Lines, rays and line segments

Before talking about parallel and intersecting lines, let us pin down what we mean by a "line".

Idea

A line is a straight collection of points that extends without end in both directions. It has no thickness and no length-limit. We name a line by any two points on it, with an arrow on each side: $\overleftrightarrow{AB}$.

A ray is half of a line. It has one starting point and goes endlessly in one direction. We write $\overrightarrow{AB}$ (starting at $A$, going through $B$ and beyond).

A line segment is the part of a line between two endpoints $A$ and $B$. It has a definite length. We write $\overline{AB}$ or simply $AB$.

When two rays share an endpoint they form an angle at that point. The shared point is the vertex; the rays are the arms. We measure angles in degrees: a full turn is $360^\circ$, a half turn (a straight angle) is $180^\circ$, a right angle is $90^\circ$.

Lines can have several relationships:

• Intersecting , they cross at exactly one point.
• Parallel , they never cross, no matter how far extended. We write $l \parallel m$.
• Perpendicular , they cross at $90^\circ$. We write $l \perp m$.
• Coincident , they lie on top of each other (one and the same line).

Worked examples

Example 1. Identify the type: a railway track viewed straight ahead.

The two rails never meet (over long distance) , they are parallel.

Example 2. What is the relationship between the two diagonals of a rectangle?

They cross at the centre , they are intersecting. (They are not perpendicular in general , only for a square.)

Example 3. Write the symbol for the line segment with endpoints $P$ and $Q$, and for the ray starting at $P$ through $Q$.

Segment: $\overline{PQ}$. Ray: $\overrightarrow{PQ}$.

Example 4. Two lines $l$ and $m$ cross at a right angle. Write this in symbols.

$l \perp m$.

Try it yourself

1. Define a line, ray and line segment in your own words.
2. Are the opposite edges of a book page parallel or intersecting?
3. What symbol do we use for parallel lines? For perpendicular lines?
4. Identify the vertex and arms of the angle $\angle ABC$.
5. Draw two intersecting lines and label the four angles.
6. Draw two parallel lines and a transversal cutting them.
7. Is it possible for two distinct lines to be both parallel and intersecting? Explain.
8. Name three real-world examples of perpendicular lines around you.

Activity

Take a square piece of paper. Fold it once to make a straight line , that is a line segment. Fold again to make two intersecting line segments. Observe the angles formed.