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Points, Lines, and Planes

Identify and describe points, lines, and planes.

Learning Objectives

After completing this section, you should be able to:

  1. Identify and describe points, lines, and planes.
  2. Express points and lines using proper notation.
  3. Determine union and intersection of sets.

Points and Lines

The first definition Euclid wrote was that of a point. He defined a point as “that which has no part.” It was later expanded to “an indivisible location which has no width, length, or breadth.” Here are the first two of the five postulates, as they are applicable to this first topic:

  1. Postulate 1: A straight line segment can be drawn joining any two points.
  2. Postulate 2: Any straight line segment can be extended indefinitely in a straight line.

Before we go further, we will define some of the symbols used in geometry in :

From , we see the variations in lines, such as line segments, rays, or half-lines. What is consistent is that two collinear points (points that lie on the same line) are required to form a line. Notice that a line segment is defined by its two endpoints showing that there is a definite beginning and end to a line segment. A ray is defined by two points on the line; the first point is where the ray begins, and the second point gives the line direction. A half-line is defined by two points, one where the line starts and the other to give direction, but an open circle at the starting point indicates that the starting point is not part of the half-line. A regular line is defined by any two points on the line and extends infinitely in both directions. Regular lines are typically drawn with arrows on each end.

Defining Lines

Try it.

For the following exercises, use this line ().

  1. Define \(\overset{\bar}{DE}\).
  2. Define \(F\).
  3. Define \(\overset{↔}{DF}\).
  4. Define \(\overset{\bar}{EF}\).
Solution
  1. The symbol \(\overset{\bar}{DE}\), two letters with a straight line above, refers to the line segment that starts at point \(D\) and ends at point \(E\).
  2. The letter \(F\) alone refers to point \(F\).
  3. The symbol \(\overset{↔}{DF}\), two letters with a line above containing arrows on both ends, refers to the line that extends infinitely in both directions and contains the points \(D\) and \(F\).
  4. The symbol \(\overset{\bar}{EF}\), two letters with a straight line above, refers to the line segment that starts at point \(E\) and ends at point \(F\).

There are numerous applications of line segments in daily life. For example, airlines working out routes between cities, where each city’s airport is a point, and the points are connected by line segments. Another example is a city map. Think about the intersection of roads, such that the center of each intersection is a point, and the points are connected by line segments representing the roads. See .

Condensed — the full section is in OpenStax Contemporary Mathematics.

Parallel Lines

Parallel lines are lines that lie in the same plane and move in the same direction, but never intersect. To indicate that the line \({l}_{1}\) and the line \({l}_{2}\) are parallel we often use the symbol \({l}_{1}∥{l}_{2}.\) The distance \(d\) between parallel lines remains constant as the lines extend infinitely in both directions. See .

Perpendicular Lines

Two lines that intersect at a \({90}^{∘}\) angle are perpendicular lines and are symbolized by \(⊥\). If \({l}_{1}\) and \({l}_{2}\) are perpendicular, we write \({l}_{1}⊥{l}_{2}.\) When two lines form a right angle, a \({90}^{∘}\) angle, we symbolize it with a little square \(□.\) See .

Identifying Parallel and Perpendicular Lines

Try it.

Identify the sets of parallel and perpendicular lines in .

Solution

Drawing these lines on a grid is the best way to distinguish which pairs of lines are parallel and which are perpendicular. Because they are on a grid, we assume all lines are equally spaced across the grid horizontally and vertically. The grid also tells us that the vertical lines are parallel and the horizontal lines are parallel. Additionally, all intersections form a \({90}^{∘}\) angle. Therefore, we can safely say the following:

\(\overset{↔}{AB}∥\overset{↔}{CD}\), the line containing the points \(A\) and \(B\) is parallel to the line containing the points \(C\) and \(D\).

\(\overset{↔}{EF}∥\overset{↔}{GH}\), the line containing the points \(E\) and \(F\) is parallel to the line containing the points \(G\) and \(H\).

\(\overset{↔}{AB}⊥\overset{↔}{EF}\), the line containing the points \(A\) and \(B\) is perpendicular to the line containing the points \(E\) and \(F\). We know this because both lines trace grid lines, and intersecting grid lines are perpendicular.

