Overview
Notes cover polygons, symmetry, angle sums, triangle theorems, and transformations including reflections, rotations, translations, and dilations, with objectives, standards, and sample problems.
Polygons and Symmetry
- Polygon: Closed plane figure with at least three straight sides.
- Regular polygon: Equilateral and equiangular.
- Convex: No diagonal has points outside the polygon.
- Concave: At least one diagonal has points outside the polygon.
- Reflection symmetry: Line splits a figure into two congruent halves; figures can have multiple lines.
Interior and Exterior Angles
- Interior angles lie inside a polygon; exterior formed by a side and extension of adjacent side.
- Sum of exterior angles of any polygon: 360 degrees.
- Sum of interior angles of a convex n-gon: (n − 2) × 180.
- Each interior angle of a regular n-gon: ((n − 2) × 180) / n.
- Each exterior angle of a regular n-gon: 360 / n.
Triangle Theorems
- Exterior angle of a triangle equals sum of the two non-adjacent interior angles.
- Proof structure uses linear pair and angle sum relationships to show m∠ext = m∠A + m∠C.
Transformations: Definitions and Rules
- Translation: (x, y) → (x + h, y + k); h left if negative, k down if negative.
- Reflection:
- Over x-axis: (x, y) → (x, −y).
- Over y-axis: (x, y) → (−x, y).
- Over y = x: (x, y) → (y, x).
- Through origin: (x, y) → (−x, −y).
- Rotation about origin:
- 90° CCW: (x, y) → (−y, x).
- 180°: (x, y) → (−x, −y).
- 270° CCW (or 90° CW): (x, y) → (y, −x).
- Composition: Order matters; sequences are not generally commutative.
Dilations
- Dilation scales a figure from a center; preimage becomes image via scale factor k.
- Coordinate rule about origin: (x, y) → (k x, k y).
- k > 1 enlargement; 0 < k < 1 reduction; k < 0 includes 180° rotation.
- Distance not preserved; angles preserved; figures become similar, not congruent.
- Scale factor from lengths: k = image / preimage (use corresponding sides).
Structured Formulas and Properties
| Concept | Formula/Rule | Notes |
|---|
| Interior Sum (n-gon) | (n − 2) × 180 | n ≥ 3 |
| Regular Interior Angle | ((n − 2) × 180) / n | Degrees |
| Exterior Sum (any polygon) | 360 | Always |
| Regular Exterior Angle | 360 / n | Degrees |
| Translation | (x, y) → (x + h, y + k) | h left if −, k down if − |
| Reflect x-axis | (x, y) → (x, −y) | Distance to axis preserved |
| Reflect y-axis | (x, y) → (−x, y) | |
| Reflect y = x | (x, y) → (y, x) | Swap coordinates |
| Rotate 90° CCW | (x, y) → (−y, x) | About origin |
| Rotate 180° | (x, y) → (−x, −y) | About origin |
| Rotate 270° CCW | (x, y) → (y, −x) | About origin |
| Dilation (origin) | (x, y) → (k x, k y) | k < 0 adds 180° rotation |
| Triangle Exterior Angle | m∠ext = m∠A + m∠C | Non-adjacent angles |
Example Calculations and Applications
- Regular polygon with exterior angle 60: n = 360/60 = 6 sides.
- Interior sum 3240: (n − 2) × 180 = 3240 → n − 2 = 18 → n = 20; each interior = 162; each exterior = 18.
- Quadrilateral exterior angles sum: 360; used to solve for unknown angles around a point.
- Determining rotation: A single rotation mapping ABCD to A'B'C'D' can be 90, 180, 270, or 360 about a center; check coordinates to match rules.
Rigid vs Non-Rigid Transformations
- Rigid (isometries): translations, reflections, rotations; preserve distance and angle; images congruent.
- Non-rigid: dilations; preserve angle only; images similar.
Common Misconceptions
- Order of transformations: Changing order can change result; follow given order.
- Center of dilation: Need not lie on the figure; can be anywhere.
Key Terms & Definitions
- Preimage: Original figure before transformation.
- Image: Resulting figure after transformation.
- Center of dilation: Fixed point from which a figure is dilated.
- Scale factor (k): Multiplier determining size change in dilation.
- Line of symmetry: Line dividing a figure into congruent halves.
- Rigid transformation: Preserves distance; isometry.
- Composition: Applying multiple transformations in sequence.