Geometry

3.1 Points, Lines, and Planes

Understanding the basic building blocks of geometry is essential. Points, lines, and planes form the foundation upon which all geometric concepts are built.

• Point: An exact location in space with no dimension—no length, width, or height. Represented by a dot and labeled with a capital letter (e.g., Point A).
• Line: A straight one-dimensional figure extending infinitely in both directions. It has length but no thickness. Represented by a line with arrows on both ends and named using two points on the line (e.g., Line AB).
• Plane: A flat, two-dimensional surface that extends infinitely in all directions. It has length and width but no thickness. Represented by a four-sided figure and named using three non-collinear points (e.g., Plane ABC).

Notation: Points are denoted by capital letters, lines by lowercase letters or pairs of points (e.g., \( \overleftrightarrow{AB} \) or \( l \)), and planes by uppercase letters (e.g., Plane P).

3.2 Angles

Angles are formed by two rays (or line segments) sharing a common endpoint called the vertex. Angles are measured in degrees (°) and are fundamental in defining the shapes and properties of geometric figures.

• Acute Angle: An angle less than 90°.
• Right Angle: An angle exactly equal to 90°.
• Obtuse Angle: An angle greater than 90° but less than 180°.
• Straight Angle: An angle exactly equal to 180°.
• Reflex Angle: An angle greater than 180° but less than 360°.

Angle Relationships:

• Adjacent Angles: Two angles that share a common vertex and a common side.
• Vertical Angles: Opposite angles formed when two lines intersect. They are equal in measure.
• Complementary Angles: Two angles whose measures add up to 90°.
• Supplementary Angles: Two angles whose measures add up to 180°.

Example:

Identify the types of angles in the following figure:

Solution: The figure shows an acute angle, a right angle, and an obtuse angle. Additionally, there are vertical angles which are equal in measure.

3.3 Triangles

Triangles are three-sided polygons that are fundamental in geometry. They are classified based on their side lengths and angle measures.

• Classification by Sides:
  • Equilateral Triangle: All three sides are equal in length, and all three angles are 60°.
  • Isosceles Triangle: Two sides are equal in length, and the base angles are equal.
  • Scalene Triangle: All three sides are of different lengths, and all three angles are of different measures.
• Classification by Angles:
  • Acute Triangle: All three angles are acute (less than 90°).
  • Right Triangle: One angle is a right angle (exactly 90°).
  • Obtuse Triangle: One angle is obtuse (greater than 90°).

Properties of Triangles:

• The sum of the interior angles in any triangle is 180°.
• The sum of any two sides of a triangle must be greater than the third side.
• The area of a triangle can be calculated using the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

3.4 Quadrilaterals

Quadrilaterals are four-sided polygons. They can be classified based on their sides, angles, and properties.

• Classification by Sides:
  • Parallelogram: Opposite sides are equal and parallel.
  • Rectangle: All angles are right angles, and opposite sides are equal.
  • Square: All sides are equal, and all angles are right angles.
  • Rhombus: All sides are equal, and opposite angles are equal.
  • Trapezoid (Trapezium): Only one pair of opposite sides is parallel.
  • Kite: Two distinct pairs of adjacent sides are equal.
• Properties of Quadrilaterals:
  • The sum of the interior angles in any quadrilateral is 360°.
  • Diagonals can help determine the type of quadrilateral and its properties.
  • Area formulas vary based on the type of quadrilateral. For example:
    • Rectangle: Area = length × width
    • Parallelogram: Area = base × height
    • Trapezoid: Area = \( \frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height} \)
    • Rhombus: Area = \( \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2 \)

3.5 Circles

Circles are fundamental in geometry, characterized by their center and radius. Key properties and elements of circles include:

• Center: The fixed point from which every point on the circle is equidistant.
• Radius: A line segment from the center to any point on the circle.
• Diameter: A line segment passing through the center, connecting two points on the circle. It is twice the length of the radius.
• Circumference: The perimeter or boundary line of the circle. Calculated using: \[ C = 2\pi r \quad \text{or} \quad C = \pi d \]
• Area: The space enclosed by the circle. Calculated using: \[ A = \pi r^2 \]

Special Segments:

• Chord: A line segment connecting two points on the circle.
• Tangent: A line that touches the circle at exactly one point.
• Secant: A line that intersects the circle at two points.
• Arc: A portion of the circumference between two points.
• Sector: A region bounded by two radii and an arc.

3.6 Polygons

Polygons are closed figures with straight sides. They are classified based on the number of sides and other properties.

• Classification by Number of Sides:
  • Triangle: 3 sides
  • Quadrilateral: 4 sides
  • Pentagon: 5 sides
  • Hexagon: 6 sides
  • Heptagon: 7 sides
  • Octagon: 8 sides
  • Nonagon: 9 sides
  • Decagon: 10 sides
• Regular vs. Irregular Polygons:
  • Regular Polygon: All sides and angles are equal.
  • Irregular Polygon: Sides and angles are not all equal.
• Convex vs. Concave Polygons:
  • Convex Polygon: All interior angles are less than 180°, and no vertices point inward.
  • Concave Polygon: At least one interior angle is greater than 180°, with at least one vertex pointing inward.

