🦕 Dino Geometry | Master Math with Dino!

📐 Angles & Lines UNIT 1

• Complementary angles: sum = 90°
• Supplementary angles: sum = 180°
• Vertical angles: equal to each other
• Angles on a straight line: sum = 180°
• Angles around a point: sum = 360°
• Parallel lines cut by transversal:
  – Alternate interior angles = equal
  – Corresponding angles = equal
  – Co-interior (same-side interior): sum = 180°

🌟 MEMORIZE!

Vertical angles are ALWAYS equal. Complementary = 90°, Supplementary = 180°.

🦕 Dino's Example

Two angles are supplementary. One is 65°. Find the other.

✅ 180° − 65° = 115°

🔺 Triangles UNIT 2

• Sum of interior angles = 180°
• Exterior angle = sum of two non-adjacent interior angles
• Area = ½ × base × height
• Pythagorean Theorem: a² + b² = c² (right triangle)
• Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17
• Isosceles: 2 equal sides → 2 equal base angles
• Equilateral: all sides equal → all angles = 60°

🌟 MEMORIZE!

3-4-5, 5-12-13 triples! Exterior angle = sum of the two REMOTE interior angles.

🦕 Dino's Example

A right triangle has legs 6 and 8. Find hypotenuse.

✅ √(6² + 8²) = √(36+64) = √100 = 10

Isosceles: 2 equal angles Equilateral: 60°-60°-60° Scalene: all different

⬡ Polygons UNIT 3

• Sum of interior angles of n-gon = (n − 2) × 180°
• Each interior angle of regular n-gon = (n−2)×180° ÷ n
• Each exterior angle of regular n-gon = 360° ÷ n
• Sum of all exterior angles (any convex polygon) = 360°
• Rectangle area = length × width
• Parallelogram area = base × height
• Trapezoid area = ½ × (b₁ + b₂) × h
• Rhombus area = ½ × d₁ × d₂ (diagonals)

🌟 MEMORIZE!

Sum exterior angles = ALWAYS 360° for any convex polygon. Interior = (n−2)×180°.

🦕 Dino's Example

Find each interior angle of a regular hexagon (6 sides).

✅ (6−2)×180° ÷ 6 = 720° ÷ 6 = 120°

⭕ Circles UNIT 4

• Circumference = 2πr = πd
• Area = πr²
• Arc length = (central angle / 360°) × 2πr
• Sector area = (central angle / 360°) × πr²
• Inscribed angle = ½ × intercepted arc
• Central angle = intercepted arc
• Tangent ⊥ radius at point of tangency
• Two tangents from external point: equal length
• Chord-chord: (part₁)(part₂) = (part₃)(part₄)

🌟 MEMORIZE!

Inscribed angle = HALF the central angle (both intercept same arc). Tangent ⊥ radius!

🦕 Dino's Example

Circle with r = 5. Find area of sector with 72° central angle.

✅ (72/360) × π(5²) = (1/5) × 25π = 5π

📦 Solid Geometry (3D) UNIT 5

• Cube: V = s³, SA = 6s²
• Rectangular prism: V = lwh, SA = 2(lw + lh + wh)
• Cylinder: V = πr²h, Lateral SA = 2πrh
• Cone: V = ⅓πr²h, Lateral SA = πrl (l = slant height)
• Sphere: V = (4/3)πr³, SA = 4πr²
• Pyramid: V = ⅓ × base area × height

🌟 MEMORIZE!

Cone & Pyramid volume = ⅓ × (base area) × h. Sphere: V=(4/3)πr³, SA=4πr².

🦕 Dino's Example

Find volume of a cylinder with r = 3, h = 7.

✅ π(3²)(7) = 63π

🔷 Similarity & Congruence UNIT 6

• Congruent (≅): same shape AND same size
Criteria: SSS, SAS, ASA, AAS, HL (right △)
• Similar (∼): same shape, different size
Criteria: AA, SSS~, SAS~
• Similar ratio k → sides scale by k, areas by k², volumes by k³
• CPCTC: Corresponding Parts of Congruent Triangles are Congruent

🌟 MEMORIZE!

AA is enough for similarity! Ratio of areas = square of ratio of sides.

🦕 Dino's Example

Two similar triangles have sides in ratio 3:5. What is ratio of areas?

✅ 3² : 5² = 9 : 25

📏 Coordinate Geometry UNIT 7

• Distance = √[(x₂−x₁)² + (y₂−y₁)²]
• Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
• Slope m = (y₂−y₁)/(x₂−x₁)
• Slope-intercept: y = mx + b
• Parallel lines: equal slopes (m₁ = m₂)
• Perpendicular lines: slopes are negative reciprocals (m₁ × m₂ = −1)
• Circle equation: (x−h)² + (y−k)² = r²

🌟 MEMORIZE!

Perpendicular slopes multiply to −1. Distance formula uses Pythagorean Theorem!

🦕 Dino's Example

Find midpoint of (2, 6) and (8, 4).

✅ ((2+8)/2, (6+4)/2) = (5, 5)