"Rawr! I'm Dino! Let's explore the wonderful world of geometry together! 20 epic questions await you!" 🦖✨
📐 Angles🔺 Triangles⭕ Circles📏 Lines🔷 Polygons📊 Coordinate🔢 Proofs🌀 3D Solids
📚 Concept Review with Dino
Master the basics before the big quiz!
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📏 Lines & Angles
🔑 Key Definitions
Line: extends infinitely in both directions
Ray: starts at a point, extends one direction
Line Segment: has two endpoints
Parallel lines: never intersect (symbol: ∥)
Perpendicular lines: meet at 90° (symbol: ⊥)
Transversal: line crossing two or more parallel lines
📐 Angle Pairs to Memorize
🧠 MUST KNOW!
Complementary: add to 90°
Supplementary: add to 180°
Vertical angles: equal (opposite angles when two lines cross)
Linear pair: adjacent angles summing to 180°
Alternate interior angles: equal (parallel lines)
Corresponding angles: equal (parallel lines)
Co-interior (same-side interior): add to 180°
📝 Example
Two parallel lines are cut by a transversal. One angle is 65°. Find its alternate interior angle.
Answer: 65° (Alternate interior angles are equal when lines are parallel)
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🔺 Triangles
🔑 Triangle Basics
Sum of angles = 180°
Exterior angle = sum of two non-adjacent interior angles
Perimeter = a + b + c
Area = (1/2) × base × height
🧠 Types of Triangles
By Angles:
Acute: all angles < 90°
Right: one angle = 90°
Obtuse: one angle > 90°
By Sides:
Scalene: all sides different
Isosceles: two sides equal → base angles equal
Equilateral: all sides equal → all angles = 60°
📐 Triangle Congruence (SSS, SAS, ASA, AAS, HL)
🧠 Congruence Shortcuts:
SSS: 3 sides equal
SAS: 2 sides + included angle equal
ASA: 2 angles + included side equal
AAS: 2 angles + non-included side equal
HL: hypotenuse + leg (right triangles only)
📐 Similarity (AA, SAS~, SSS~)
AA: 2 pairs of equal angles → similar triangles
SAS~: 2 sides proportional + included angle equal
SSS~: all 3 sides proportional
📝 Example
A triangle has angles 45° and 75°. Find the third angle.
180° - 45° - 75° = 60° ✓
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📐 Pythagorean Theorem
🔑 The Theorem
a² + b² = c²
(c = hypotenuse, the longest side)
Distance Formula:
d = √[(x₂-x₁)² + (y₂-y₁)²]
🧠 Special Right Triangles
30-60-90 Triangle:
sides: x, x√3, 2x
opposite: 30°, 60°, 90°
45-45-90 Triangle:
sides: x, x, x√2
opposite: 45°, 45°, 90°
🔢 Common Pythagorean Triples
3-4-55-12-138-15-177-24-256-8-109-12-15
📝 Example
In a 45-45-90 triangle, one leg = 6. Find the hypotenuse.
hypotenuse = 6√2 ≈ 8.49
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🔷 Quadrilaterals & Polygons
🔑 Key Formulas
Sum of interior angles of n-gon = (n-2) × 180°
Each interior angle (regular n-gon) = (n-2)×180° / n
Each exterior angle (regular n-gon) = 360° / n
Sum of exterior angles (any convex polygon) = 360°
Circumference = 2πr = πd
Area = πr²
Arc length = (θ/360°) × 2πr
Sector area = (θ/360°) × πr²
🧠 Circle Angle Theorems
Inscribed Angle Theorem:
Inscribed angle = (1/2) × intercepted arc
Central Angle:
Central angle = intercepted arc
Tangent-Chord Angle:
= (1/2) × intercepted arc
Two Chords Inside:
angle = (1/2)(arc1 + arc2)
Two Secants/Tangents Outside:
angle = (1/2)|arc1 - arc2|
📐 Chord & Tangent Rules
Two chords: a×b = c×d (products of segments)
Tangent² = external × whole secant
Tangent from external point: both tangents equal in length
📝 Example
A circle has radius 5. Find its area.
Area = π(5)² = 25π ≈ 78.54 sq units
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📊 Coordinate Geometry
🔑 Essential Formulas
Slope: m = (y₂-y₁)/(x₂-x₁)
Distance: d = √[(x₂-x₁)²+(y₂-y₁)²]
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope-intercept: y = mx + b
Point-slope: y-y₁ = m(x-x₁)