# Lines and Angles

## Big Picture

Some lines intersect and some lines don’t. Imagine a set of railroad tracks. The two tracks in the set never intersect, so they’re parallel. At a crossroad, however, another pair of tracks intersects these tracks, forming 90° angles. The two sets of tracks are perpendicular. There’s a power line that crosses above the tracks, but the power line and the tracks never intersect because the power line is above the tracks - they’re on different planes. Lines work the same way as railroad tracks, except that lines can intersect in many different ways to produce angles with various properties.

## Key Terms

Parallel Lines: Lines that lie in the same plane and never intersect.

Parallel Planes: Planes that lie in the same plane and never intersect.

Perpendicular Lines: Lines that intersect to form four right angles.

Perpendicular Planes: Planes that intersect to form four right angles.

Skew Lines: Lines that lie in different planes and never intersect.

Perpendicular Bisector: A line, ray, or segment that passes through the midpoint of a segment at a right angle.

## Notation

### Diagram

Parallel lines

$$\overleftrightarrow{AB} || \overleftrightarrow{MN} \text{ or / } || m$$

Perpendicular lines

$$\overleftrightarrow{AS} \perp \overleftrightarrow{TC}$$

• To mark parallel lines in a diagram, draw arrows (>) on each parallel line. Lines that are parallel will have the same number of arrows.
• To mark perpendicular lines in a diagram, draw a half-square marking the angle.
• Only one of the right angles needs to be marked with the half-square.

## Parallel, Perpendicular, and Skew

Think of each segment in the figure as part of a line. Examples:

Parallel Lines

• $$\overleftrightarrow{AB} || \overleftrightarrow{EF}, \overleftrightarrow{AC} || \overleftrightarrow{BD}$$

Parallel Planes

• Plane $$\overleftrightarrow{AEB} ||$$ Plane $$\overleftrightarrow{CGH}$$, Plane $$\overleftrightarrow{AEG} ||$$ Plane $$\overleftrightarrow{BFH}$$

Perpendicular Lines

• $$\overleftrightarrow{AB} \perp \overleftrightarrow{AC}, \overleftrightarrow{BD} \perp \overleftrightarrow{DH}$$

Perpendicular Planes

• plane $$\overleftrightarrow{AEB} \perp$$ plane $$\overleftrightarrow{CAD}$$, plane $$\overleftrightarrow{CGD} \perp$$  plane $$\overleftrightarrow{AEB}$$

Skew Lines

• $$\overleftrightarrow{AB}$$ and $$\overleftrightarrow{EG}$$, $$\overleftrightarrow{BD}$$ and $$\overleftrightarrow{CG}$$

# Lines and Angles cont.

## Postulates and Theorems

### For Parallel Lines

Parallel Line Postulate: Given a line and a point not on the line, there is exactly one line parallel to this line that goes through the point.

• There are infinitely many lines that pass through point P, but only one of those lines will be parallel to line l.

Transitive Property of Parallel Lines: If two lines are parallel to the same line, then they are parallel to each other.

• If lines l || m and m || n, then l || n.

### For Perependicular Lines

Perpendicular Line Postulate: Given a line and a point not on the line, there is exactly one line perpendicular to this line that goes through the point.

• There are infinitely many lines that pass through point P, but only one of those lines will be perpendicular to line l.

A similar postulate is the Perpendicular Bisector Postulate.

Perpendicular Bisector Postulate: For every line segment, there is one perpendicular bisector that passes through the midpoint.

• $$\overleftrightarrow{CD}$$ is the perpendicular bisector of AB and divides the segments into $$\overline{AD}$$ and $$\overline{DB}$$.  AD = DB

Theorems:

• If two lines are perpendicular, they intersect to form four right angles.
• If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
a. If $$m \perp l$$, then  ∠∠1 and  2 are complementary (add up to 90°).