Geometry Basics Lecture Notes

Lines, Rays, and Segments

• Line:
  • A straight one-dimensional figure having no thickness and extending infinitely in both directions.
  • Often represented by two points on it, e.g., line AB.
• Ray:
  • A line with a single endpoint (or point of origin) that extends infinitely in one direction.
  • Named with two points, starting with the endpoint, e.g., ray AB.
• Segment:
  • A part of a line that is bounded by two distinct endpoints.
  • Can be named as segment AB or segment BA.

Angles

• Acute Angle:
  • Measures between 0 and 90 degrees.
• Right Angle:
  • Measures exactly 90 degrees.
• Obtuse Angle:
  • Measures greater than 90 degrees but less than 180 degrees.
• Straight Angle:
  • Measures exactly 180 degrees, resembling a straight line.

Midpoints and Bisectors

• Midpoint:
  • The point that divides a segment into two equal parts.
  • If B is the midpoint of AC, then AB is congruent to BC.
• Segment Bisector:
  • A line, segment, or ray that divides a segment into two equal parts.
• Angle Bisector:
  • A ray that divides an angle into two congruent angles.

Parallel and Perpendicular Lines

• Parallel Lines:
  • Lines in a plane that do not intersect.
  • Have the same slope; denoted as line A || line B.
• Perpendicular Lines:
  • Lines that intersect at a right angle (90 degrees).
  • Slopes are negative reciprocals of each other; denoted as line A ⊥ line B.

Complementary and Supplementary Angles

• Complementary Angles:
  • Two angles whose measures add up to 90 degrees.
• Supplementary Angles:
  • Two angles whose measures add up to 180 degrees.

Transitive Property

• If angle 1 is congruent to angle 2 and angle 3 is congruent to angle 2, then angle 1 is congruent to angle 3.

Vertical Angles

• Formed when two lines intersect.
• Opposite angles are congruent, e.g., angle 1 is congruent to angle 3.

Medians and Altitudes

• Median:
  • A segment from a vertex to the midpoint of the opposite side.
• Altitude:
  • A perpendicular segment from a vertex to the opposite side or line containing the opposite side.

Perpendicular Bisectors

• A line that is perpendicular to a segment at its midpoint, dividing it into two equal parts.
• Any point on the perpendicular bisector is equidistant to the endpoints of the segment.

Congruence Postulates

• SSS (Side-Side-Side): If all three sides of one triangle are congruent to three sides of another, the triangles are congruent.
• SAS (Side-Angle-Side): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another, the triangles are congruent.
• ASA (Angle-Side-Angle): If two angles and the included side of one triangle are congruent to two angles and the included side of another, the triangles are congruent.
• AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another, the triangles are congruent.
• CPCTC: Corresponding parts of congruent triangles are congruent.

Additional Concepts

• Hypotenuse Leg Postulate: For right triangles, if the hypotenuse and one leg are congruent to the hypotenuse and one leg of another, the triangles are congruent.

This lecture covered essential concepts in geometry, including definitions and properties of shapes and angles, congruence postulates, and reasoning strategies. Be sure to review these notes and practice using the examples provided.