Plane Geometry: Lines, Angles, Polygons, and Circles
Plane Geometry
System Representation
Allows us to represent elements in a plane.
Point
Determined by its coordinates and is usually given a name.
Line
An infinite set of points in the same direction.
Ray
Each of the two parts formed on a line when divided by a point.
Line Segment
A piece of a line between two points.
Types of Line Segments:
- Null Segment: Its ends meet at the same point.
- Consecutive Segments: Share a common endpoint.
- Aligned Segments: Lie on the same line.
Angle
Region of the plane determined by two rays that share an origin. Measured in degrees or radians.
Types of Angles:
| Name | Measure |
|---|---|
| Acute | < 90° |
| Right | = 90° |
| Obtuse | > 90° and < 180° |
| Convex | < 180° |
| Straight | = 180° |
| Concave/Reflex | > 180° and <360° |
| Zero | = 0° |
| Full | = 360° |
| Negative | -x° |
Angle Relationships:
- Consecutive Angles: Share the same vertex and one side.
- Adjacent Angles: Are consecutive and add up to 180°.
- Vertical Angles: Share a vertex, and the sides of one are extensions of the other.
- Supplementary Angles: Add up to 180°.
Angles formed between two perpendicular lines and a transversal:
- Corresponding Angles: (1, 3), (2, 4), (5, 7), (6, 8) are equal.
- Alternate Interior Angles: (2, 6), (3, 7) are equal.
- Alternate Exterior Angles: (1, 5), (8, 4) are equal.
Polygons
A region of the plane bounded by a closed polygonal chain.
- Side: Each segment forming the polygon.
- Vertex: Point where two sides meet.
- Diagonal: Segment joining two non-consecutive vertices.
- Interior Angle: Angle formed by two adjacent sides inside the polygon.
Polygon Classification:
- Convex: All interior angles are less than 180°.
- Concave: At least one interior angle is greater than 180°.
Regular Polygon Elements:
- Center: Point equidistant from all vertices.
- Radius: Distance between the center and any vertex.
- Circumcircle: Circle passing through all vertices of the polygon.
- Apothem: Distance between the center and the midpoint of any side.
- Incircle: Circle tangent to all sides of the polygon.
Triangles
Three-sided polygons with no diagonals. The sum of interior angles is 180°.
- Altitude: Line segment from a vertex perpendicular to the opposite side. The intersection of altitudes is the orthocenter.
- Median: Line segment from a vertex to the midpoint of the opposite side. The intersection of medians is the centroid.
- Perpendicular Bisector: Line perpendicular to a side at its midpoint. The intersection of perpendicular bisectors is the circumcenter.
- Angle Bisector: Line that divides an angle into two equal parts. The intersection of angle bisectors is the incenter.
Quadrilaterals
Four-sided polygons with two diagonals. The sum of interior angles is 360°.
Types of Quadrilaterals:
- Parallelograms:
- Square: All sides and angles are equal.
- Rectangle: Opposite sides are equal, and all angles are equal.
- Rhombus: All sides are equal, and opposite angles are equal.
- Rhomboid: Opposite sides and angles are equal.
- Non-Parallelograms:
- Trapezoid: At least one pair of parallel sides.
- Trapezium/Irregular Quadrilateral: No parallel sides.
Circular Figures
Circumference: Set of points equidistant from a center. The distance is called the radius.
Elements of a Circle:
- Chord: Line segment connecting two points on the circumference.
- Diameter: Chord passing through the center.
- Arc: Part of the circumference between two points.
- Semicircle: Arc formed by a diameter.
- Circle/Disk: Region of the plane enclosed by the circumference.
Figures within a Circle:
- Circular Segment: Region bounded by a chord and its corresponding arc.
- Circular Sector: Region bounded by two radii and their corresponding arc.
- Annulus: Region between two concentric circles.
Other Formulas and Concepts
- Area of a Circle: πr²
- Length of an Arc: (θ/360) * 2πr
- Circumference of a Circle: 2πr
- Number of Diagonals of a Polygon: n(n-3)/2
- Sum of Interior Angles of a Polygon: (n-2) * 180°
- Area of a Regular Polygon: (Perimeter * Apothem) / 2
- Area of a Triangle: (Base * Height) / 2
- Pythagorean Theorem: a² + b² = c² (for right triangles)
- Area of Quadrilaterals:
Square A = s² Rectangle A = l * w Rhombus A = (d1 * d2) / 2 
Rhomboid A = b * h Trapezoid A = ((b1 + b2) * h) / 2 
- Central Angle of a Regular Polygon: 360°/n
- Interior Angle of a Regular Polygon: ((n-2) * 180°)/n