven using more fundamental postulates.

A theorem is a statement that can be proven true using a logical argument based on facts and statements that are accepted to be true. If points, lines, and planes are the building blocks of geometry, then theorems are the cement that binds them together. Theorems often express the relationships among the geometric figures and their measures that are the heart of geometry. An example of a theorem is “the diagonals of a square are perpendicular.” When proving a theorem, we may call upon previously proven theorems, postulates, and definitions.

Congruent
The term congruent is similar to the term equal. However, congruent applies to geometric figures while equal applies to numbers. Figures that have the same size and shape are said to be congruent. The symbol for congruent is ≅.

As often happens in mathematics, there are different approaches to determining if two figures are congruent. Since congruent figures have the same size and shape, we can compare lengths and angle measures. Two segments are congruent if their lengths are equal. Two angles are congruent if their angle measures are equal. Polygons are congruent if all pairs of corresponding angles and sides have the same measure. Circles are congruent if their radii are congruent. Alternatively, congruence can be established through transformations. Two figures are congruent if a set of rigid motion transformations map one figure onto the other. 

Keep in mind the difference in notation between congruent and equality. If two segments,

and
,

are congruent, we state that fact with
.

Since the segments are congruent, we know their lengths are equal, which we state with CD = EF. Note the difference in symbol, ≅ versus =. In addition, we use the overbar when referring to the segment and just the endpoints when referring to its length.

Congruence of segments and angles can be specified in a sketch using tick marks for segments and arcs for angles. Sides with the same number of tick marks are congruent to one another, and angles with the same number of arcs are congruent to one another. Figure 1.10 shows a parallelogram with two pairs of congruent sides and two pairs of congruent angles. The pair of long sides each have one tick mark and are congruent, while the pair of short sides each have two tick marks and are congruent. The same is true for the two pairs of angles but using arcs.

Figure below shows the congruent markings for a square. All four sides are congruent, so each side has one tick mark. The four angles are congruent, but they are also right angles, so the right angle marking can be used in place of the arcs.

Congruent markings in a parallelogram

Congruent and right angle markings in a square

Collinear and Coplanar
A set of points that all lie on the same line is described as collinear. Figure below illustrates collinear points L, M, N, O. Points that are not collinear are described as noncollinear. Points R, S, and T in next Figure (b) are noncollinear. Any two given points will always be collinear since a straight line can always be drawn through two points. This is a consequence of our first postulate.

Collinear and noncollinear points

Postulate 1
There is one, and only one, line that contains two given points.

Extending to three dimensions, a set of points that all lie on the same plane is described as coplanar. Figure below illustrates coplanar points L, M, N, O. Points that do not lie on the same plane are noncoplanar. Any three given points will always be coplanar.

Coplanar points L, M, N, O

Postulate 2
There is one, and only one, plane that contains three given points.
In addition to points, lines may also be coplanar. Coplanar lines are lines that are completely contained within the same plane. Remember, both the plane and the lines continue forever in their respective dimensions.

Intersecting, Parallel, Perpendicular, and Skew
Coincide simply means to lie on top of one another. Two lines or planes that coincide are essentially the same. Intersecting means to cross one another. Intersecting lines always cross at a single point, called the point of intersection.

Figure shows lines r and s intersecting at point M. The intersection of two planes is always a single line, called the line of intersection.

The 2nd Figure below shows planes ABC and ABD intersecting at
.

Intersecting lines


Intersecting planes

Postulate 3
The intersection of two lines is a point.

Postulate 4
The intersection of two planes is a line.

Postulate 5
Intersecting lines are always coplanar.

Perpendicular
Perpendicular is a special case of intersecting, where the lines or planes intersect at right angles. The symbol for perpendicular is ⊥. In the Figure below, line r ⊥ line s. In Figure 1.17, plane R ⊥ plane S. The small square at the right angle in Figure 1.16 is a symbol for a right angle. Segments and rays are perpendicular if the lines that contain them are perpendicular. Note that our definition of perpendicular involves right angles, not a 90° measure. Perpendicular lines lead us to right angles, and the right angles lead us to the 90° measure.

Perpendicular lines

Perpendicular planes

Parallel
Parallel lines are lines that never intersect and are coplanar. You can recognize parallel lines by the way they run in the same directions like a pair of train tracks. We use the symbol || for parallel. In Figure 1.18, line r || line s. Segments and rays are parallel if the lines that contain them are parallel. The “and are coplanar” part of the definition is important because it distinguishes parallel from skew. Planes can be parallel as well, as shown in this figure. Parallel planes never intersect.


Line r || line s


Plane R || Plane S

Skew Lines
Skew lines are lines that are not coplanar. Like parallel lines, skew lines will never intersect. However, unlike parallel lines, skew lines run in different directions.

In this figure, line

and line

are skew.

We do not have a special symbol for skew.

Skew lines

and



Math Fact
Even though

and

are not connected with arrows in this figure, a line still exists that passes through each of the two pairs of points. Any two points can be used to specify a line. The same goes for planes. Plane ACGE slices diagonally through the prism even though we do not see the points connected in the manner seen in plane EFG. Any three points can be used to specify a plane.

If you look at any pair of lines, one and only one of the following must be true. They can coincide, intersect, be parallel, or be skew. Any pair of planes will coincide, be parallel, or intersect. We do not use the word skew to describe planes.

Examples
For examples 1–5, use the figure of the cube below.


Identify 3 segments parallel to
.
Identify 4 segments perpendicular to
.
Identify 4 segments skew to
.
Identify 1 plane parallel to plane EFG.
Identify 4 planes perpendicular to plane EFG.

Solutions to examples 1–5:

,
, and

are parallel to
.
,
,
, and

are perpendicular to
.
,
,
, and

are skew to
.
Plane ABC is parallel to plane EFG.
Planes EAB, FBC, GCD, and HDA are perpendicular to plane EFG.