Lines and Angles Intersections

Study Notes

Understanding lines and angles are crucial for solving practical problems like determining the height of towers or the distance of ships.
Lines extending infinitely in both directions are called lines, while parts with two endpoints are called line segments.
Rays originate from a point and extend indefinitely in one direction.
Collinear points lie on the same line, while non-collinear points do not.
When two rays originate from the same point, they form an angle.
Different types of angles exist:
- Acute angles: Less than 90 degrees.
- Right angles: Exactly 90 degrees.
- Obtuse angles: Greater than 90 degrees but less than 180 degrees.
- Straight angles: Exactly 180 degrees.
- Reflex angles: Greater than 180 degrees but less than 360 degrees.

Intersecting lines cross at a single point.
Non-intersecting lines are parallel, always maintaining the same distance between them.

Complementary angles: Two angles whose sum is 90 degrees.
Supplementary angles: Two angles whose sum is 180 degrees.
Adjacent angles: Two angles that share a common vertex and side but don't overlap.
Linear pairs: Two adjacent angles that form a straight angle (180 degrees).
Vertically opposite angles: Two angles formed by two intersecting lines, where the two angles share a common vertex but are on opposite sides of their common ray. They are always equal.

When a ray stands on a line, it forms two adjacent angles that are supplementary (linear pair).
If two lines intersect, the vertically opposite angles are equal.

If two distinct lines are each parallel to the same line, they must also be parallel to each other.
This is based on considering corresponding angles formed by a transversal intersecting these lines.