We can also state that \(\overset{↔}{AB}⊥\overset{↔}{GH}\); the line containing the points \(A\) and \(B\) is perpendicular to the line containing the points \(G\) and \(H\) because both lines trace grid lines, which are perpendicular by definition.

We also have \(\overset{↔}{CD}⊥\overset{↔}{EF}\); the line containing the points \(C\) and \(D\) is perpendicular to the line containing the points \(E\) and \(F\) because both lines trace grid lines, which are perpendicular by definition.

Finally, we see that \(\overset{↔}{CD}⊥\overset{↔}{GH}\); the line containing the points \(C\) and \(D\) is perpendicular to the line containing the points \(G\) and \(H\) because both lines trace grid lines, which are perpendicular by definition.

Defining Union and Intersection of Sets

Union and intersection of sets is a topic from set theory that is often associated with points and lines. So, it seems appropriate to introduce a mini-version of set theory here. First, a set is a collection of objects joined by some common criteria. We usually name sets with capital letters. For example, the set of odd integers between 0 and 10 looks like this: \(A=\{1,3,5,7,9\}.\) When it involves sets of lines, line segments, or points, we are usually referring to the union or intersection of set.

The union of two or more sets contains all the elements in either one of the sets or elements in all the sets referenced, and is written by placing this symbol \(\cup\) in between each of the sets. For example, let set \(A=\{1,2,3\},\) and let set \(B=\{4,5,6\}.\) Then, the union of sets A and B is \(A\cup B=\{1,2,3,4,5,6\}.\)

The intersection of two or more sets contains only the elements that are common to each set, and we place this symbol \(\cap\) in between each of the sets referenced. For example, let’s say that set \(A=\{1,3,5\},\) and let set \(B=\{5,7,9\}.\) Then, the intersection of sets \(A\) and \(B\) is \(A\cap B=\{5\}.\)

Condensed — the full section is in OpenStax Contemporary Mathematics.

Planes

A plane, as defined by Euclid, is a “surface which lies evenly with the straight lines on itself.” A plane is a two-dimensional surface with infinite length and width, and no thickness. We also identify a plane by three noncollinear points, or points that do not lie on the same line. Think of a piece of paper, but one that has infinite length, infinite width, and no thickness. However, not all planes must extend infinitely. Sometimes a plane has a limited area.

We usually label planes with a single capital letter, such as Plane \(P\), as shown in , or by all points that determine the edges of a plane. In the following figure, Plane \(P\) contains points \(A\) and \(B\), which are on the same line, and point \(C\), which is not on that line. By definition, \(P\) is a plane. We can move laterally in any direction on a plane.

One way to think of a plane is the Cartesian coordinate system with the \(x\)-axis marked off in horizontal units, and \(y\)-axis marked off in vertical units. In the Cartesian plane, we can identify the different types of lines as they are positioned in the system, as well as their locations. See .

This plane contains points \(S\), \(T\), and \(R\). Points \(T\) and \(R\) are colinear and form a line segment. Point \(S\) is not on that line segment. Therefore, this represents a plane.

To give the location of a point on the Cartesian plane, remember that the first number in the ordered pair is the horizontal move and the second number is the vertical move. Point \(R\) is located at \((4,2);\) point \(S\) is located at \((-3,4);\) and point \(T\) is located at \((-1,-1).\) We can also identify the line segment as \(\overset{\bar}{TR}.\)

Two other concepts to note: Parallel planes do not intersect and the intersection of two planes is a straight line. The equation of that line of intersection is left to a study of three-dimensional space. See .

To summarize, some of the properties of planes include:

  • Three points including at least one noncollinear point determine a plane.
  • A line and a point not on the line determine a plane.
  • The intersection of two distinct planes is a straight line.
Intersecting Planes

Try it.

Name two pairs of intersecting planes on the shower enclosure illustration ().

Solution

The plane \(\text{ABCD}\) intersects plane \(\text{CDEF}\), and plane \(\text{CDEF}\) intersects plane \(\text{EFGH}\).

