Introduction to Geometry

• Geometry is a branch of mathematics dealing with shapes, sizes, relative positions of figures, and the properties of space.
• Geometry is one of the oldest branches of mathematics

Basic Geometric Figures

• A point is an exact location in space, possessing no size, only position.
• A line is a straight, continuous arrangement of infinitely many points with infinite length but no width or thickness.
• A plane is a flat, two-dimensional surface extending infinitely far.

Angles

• An angle is formed by two rays sharing a common endpoint called the vertex.
• Angles are typically measured in degrees.
• An acute angle measures less than 90 degrees.
• A right angle measures exactly 90 degrees.
• An obtuse angle measures greater than 90 degrees but less than 180 degrees.
• A straight angle measures exactly 180 degrees.
• A reflex angle measures greater than 180 degrees but less than 360 degrees.
• Complementary angles are two angles whose measures add up to 90 degrees.
• Supplementary angles are two angles whose measures add up to 180 degrees.

Lines

• Parallel lines lie in a plane, do not intersect, and have the same slope.
• Perpendicular lines intersect at a right angle (90 degrees), and the product of their slopes is -1.
• A transversal intersects two or more other lines.
• When a transversal intersects two parallel lines: Corresponding angles are equal, Alternate interior angles are equal, Alternate exterior angles are equal, Consecutive interior angles are supplementary.

Polygons

• A polygon is a closed, two-dimensional shape formed by straight line segments.
• A triangle is a polygon with three sides and three angles.
• A quadrilateral is a polygon with four sides and four angles.
• A pentagon is a polygon with five sides and five angles.
• A hexagon is a polygon with six sides and six angles.
• A heptagon is a polygon with seven sides and seven angles.
• An octagon is a polygon with eight sides and eight angles.
• A decagon is a polygon with ten sides and ten angles.
• A regular polygon has all sides equal in length and all angles equal in measure.
• An irregular polygon has sides that are not all the same length or angles that are not all the same measure.
• A convex polygon has all interior angles less than 180 degrees.
• A concave polygon has at least one interior angle greater than 180 degrees.

Triangles

• An equilateral triangle has three equal sides and three equal angles (each 60 degrees).
• An isosceles triangle has two equal sides and two equal angles.
• A scalene triangle has no equal sides and no equal angles.
• A right triangle has one right angle (90 degrees); the side opposite the right angle is the hypotenuse, and the other two sides are legs.
• An acute triangle has all three angles acute (less than 90 degrees).
• An obtuse triangle has one obtuse angle (greater than 90 degrees).
• The sum of the angles in any triangle is always 180 degrees.
• Pythagorean theorem: In a right triangle, a² + b² = c², where c is the hypotenuse and a and b are the other two sides.

Quadrilaterals

• A square is a quadrilateral with four equal sides and four right angles.
• A rectangle is a quadrilateral with four right angles and opposite sides equal.
• A parallelogram is a quadrilateral with two pairs of parallel sides; opposite sides and angles are equal.
• A rhombus is a quadrilateral with four equal sides and opposite angles equal.
• A trapezoid is a quadrilateral with at least one pair of parallel sides.
• A kite is a quadrilateral with two pairs of adjacent sides that are equal in length.

Circles

• A circle consists of all points in a plane at a fixed distance from a center point.
• The radius is the distance from the center of the circle to any point on the circle.
• The diameter is the distance across the circle through the center and is twice the radius.
• The circumference is the distance around the circle, given by C = 2πr, where r is the radius.
• The area is the amount of space enclosed by the circle, given by A = πr², where r is the radius.
• A chord is a line segment connecting two points on a circle.
• A tangent is a line that touches the circle at only one point.

3-Dimensional Figures (Solids)

• A polyhedron is a 3-dimensional solid bounded by polygons, called faces.
• A prism is a polyhedron with two parallel and congruent bases and other faces that are parallelograms.
• A pyramid is a polyhedron with a polygonal base and triangular faces meeting at a common vertex (apex).
• A cube is a prism with six square faces.
• A rectangular prism is a prism with six rectangular faces.
• A cylinder is a solid with two parallel and congruent circular bases connected by a curved surface.
• A cone is a solid with a circular base and a curved surface tapering to a point (apex).
• A sphere is a solid consisting of all points in space at a fixed distance from a center point.
• Volume is the amount of space a 3-dimensional object occupies.
• Surface area is the total area of all the surfaces of a 3-dimensional object.

Coordinate Geometry

• The coordinate plane is formed by two perpendicular number lines (x-axis and y-axis) intersecting at the origin (0,0).
• Coordinates are ordered pairs (x, y) specifying a point's location in the coordinate plane.
• Distance formula: √((x₂ - x₁)² + (y₂ - y₁)²) calculates the distance between points (x₁, y₁) and (x₂, y₂).
• Midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2) finds the midpoint of the line segment connecting points (x₁, y₁) and (x₂, y₂).
• Slope (m) measures a line's steepness and is calculated as m = (y₂ - y₁) / (x₂ - x₁) given points (x₁, y₁) and (x₂, y₂).
• Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.

Transformations

• Translation slides a figure along a straight line without changing size or shape.
• Reflection flips a figure over a line (the line of reflection).
• Rotation turns a figure around a fixed point (the center of rotation).
• Dilation changes the size of a figure by a scale factor: Enlargement if the factor is greater than 1; reduction if less than 1.

Congruence and Similarity

• Congruent figures have the same shape and size; corresponding sides and angles are equal.
• Similar figures have the same shape but different sizes; corresponding angles are equal, and corresponding sides are proportional.
• Similarity ratio is the ratio of the lengths of corresponding sides in similar figures.