Properties of Polygons:

• The sum of the interior angles of an n-sided polygon is: \[ (n - 2) \times 180° \]
• Each interior angle of a regular n-sided polygon is: \[ \frac{(n - 2) \times 180°}{n} \]
• The sum of the exterior angles of any polygon is 360°.

3.7 Solid Geometries

Solid geometries deal with three-dimensional figures, exploring their properties and relationships.

• Polyhedrons: Three-dimensional shapes with flat polygonal faces, straight edges, and sharp vertices. Examples include cubes, prisms, pyramids, and dodecahedrons.
• Cylinders: Surfaces generated by parallel lines (generators) moving along a closed curve (base), typically circular.
• Cones: Surfaces generated by lines (generators) connecting a fixed point (vertex) to every point on a base curve.
• Spheres: Perfectly symmetrical three-dimensional shapes where every point on the surface is equidistant from the center.
• Tori: Donut-shaped surfaces generated by rotating a circle around an axis outside the circle.

Properties of Solid Figures:

• Volume: The amount of space enclosed within a solid figure.
• Surface Area: The total area of all the faces or surfaces of a solid figure.

4.1 Similarity and Congruence

Similarity and congruence are fundamental concepts in geometry that describe the relationship between two shapes.

• Congruent Figures: Figures that are identical in shape and size. All corresponding sides and angles are equal.
• Similar Figures: Figures that have the same shape but different sizes. Corresponding angles are equal, and corresponding sides are proportional.

Criteria for Triangle Congruence:

• SAS (Side-Angle-Side): Two sides and the included angle are equal.
• ASA (Angle-Side-Angle): Two angles and the included side are equal.
• SSS (Side-Side-Side): All three sides are equal.
• AAS (Angle-Angle-Side): Two angles and a non-included side are equal.
• HL (Hypotenuse-Leg): For right triangles, the hypotenuse and one leg are equal.

Similarity Criteria for Triangles:

• AA (Angle-Angle): Two angles of one triangle are equal to two angles of another triangle.
• SAS (Side-Angle-Side): Two sides are proportional, and the included angles are equal.
• SSS (Side-Side-Side): All three sides are proportional.

Example:

Determine if the following triangles are congruent:

Solution:

• Given that both triangles have two sides and the included angle equal (SAS), they are congruent.

Answer: The triangles are congruent by the SAS criterion.

4.2 Trigonometry

Trigonometry deals with the relationships between the sides and angles of triangles, especially right-angled triangles. It is essential in various applications, including physics, engineering, and navigation.

• Basic Trigonometric Ratios:
  • Sine (sin): \( \sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \)
  • Cosine (cos): \( \cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \)
  • Tangent (tan): \( \tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}} \)
• Inverse Trigonometric Functions: Used to find angles when the sides are known.
• Pythagorean Theorem: In right-angled triangles, the square of the hypotenuse is equal to the sum of the squares of the other two sides. \[ c^2 = a^2 + b^2 \]
• Trigonometric Identities: Fundamental identities such as: \[ \sin^2 \theta + \cos^2 \theta = 1 \] \[ 1 + \tan^2 \theta = \sec^2 \theta \]

4.3 Coordinate Geometry

Coordinate Geometry, also known as Analytic Geometry, combines algebra and geometry using a coordinate system to describe geometric shapes and solve geometric problems.

• Distance Formula: Calculates the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \): \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
• Midpoint Formula: Finds the midpoint between two points \( (x_1, y_1) \) and \( (x_2, y_2) \): \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
• Slope Formula: Determines the slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
• Equation of a Line: Can be expressed in various forms:
  • Slope-Intercept Form: \( y = mx + c \)
  • Point-Slope Form: \( y - y_1 = m(x - x_1) \)
  • Standard Form: \( Ax + By = C \)
• Conic Sections: Graphs of quadratic equations in two variables, including circles, ellipses, parabolas, and hyperbolas.

4.4 Vectors

Vectors are quantities that have both magnitude and direction. They are essential in physics and engineering for representing forces, velocities, and other directional quantities.

• Representation: Vectors are typically represented by arrows in diagrams, where the length signifies magnitude and the arrow points in the direction.
• Components of Vectors: A vector in a two-dimensional space can be broken down into horizontal (x) and vertical (y) components. \[ \vec{A} = \langle A_x, A_y \rangle \]
• Magnitude of a Vector: The length of the vector, calculated using the Pythagorean theorem. \[ |\vec{A}| = \sqrt{A_x^2 + A_y^2} \]
• Vector Addition: Combining two vectors to form a resultant vector. \[ \vec{A} + \vec{B} = \langle A_x + B_x, A_y + B_y \rangle \]
• Dot Product: A scalar product of two vectors, calculated as: \[ \vec{A} \cdot \vec{B} = A_xB_x + A_yB_y \]
• Cross Product: A vector product of two vectors, resulting in a vector perpendicular to both. \[ \vec{A} \times \vec{B} = (A_yB_z - A_zB_y) \hat{i} - (A_xB_z - A_zB_x) \hat{j} + (A_xB_y - A_yB_x) \hat{k} \] *Note: Applicable in three-dimensional space.*