Review Notes(ANTONIO)

4.1 Axiomatic System of Geometry

  • GEOMETRY- The branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids.

    • Point: Indicates a location, without size, length, width, or depth.

    • Line: A straight continuous arrangement of infinitely many points extending in both directions.

    • Plane: A flat, two-dimensional surface that extends infinitely.

  • Postulate (Axiom): A statement accepted as true without proof.

  • Theorem: A mathematical statement that can be proven true using definitions, postulates, and other theorems.

  • Lemma: A theorem used as a steppingstone to prove other theorems.

  • Corollary: A theorem that follows directly from another theorem.

4.2 Conditional and Biconditional Statements

  • Conditional Statement: A statement that can be expressed in an "if-then" format.

    • Hypothesis: The part of the statement that follows the word "if."

    • Conclusion: The part of the statement that follows the word "then."

4.3 Triangle Congruence

  • Equilateral Triangle: A triangle with three equal sides.

  • Isosceles Triangle: A triangle with two equal sides.

  • Scalene Triangle: A triangle with no equal sides.

  • Corresponding Parts of Congruent Triangles are Congruent (CPCTC).

  • Similar Triangles: Triangles with the same shape but different sizes.

4.4 Geometric Reasoning

  • Geometry: The branch of mathematics dealing with measurement, properties, and relationships of points, lines, angles, surfaces, and solids.

  • Analogy: The ability to reason with relational patterns.

  • Intuition: Drawing conclusions based on incomplete information or feelings rather than facts.

4.5 Proving Triangle Congruence

  • Reflexive Property: Anything is equal to itself (e.g., a=aa = a).

  • Vertical Angles: Pairs of opposite angles made by intersecting lines, which are always congruent.

  • Supplementary Angles: Two angles that add up to 180 degrees.

  • Right Angle: An angle that measures exactly 90 degrees.

  • Corresponding Angles: Angles that appear at the same relative position on parallel lines.

  • Alternate Exterior Angles: Angles that lie outside the parallel lines on opposite sides of a transversal.

  • Parallel Lines: Two lines that never intersect and are always the same distance apart.

  • Ray: A part of a line that has one endpoint and extends infinitely in one direction.

  • Perpendicular Bisector: A line, line segment, or ray that bisects a given line segment or side of a polygon at a right angle such that the two segments have equal length.

  • Midpoint: Any point, line segment, ray or plane that intersects a line segment to create two smaller segments that have equal length.