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

  • Modern-day geometry began in approximately 300 BCE with Euclid’s Elements, where he defined the principles associated with the line, the point, and the plane.
  • Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other.
  • The union of two sets, \(A\) and \(B\), contains all points that are in both sets and is symbolized as \(A\cup B.\)
  • The intersection of two sets \(A\) and \(B\) includes only the points common to both sets and is symbolized as \(A\cap B.\)

Practice (5)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. For the following exercises, use this line ().

    1. Define \(\overset{\bar}{DE}\).
    2. Define \(F\).
    3. Define \(\overset{↔}{DF}\).
    4. Define \(\overset{\bar}{EF}\).
    Mutasd meg a választ!
    1. The symbol \(\overset{\bar}{DE}\), two letters with a straight line above, refers to the line segment that starts at point \(D\) and ends at point \(E\).
    2. The letter \(F\) alone refers to point \(F\).
    3. The symbol \(\overset{↔}{DF}\), two letters with a line above containing arrows on both ends, refers to the line that extends infinitely in both directions and contains the points \(D\) and \(F\).
    4. The symbol \(\overset{\bar}{EF}\), two letters with a straight line above, refers to the line segment that starts at point \(E\) and ends at point \(F\).

  2. Identify the sets of parallel and perpendicular lines in .

    Mutasd meg a választ!

    Drawing these lines on a grid is the best way to distinguish which pairs of lines are parallel and which are perpendicular. Because they are on a grid, we assume all lines are equally spaced across the grid horizontally and vertically. The grid also tells us that the vertical lines are parallel and the horizontal lines are parallel. Additionally, all intersections form a \({90}^{∘}\) angle. Therefore, we can safely say the following:

    \(\overset{↔}{AB}∥\overset{↔}{CD}\), the line containing the points \(A\) and \(B\) is parallel to the line containing the points \(C\) and \(D\).

    \(\overset{↔}{EF}∥\overset{↔}{GH}\), the line containing the points \(E\) and \(F\) is parallel to the line containing the points \(G\) and \(H\).

    \(\overset{↔}{AB}⊥\overset{↔}{EF}\), the line containing the points \(A\) and \(B\) is perpendicular to the line containing the points \(E\) and \(F\). We know this because both lines trace grid lines, and intersecting grid lines are perpendicular.

    We can also state that \(\overset{↔}{AB}⊥\overset{↔}{GH}\); the line containing the points \(A\) and \(B\) is perpendicular to the line containing the points \(G\) and \(H\) because both lines trace grid lines, which are perpendicular by definition.

    We also have \(\overset{↔}{CD}⊥\overset{↔}{EF}\); the line containing the points \(C\) and \(D\) is perpendicular to the line containing the points \(E\) and \(F\) because both lines trace grid lines, which are perpendicular by definition.

    Finally, we see that \(\overset{↔}{CD}⊥\overset{↔}{GH}\); the line containing the points \(C\) and \(D\) is perpendicular to the line containing the points \(G\) and \(H\) because both lines trace grid lines, which are perpendicular by definition.

