5. Parallel and Intersecting lines

Chapter Topics and Important Facts

When two lines intersect, they form four angles.
Opposite angles formed by the intersection are called vertically opposite angles, and they are always equal to each other.
Adjacent angles that form a straight line are called linear pairs, and they always add up to 180°.
Justifying geometric conclusions without relying strictly on physical measurement is called a "proof" in mathematics.
Physical measurements of angles might sometimes appear inaccurate due to measurement errors (like improper protractor use) or the thickness of the drawn lines.

When two intersecting lines form angles of exactly 90°, they are perpendicular to each other.
In geometric figures, perpendicular lines are indicated by a small square angle mark between them.

Parallel lines are defined as a pair of lines that lie on the same plane and never meet, no matter how far they are extended at both ends.
In diagrams, an arrow mark ( $>$ ) is used on lines to indicate that they are parallel.
If multiple sets of parallel lines exist in one figure, double arrows ( $>>$ ) are used for the second set, and so on.

A line that intersects another pair of lines is called a transversal, and it creates two distinct sets of four angles (eight angles in total).
Corresponding Angles: These are angles that occupy the same relative position at each intersection. When a transversal intersects parallel lines, the corresponding angles are equal. Conversely, if corresponding angles are equal, the pair of lines must be parallel.
Alternate Angles: Alternate angles formed by a transversal intersecting parallel lines are always equal to each other.
Interior Angles: The interior angles on the same side of a transversal intersecting parallel lines always add up to 180°.

Question: In a figure showing two intersecting lines, if $\angle a$ is 120°, can you figure out the measurements of $\angle b$ , $\angle c$ , and $\angle d$ without drawing and measuring them?
Answer:
- $\angle b$ and $\angle a$ form a linear pair (adding to 180°), so $\angle b = 180° - 120° = 60°$ .
- $\angle c$ and $\angle b$ form a linear pair, so $\angle c = 120°$ (it is also vertically opposite to $\angle a$ ).
- $\angle d$ is vertically opposite to $\angle b$ , so $\angle d = 60°$ .

Question: Parallel lines $l$ and $m$ are intersected by the transversal $t$ . If $\angle 6$ is 135°, what are the measures of the other angles ( $\angle 1$ through $\angle 8$ )?
Answer:
- $\angle 2 = 135°$ because it corresponds to $\angle 6$ .
- $\angle 8 = 135°$ because it is vertically opposite to $\angle 6$ .
- $\angle 4 = 135°$ because it corresponds to $\angle 8$ (and is vertically opposite to $\angle 2$ ).
- $\angle 5 = 45°$ because it forms a linear pair with $\angle 6$ ( $180° - 135° = 45°$ ).
- Consequently, $\angle 1$ , $\angle 3$ , and $\angle 7$ also measure 45° due to their corresponding and vertically opposite relationships.

Question: How can you make a line parallel to a crease $l$ such that it passes through a specific point $A$ using paper folding?
Answer: First, fold the piece of paper to get a crease that is perpendicular to $l$ . Then, fold a new line that is perpendicular to this new crease, ensuring the fold passes exactly through point $A$ .