What is Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. 

🔵 What Are Spaces and Subspaces in Geometry?

Let’s break it down in simple terms:

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🟦 Space (in Geometry)

In geometry, a space is a set where geometric figures exist.

• The most common space is 3-dimensional space (3D space), which includes length, width, and height — the space we live in!
• We also talk about 2-dimensional space (like a flat plane — just length and width), and even 1-dimensional space (a straight line).
• Mathematically, space is like a container or background that holds points, lines, planes, and shapes.

Example:

• A piece of paper is a 2D space.
• A room is a 3D space.
• A number line is a 1D space.

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🟩 Subspace

A subspace is a smaller part of a space that still follows the rules of that space.

• It’s like a "space inside a space".
• In 2D, a line on a plane is a subspace.
• In 3D, a flat surface like a wall is a 2D subspace of the 3D room.

Key idea:
A subspace must contain all points, lines, or objects that follow the same geometry rules as the bigger space it’s in.

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🎨 Visual Analogy:

Space Type	Example	Subspace Example
3D Space	A room	A wall inside the room (plane)
2D Space (plane)	A piece of paper	A line drawn on the paper
1D Space (line)	A number line	A segment of the line

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🧠 Quick Summary:

• Space = The whole geometric world you're working in.
• Subspace = A smaller world inside it, following the same geometric rules.

📍 Points, Lines, Rays, Line Segments, and Planes

• Point: A location in space. It has no size, only position. Represented with a dot and named with a capital letter (e.g., Point A).
• Line: A straight path of points that goes on forever in both directions. Named using two points on the line (e.g., line AB).
• Ray: A part of a line that starts at a point and goes on forever in one direction. (e.g., ray AB starts at A and passes through B).
• Line Segment: A part of a line with two endpoints. (e.g., segment AB has a start at A and ends at B).
• Plane: A flat surface that extends forever in all directions. It is usually represented as a parallelogram in diagrams.

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🔺 Angles

• Angle: Formed when two rays meet at a common endpoint (called the vertex).

• Types of Angles:

1. Straight Angle: Exactly 180°
2. Obtuse Angle: More than 90° but less than 180°
3. Right Angle: Exactly 90°
4. Acute Angle: Less than 90°

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📐 Perpendicular Lines

1. Perpendicular Lines: Two lines that intersect to form right angles (90°).
2. Symbol: ⟂

• Example: Line AB ⟂ Line CD

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➗ Transversals

• Transversal: A line that cuts across two or more other lines.

• When a transversal cuts parallel lines, it forms special angle pairs:

  • Corresponding Angles (equal)

  • Alternate Interior Angles (equal)

  • Alternate Exterior Angles (equal)

  • Consecutive Interior Angles (add up to 180°)

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🔷 Polygons and Polygonal Regions

• Polygon: A closed figure made up of line segments.

• Polygonal Region: The inside area of a polygon.

• Common Polygons:

  • Triangle (3 sides)

  • Quadrilateral (4 sides)

  • Pentagon (5 sides)

  • Hexagon (6 sides), etc.

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🔺 Triangles

• Classified by sides:

  • Equilateral: All sides are equal

  • Isosceles: Two sides are equal

  • Scalene: All sides are different

• Classified by angles:

  • Acute Triangle: All angles < 90°

  • Right Triangle: One angle = 90°

  • Obtuse Triangle: One angle > 90°

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🟩 Quadrilaterals

• Quadrilateral: A polygon with 4 sides.

• Types:

  • Square: 4 equal sides, 4 right angles

  • Rectangle: Opposite sides equal, 4 right angles

  • Rhombus: 4 equal sides, opposite angles equal

  • Parallelogram: Opposite sides equal and parallel

  • Trapezoid: Only one pair of parallel sides

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⚪ Circles

• Circle: A set of points that are all the same distance from a center point.

• Parts of a circle:

  • Radius: From the center to any point on the circle

  • Diameter: Across the circle through the center (2× radius)

  • Chord: A line segment inside the circle that connects two points

  • Arc: A part of the circle's curve

  • Circumference: The perimeter of the circle

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📓 End of Notes - Stay tuned for more Geometry lessons!

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