Geometry: Lines and Angles

• Geometry of straight lines involves the study of angles, lines, and their relationships.
• It forms the basis for understanding more complex geometric figures and proofs.

Basic Definitions

• Point: A location in space, having no dimension.
• Line: A straight path that extends infinitely in both directions. It has one dimension.
• Line Segment: A part of a line with two endpoints.
• Ray: A part of a line that starts at one endpoint and extends infinitely in one direction.
• Plane: A flat, two-dimensional surface that extends infinitely in all directions.
• Angle: Formed by two rays sharing a common endpoint (vertex).

Types of Angles

• Acute Angle: An angle measuring between 0° and 90°.
• Right Angle: An angle measuring exactly 90°.
• Obtuse Angle: An angle measuring between 90° and 180°.
• Straight Angle: An angle measuring exactly 180°. It forms a straight line.
• Reflex Angle: An angle measuring between 180° and 360°.
• Complete Angle: An angle measuring exactly 360°.

Angle Relationships

• Complementary Angles: Two angles whose measures add up to 90°.
• Supplementary Angles: Two angles whose measures add up to 180°.
• Adjacent Angles: Two angles that share a common vertex and a common side but do not overlap.
• Linear Pair: A pair of adjacent angles that form a straight line (sum to 180°).
• Vertical Angles: Two non-adjacent angles formed by intersecting lines. Vertical angles are congruent (equal in measure).

Lines

• Intersecting Lines: Lines that cross each other at a point.
• Parallel Lines: Lines in the same plane that never intersect. The distance between them remains constant.
• Perpendicular Lines: Lines that intersect at a right angle (90°).
• Transversal: A line that intersects two or more other lines.

Angles Formed by a Transversal

• When a transversal intersects two lines, it forms several angles with specific relationships.
• Corresponding Angles: Angles in the same relative position at each intersection. If the lines are parallel, corresponding angles are congruent.
• Alternate Interior Angles: Angles on opposite sides of the transversal and inside the two lines. If the lines are parallel, alternate interior angles are congruent.
• Alternate Exterior Angles: Angles on opposite sides of the transversal and outside the two lines. If the lines are parallel, alternate exterior angles are congruent.
• Consecutive Interior Angles (Same-Side Interior Angles): Angles on the same side of the transversal and inside the two lines. If the lines are parallel, consecutive interior angles are supplementary (add up to 180°).

Parallel Lines and Transversals

• If two parallel lines are cut by a transversal, then:
  • Corresponding angles are congruent.
  • Alternate interior angles are congruent.
  • Alternate exterior angles are congruent.
  • Consecutive interior angles are supplementary.
• Conversely, if any of the above conditions are met, then the two lines are parallel.

Theorems and Postulates

• Vertical Angle Theorem: Vertical angles are congruent.
• Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then corresponding angles are congruent.
• Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
• Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.
• Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary.
• Linear Pair Postulate: If two angles form a linear pair, then they are supplementary.

Angle Bisector

• A line or ray that divides an angle into two congruent angles.

Perpendicular Bisector

• A line that is perpendicular to a line segment and passes through its midpoint.

Properties of Parallel Lines

• Parallel lines have the same slope.
• The distance between two parallel lines is constant.

Slope of a Line

• The slope (m) of a line is a measure of its steepness.
• Given two points (x₁, y₁) and (x₂, y₂) on a line, the slope is calculated as: m = (y₂ - y₁) / (x₂ - x₁)
• A positive slope indicates an increasing line (from left to right).
• A negative slope indicates a decreasing line (from left to right).
• A zero slope indicates a horizontal line.
• An undefined slope indicates a vertical line.

Equations of Lines

• Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis).
• Point-Slope Form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line.
• Standard Form: Ax + By = C, where A, B, and C are constants.

Distance

• Distance Between Two Points:
  • Given two points (x₁, y₁) and (x₂, y₂), the distance (d) between them is: d = √((x₂ - x₁)² + (y₂ - y₁)²)
• Distance from a Point to a Line:
  • The shortest distance from a point to a line is the perpendicular distance.

Concurrency of Lines

• Concurrent Lines: Three or more lines that intersect at a single point.

Examples of Geometric Proofs

• Geometric proofs involve using definitions, postulates, and theorems to logically demonstrate that a certain statement is true.
• Proofs typically involve:
  • Given: The information that is provided.
  • Prove: The statement that needs to be proven.
  • Statements: Logical steps used to reach the conclusion.
  • Reasons: Justifications (definitions, postulates, theorems) for each step.

Common Geometric Constructions

• Bisecting an angle using a compass and straightedge.
• Constructing a perpendicular bisector of a line segment.
• Constructing a line parallel to a given line through a point not on the line.
• Constructing a perpendicular line to a given line through a point on or off the line.

Coordinate Geometry

• Using the coordinate plane to analyze geometric figures.
• Finding the midpoint of a line segment: Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
• Determining if lines are parallel or perpendicular based on their slopes.
• Calculating areas and perimeters of geometric figures using coordinates.