  3. Use the line () for the following exercises. Draw each answer over the main drawing.

    1. Find \(\overset{\to}{BD}\cap \overset{←}{CA}\).
    2. Find \(\overset{\bar}{AB}\cup \overset{\bar}{AD}\).
    3. Find \(\overset{↔}{AD}\cup \overset{\bar}{BC}\).
    4. Find \(\overset{↔}{AD}\cap \overset{\bar}{BC}\).
    5. Find \(\overset{←}{BA}\cap \overset{\bar}{CD}\).
    Mutasd meg a választ!
    1. Find \(\overset{\to}{BD}\cap \overset{←}{CA}\). This is the intersection of the ray \(\overset{\to}{BD}\) and the ray \(\overset{←}{CA}.\) Intersection includes only the elements that are common to both lines. For this intersection, only the line segment \(\overset{\bar}{BC}\) is common to both rays. Thus, \(\overset{\to}{BD}\cap \overset{←}{CA}=\overset{\bar}{BC}.\)
    2. Find \(\overset{\bar}{AB}\cup \overset{\bar}{AD}\). The problem is asking for the union of two line segments, \(\overset{\bar}{AB}\) and \(\overset{\bar}{AD}.\) Union includes all elements in the first line and all elements in the second line. Since \(\overset{\bar}{AB}\) is part of \(\overset{\bar}{AD}\), \(\overset{\bar}{AB}\cup \overset{\bar}{AD}=\overset{\bar}{AD}.\)
    3. Find \(\overset{↔}{AD}\cup \overset{\bar}{BC}\). This is the union of the line \(\overset{↔}{AD}\) with the line segment \(\overset{\bar}{BC}\). As the line segment \(\overset{\bar}{BC}\) is included on the line \(\overset{↔}{AD},\) then the union of these two lines equals the line \(\overset{↔}{AD}\).
    4. Find \(\overset{↔}{AD}\cap \overset{\bar}{BC}\). The intersection of the line \(\overset{↔}{AD}\) with line segment \(\overset{\bar}{BC}\) is the set of elements common to both lines. In this case, the only element in common is the line segment \(\overset{\bar}{BC}.\)
    5. Find \(\overset{←}{AB}\cap \overset{\bar}{CD}\). This is the intersection of the ray \(\overset{←}{AB}\) and the line segment \(\overset{\bar}{CD}.\) Intersection includes elements common to both lines. There are no elements in common. The intersection yields the empty set, as shown in . Therefore, \(\overset{←}{AB}\cap \overset{\bar}{CD}=\emptyset\).

  4. For the following exercises, refer to

    1. Identify the location of points \(A\), \(B\), \(C\), and \(D\).
    2. Describe the line from point \(A\) to point \(D\).
    3. Describe the line from point \(C\) containing point \(B\).
    4. Does this figure represent a plane?
    Mutasd meg a választ!
    1. Point \(A\) is located at \((4,3);\) point \(B\) is located at \((-4,1);\) point \(C\) is located at \((-2,-3);\) point \(D\) is located at \((1,1).\)
    2. The line from point \(A\) to point \(D\) is a line segment \(\overset{\bar}{AD.}\)
    3. The line from point \(C\) containing point \(B\) is a ray \(\overset{\to}{CB}\) starting at point \(C\) in the direction of \(B\).
    4. Yes, this figure represents a plane because it contains at least three points, points \(A\) and \(D\) form a line segment, and neither point \(B\) nor point \(C\) is on that line segment.

  5. Name two pairs of intersecting planes on the shower enclosure illustration ().

    Mutasd meg a választ!

    The plane \(\text{ABCD}\) intersects plane \(\text{CDEF}\), and plane \(\text{CDEF}\) intersects plane \(\text{EFGH}\).

Symbols used here

A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
\theta
theta
The usual name for an angle.
^\circ
degrees
1/360 of a full turn. 180° = π radians.
\sin,\ \cos,\ \tan
sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.
\angle ABC,\ \triangle ABC
angle, triangle
The angle at B between BA and BC; the triangle with those vertices.
\parallel,\ \perp,\ \cong,\ \sim
parallel, perpendicular, congruent, similar
Never meet; meet at 90°; identical shape and size; same shape.

How to: Points, Lines, and Planes

  1. Identify and describe points, lines, and planes.
  2. Express points and lines using proper notation.
  3. Determine union and intersection of sets.
  4. Define
  5. Define
  6. Define
  7. Define
  8. The symbol

Questions people ask

Why does every triangle have angles adding to 180°?

Draw a line through one vertex parallel to the opposite side: the two alternate angles equal the other two angles of the triangle, and the three angles at the vertex lie on a straight line. That is Euclid's proof, and it is why the fact holds only on a flat plane.

When do I use the law of sines versus the law of cosines?

Cosines when you know three sides, or two sides and the angle between them. Sines when you know an angle and the side opposite it, plus one more piece.

What is the difference between area and perimeter?

Perimeter is the length of the boundary (one dimension, measured in metres); area is the amount of surface inside (two dimensions, square metres). Doubling every side doubles the perimeter but quadruples the area.

Próbáld a sajátodat.

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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