Parallel and perpendicular lines Knowledge-Based Study

Study notes

1. In geometry, two straight lines lying in the same plane (coplanar lines) can relate to each other in distinct ways based on whether they meet and the angles they form.

   Basic Classifications

   • Intersecting Lines: Lines that cross each other at exactly one point.
   • Parallel Lines: Lines that lie in the same plane and never meet, no matter how far they are extended.
   • Perpendicular Lines: A special case of intersecting lines that meet at an angle of exactly $90^\circ$.

   Two lines are parallel if they are in the same plane and never intersect.

   Symbol and Notation

   We use the symbol $\parallel$ to indicate that two lines are parallel. For instance, if line $AB$ is parallel to line $CD$, we write:

   $$AB \parallel CD$$

   Essential Characteristics

   • Equidistance: The perpendicular distance between two parallel lines remains constant along their entire length.
   • No Common Points: Parallel lines share zero intersection points.
   • Direction: In coordinate geometry, parallel lines have identical slopes ($m_1 = m_2$).

   Two lines are perpendicular if they intersect to form right angles ($90^\circ$).

   Symbol and Notation

   We use the symbol $\perp$ to denote perpendicular lines. If line $PQ$ is perpendicular to line $RS$, we write:

   $$PQ \perp RS$$

   Essential Characteristics

   • Four Right Angles: When two lines intersect perpendicularly, they create four adjacent angles, each measuring $90^\circ$.
   • Square Marker: In geometric diagrams, a small square drawn at the point of intersection indicates a right angle.
   • Coordinate Relationship: In coordinate geometry, the slopes of two perpendicular lines are negative reciprocals of one another ($m_1 \cdot m_2 = -1$), provided neither line is vertical.

   When analyzing two lines in a plane, determine their relationship by checking their intersection and angle:

   • Intersection Count: Parallel lines have $0$ intersection points, whereas perpendicular lines have exactly $1$ intersection point.
   • Intersection Angle: Perpendicular lines intersect at strictly $90^\circ$. If two lines intersect at any other angle (such as $45^\circ$ or $120^\circ$), they are intersecting, but neither parallel nor perpendicular.

   Geometric Example: A Rectangle

   Consider rectangle $ABCD$:

   • Opposite sides are parallel: $AB \parallel CD$ and $BC \parallel DA$.
   • Adjacent sides are perpendicular: $AB \perp BC$, $BC \perp CD$, $CD \perp DA$, and $DA \perp AB$.