Basic Geometric Elements

• Geometry is a branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• It is one of the oldest mathematical sciences.
• It is dealing with shapes, sizes, relative positions of figures, and the properties of space.

Basic Geometric Elements

• Point: a location in space, having no size or dimension.
• Line: a straight, one-dimensional figure extending infinitely in both directions.
• Plane: a flat, two-dimensional surface that extends infinitely far.
• Line Segment: a part of a line between two endpoints.
• Ray: a part of a line that starts at one endpoint and extends infinitely in one direction.
• Angle: the figure formed by two rays sharing a common endpoint (vertex).

Types of Angles

• Acute Angle: an angle measuring less than 90 degrees.
• Right Angle: an angle measuring exactly 90 degrees.
• Obtuse Angle: an angle measuring greater than 90 degrees but less than 180 degrees.
• Straight Angle: an angle measuring exactly 180 degrees.
• Reflex Angle: an angle measuring greater than 180 degrees but less than 360 degrees.

Angle Pairs

• Complementary Angles: two angles whose measures add up to 90 degrees.
• Supplementary Angles: two angles whose measures add up to 180 degrees.
• Vertical Angles: a pair of opposite angles formed by intersecting lines. They are always congruent (equal in measure).
• Adjacent Angles: angles that share a common vertex and a common side but do not overlap.

Geometric Shapes

• Triangle: a polygon with three sides and three angles.
  • Equilateral Triangle: all three sides are equal in length.
  • Isosceles Triangle: two sides are equal in length.
  • Scalene Triangle: no sides are equal in length.
  • Right Triangle: one angle is a right angle (90 degrees).
• Quadrilateral: a polygon with four sides and four angles.
  • Square: a quadrilateral with four equal sides and four right angles.
  • Rectangle: a quadrilateral with four right angles and opposite sides equal in length.
  • Parallelogram: a quadrilateral with opposite sides parallel and equal in length.
  • Rhombus: a quadrilateral with four equal sides and opposite angles equal.
  • Trapezoid: a quadrilateral with at least one pair of parallel sides.
• Circle: a set of points equidistant from a central point.
  • Radius: the distance from the center of the circle to any point on the circle.
  • Diameter: the distance across the circle through the center (twice the radius).
  • Circumference: the distance around the circle.
  • Area: the amount of space enclosed by the circle.

Congruence and Similarity

• Congruent: figures that have the same shape and size.
• Similar: figures that have the same shape but may have different sizes.

Theorems and Postulates

• Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
• Angle Sum Theorem for Triangles: The sum of the measures of the angles in a triangle is always 180 degrees.

Coordinate Geometry

• Using a coordinate system (usually Cartesian) to represent geometric figures and solve geometric problems using algebraic techniques.
• Distance Formula: Used to find the distance between two points in a coordinate plane.
• Midpoint Formula: Used to find the midpoint of a line segment in a coordinate plane.
• Slope: Used to measure the steepness and direction of a line.

Conditional Statements

• A conditional statement is a statement that can be written in "if-then" form.
• It asserts that if a certain condition (the hypothesis) is true, then a certain consequence (the conclusion) is also true.

Structure of a Conditional Statement

• "If p, then q" where p is the hypothesis and q is the conclusion.
• Hypothesis (p): The condition that must be met.
• Conclusion (q): The result that follows if the hypothesis is true.
• Example: "If it is raining (p), then the ground is wet (q)."

Variations of Conditional Statements

• Converse: Formed by switching the hypothesis and the conclusion ("If q, then p").
  • From "If it is raining, then the ground is wet," the converse is "If the ground is wet, then it is raining."
• Inverse: Formed by negating both the hypothesis and the conclusion ("If not p, then not q").
  • From "If it is raining, then the ground is wet," the inverse is "If it is not raining, then the ground is not wet."
• Contrapositive: Formed by switching and negating the hypothesis and the conclusion ("If not q, then not p").
  • From "If it is raining, then the ground is wet," the contrapositive is "If the ground is not wet, then it is not raining."

Logical Equivalence

• Contrapositive: A conditional statement and its contrapositive are logically equivalent. If one is true, the other is also true.
• Converse and Inverse: The converse and inverse of a conditional statement are logically equivalent. If one is true, the other is also true.

Truth Value of Conditional Statements

• A conditional statement is considered false only when the hypothesis (p) is true and the conclusion (q) is false.
• In all other cases (p true, q true; p false, q true; p false, q false), the conditional statement is considered true.

Examples of Conditional Statements in Geometry

• "If two angles are vertical angles, then they are congruent."
  • Hypothesis: Two angles are vertical angles.
  • Conclusion: They are congruent.
• "If a triangle has three equal sides, then it is an equilateral triangle."
  • Hypothesis: A triangle has three equal sides.
  • Conclusion: It is an equilateral triangle.

Biconditional Statements

• Biconditional statements combine a conditional statement and its converse.
• They are written in the form "p if and only if q" (often abbreviated as "p iff q").
• A biconditional statement is true only when both the conditional statement (if p, then q) and its converse (if q, then p) are true.
• Example: "A triangle is equilateral if and only if it has three equal sides."