Fix the mistake: Basics of Plane Figures

Plane Figures ① Lines, Line Segments, and Rays

In the unit on plane figures, we first organize the basic terms line, line segment, ray, and angle. These words are used repeatedly in later geometry lessons, so learn them accurately.

Lines, Line Segments, and Rays

Let us distinguish the three figures.

  • Line AB: A straight line extending without end in both directions. Its notation is “line AB.”
  • Line segment AB: A “bounded” line with points A and B as its endpoints. It has a length.
  • Ray AB: A line that starts at A and extends without end toward B.

The length of line segment AB is called the distance between A and B and is written as AB.

Relationships Between Two Lines

The positional relationships between two lines are as follows.
　・Parallel: They never intersect, no matter how far they are extended. Symbol: ℓ // m
　・Intersecting: They intersect at one point.
　・Perpendicular: They intersect at a right angle (90°). Symbol: ℓ ⊥ m
There is exactly one line passing through two points.

Example ①
Select all the correct statements below.
ア) Line AB and line BA are the same line.
イ) Line segment AB and ray BA are the same figure.
ウ) There are many lines passing through two points A and B.
エ) Two parallel lines never intersect.
Solution: The correct statements are ア、エ.
イ: A line segment has two endpoints, while a ray has only one endpoint and extends infinitely in the opposite direction. They are different figures.
ウ: Exactly one line passes through two points.

Distance

The distance between two points A and B is the length of line segment AB.
The distance from a point to line ℓ is the length of the perpendicular drawn from the point to the line (the shortest length).

Example ②
There are two parallel lines ℓ and m. From point A on ℓ, draw a perpendicular to m, meeting it at H. If AH = 5 cm, find the distance between lines ℓ and m.
Solution: The distance between two parallel lines is the length of the perpendicular between them. Since AH = 5 cm, the answer is 5 cm.
Key Points
  • Line: no endpoints / line segment: two endpoints / ray: one endpoint.
  • Exactly one line passes through two points.
  • Parallel: ℓ // m; perpendicular: ℓ ⊥ m.
  • Distance from a point to a line = length of the perpendicular from the point to the line.

Practice Problems

  1. Line segment AB has length 8 cm, and line segment BC has length 6 cm. Points A, B, and C lie on one straight line, with B between A and C. Find the length of line segment AC.
  2. The opposite sides of a parallelogram are parallel. Do the two corresponding lines intersect?
  3. The distance from point P to line ℓ is 4 cm, and the distance from point Q to line ℓ is 7 cm. P and Q are on the same side of ℓ. Which type of figure is the segment connecting P and Q: a line, line segment, or ray?

Answers and Explanations

  1. Answer: 14 cm
    Explanation: Since B is between A and C, AC = AB + BC = 8 + 6 = 14 cm.
  2. Answer: They do not intersect.
    Explanation: Two parallel lines do not intersect even when extended indefinitely.
  3. Answer: A line segment
    Explanation: A figure connecting two points with two endpoints is line segment PQ.

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