Introduction to Geometry

• Geometry is a branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.

Core Concepts

• Point: A location in space with no dimension.
• Line: An infinitely long, straight path with no thickness.
• Plane: A flat, two-dimensional surface that extends infinitely far.
• Angle: The measure of the rotation between two lines or surfaces that meet at a common point.

Basic Geometric Shapes

• Triangle: A polygon with three sides and three angles.
• Square: A quadrilateral with four equal sides and four right angles.
• Rectangle: A quadrilateral with two pairs of equal sides and four right angles.
• Circle: The set of all points in a plane that are equidistant from a center point.

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.

Triangles

• Equilateral Triangle: A triangle with all three sides equal in length and all three angles equal to 60 degrees.
• Isosceles Triangle: A triangle with two sides of equal length and two equal angles.
• Scalene Triangle: A triangle with all three sides of different lengths and all three angles of different measures.
• Right Triangle: A triangle with one right angle (90 degrees).

Quadrilaterals

• Parallelogram: A quadrilateral with two pairs of parallel sides.
• Rhombus: A parallelogram with all four sides equal in length.
• Trapezoid: A quadrilateral with at least one pair of parallel sides.

Circles

• Radius: The distance from the center of the circle to any point on the circle.
• Diameter: The distance across the circle through the center. It is twice the radius.
• Circumference: The distance around the circle, calculated as 2πr or πd, where r is the radius and d is the diameter.
• Area: The measure of the region enclosed by the circle, calculated as πr².

3-Dimensional Shapes

• Cube: A solid with six square faces.
• Rectangular Prism: A solid with six rectangular faces.
• Sphere: The set of all points in space that are equidistant from a center point.
• Cylinder: A solid with two parallel circular bases connected by a curved surface.
• Cone: A solid with a circular base and a single vertex.
• Pyramid: A solid with a polygonal base and triangular faces that meet at a common vertex.

Volume and Surface Area

• Volume: The amount of space a 3-dimensional object occupies.
• Surface Area: The total area of the surfaces of a 3-dimensional object.
• Volume of a Cube: s³, where s is the side length.
• Surface Area of a Cube: 6s², where s is the side length.
• Volume of a Rectangular Prism: lwh, where l is the length, w is the width, and h is the height.
• Surface Area of a Rectangular Prism: 2(lw + lh + wh), where l is the length, w is the width, and h is the height.
• Volume of a Sphere: (4/3)πr³, where r is the radius.
• Surface Area of a Sphere: 4πr², where r is the radius.
• Volume of a Cylinder: πr²h, where r is the radius and h is the height.
• Surface Area of a Cylinder: 2πr² + 2πrh, where r is the radius and h is the height.
• Volume of a Cone: (1/3)πr²h, where r is the radius and h is the height.
• Surface Area of a Cone: πr² + πrl, where r is the radius and l is the slant height.
• Volume of a Pyramid: (1/3)Bh, where B is the area of the base and h is the height.

Coordinate Geometry

• Cartesian Coordinate System: A system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length.
• Distance Formula: The distance between two points (x₁, y₁) and (x₂, y₂) is √((x₂ - x₁)² + (y₂ - y₁)²).
• Midpoint Formula: The midpoint of the line segment joining two points (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).
• Slope of a Line: The measure of the steepness of a line, calculated as the change in y divided by the change in x (rise over run). Represented as m = (y₂ - y₁)/(x₂ - x₁).
• Slope-Intercept Form: A way to represent the equation of a line as y = mx + b, where m is the slope and b is the y-intercept.

Transformations

• Translation: Moving a shape without rotating or resizing it.
• Rotation: Turning a shape around a fixed point.
• Reflection: Creating a mirror image of a shape across a line.
• Dilation: Enlarging or reducing the size of a shape.

Congruence and Similarity

• Congruent Shapes: Shapes that have the same size and shape.
• Similar Shapes: Shapes that have the same shape but can be different sizes.
• Criteria for Triangle Congruence: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS).
• Criteria for Triangle Similarity: Angle-Angle (AA), Side-Side-Side (SSS), Side-Angle-Side (SAS).

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: a² + b² = c².
• Triangle Sum Theorem: The sum of the angles in any triangle is always 180 degrees.
• Vertical Angles Theorem: Vertical angles (angles opposite each other when two lines intersect) are congruent.
• Parallel Postulate: Through a point not on a line, there is exactly one line parallel to the given line.

Trigonometry

• Trigonometric Ratios: Ratios relating the angles and sides of a right triangle. Sine (sin), Cosine (cos), and Tangent (tan).
• Sine (sin): Opposite / Hypotenuse.
• Cosine (cos): Adjacent / Hypotenuse.
• Tangent (tan): Opposite / Adjacent.

Geometric Proofs

• Direct Proof: Starts with known facts and uses logical steps to arrive at the desired conclusion.
• Indirect Proof (Proof by Contradiction): Assumes the opposite of what you want to prove and shows that this assumption leads to a contradiction, thus proving the original statement.

Constructions

• Using only a compass and straightedge to create geometric figures.
• Examples: Bisecting an angle, constructing a perpendicular bisector, constructing parallel lines.