Geometry Made Easy! 📐✨ Here’s a simple, emoji-filled summary of all the key geometry concepts from your PDF, perfect for SSC CHSL Quant Aptitude prep! 12 Basic Concepts 🟢 Point: An exact location, no size—just a dot! ⚫ Line Segment: Straight path between two points (A & B) with a fixed length. 📏 Ray: Starts at a point and goes on forever in one direction. ➡️ Intersecting Lines: Cross at a point. ✖️ Concurrent Lines: 3+ lines meeting at the same point. 🌟 Angles 🔺 Angle: Formed when two lines meet at a point. (∠AOB) 🧭 Right Angle: 90° ⬛ Acute Angle: Less than 90° 🔽 Obtuse Angle: Between 90° and 180° 🔼 Reflex Angle: Between 180° and 360° 🔄 Complementary Angles: Add up to 90°. 60° + 30° = 90° 🤝 Supplementary Angles: Add up to 180°. 130° + 50° = 180° 👫 Vertically Opposite Angles: Equal when two lines cross. 🔁 Bisector: Divides an angle into two equal parts. ✂️ Lines & Parallelism 🟦 Parallel Lines: Never meet, always same distance apart. 〰️〰️ Transversal: Cuts across two or more lines. ➖ Alternate Angles: Equal when lines are parallel. 🔄 Corresponding Angles: Equal when lines are parallel. 🟰 Sum of Angles on a Straight Line: 180° ➡️ Triangles 🔺 Triangle: 3 sides, 3 angles. △ Sum of Angles: Always 180° 🔢 Side Rule: Any two sides’ sum > third side. ➕ Exterior Angle: Equals sum of two opposite interior angles. 🔄 Area of Triangle 📏 $ Area = \frac{1}{2} \times base \times height $ 🏕️ $ Area = \frac{1}{2} ab \sin C $ (using sine) 🟢 Heron’s Formula: $ \sqrt{s(s-a)(s-b)(s-c)} $, where $ s = \frac{a+b+c}{2} $ 🧮 Special Segments in Triangles 📐 Median: Vertex to midpoint of opposite side. Intersect at centroid (divides median 2:1). 🔹 Altitude: Perpendicular from vertex to opposite side. Meet at orthocenter. ⬆️ Angle Bisector: Divides angle into two equal parts, meet at incentre. ✂️ Perpendicular Bisector: Cuts side into two equal parts at 90°, meet at circumcentre. ➕ Similarity & Area 🟠 Similar Triangles: Angles equal, sides in proportion. 🔁 Area Ratio: Ratio of areas = (ratio of sides)² 📊 Polygons 🟣 Regular Polygon: All sides & angles equal. 🛑 Sum of Angles: $ 180(n-2) $, where n = number of sides. 🔢 Interior Angle: $ \frac{180(n-2)}{n} $ 🧮 Example: Octagon (8 sides): Each angle = 135° 🛑 Quadrilaterals 🟤 Square: All sides equal, all angles 90°. Area = $ a^2 $, Diagonal = $ a\sqrt{2} $ ⬛ Rectangle: Opposite sides equal, all angles 90°. Area = length × breadth 🟦 Parallelogram: Opposite sides equal & parallel. Area = base × height 🔲 Rhombus: All sides equal, diagonals perpendicular. Area = $ \frac{1}{2} \times d_1 \times d_2 $ 💎 Trapezium: One pair of parallel sides. Area = $ \frac{1}{2} \times (sum of parallel sides) \times height $ 🟫 Circle ⚪ Center: Fixed point (O) 🟢 Radius (r): Distance from center to edge. 📏 Diameter (d): $ 2r $ ⬅️➡️ Circumference: $ 2\pi r $ 🔵 Area: $ \pi r^2 $ 🧮 Chord: Line joining two points on the circle. ➖ Arc: Part of circumference. 🟠 Sector: ‘Pizza slice’ of the circle. Area = $ \frac{\theta}{360} \times \pi r^2 $ 🍕 Tangent: Touches circle at one point, perpendicular to radius. 🚏 Key Circle Properties 🟡 Equal chords = equal distance from center 🟰 Angle in a semicircle = 90° ⬛ Angles in same arc = equal 🔁 Angle at center = 2 × angle at circumference (same arc) 🔄 Intersecting Chords Rule: $ AP \times PB = CP \times PD $ ✖️ Tangents from same point are equal 🟰 Important Examples & Tricks 🏆 Sum of Third Side in Triangle: If two sides are 12 & 7, possible third sides = 6 to 18 (integers), sum = 156. Area of Hexagon: $ \frac{3\sqrt{3}}{2} a^2 $ for side a. 🟩 Circle Chord Example: Chord of length 8 cm is twice as far from center as chord of 10 cm. Use Pythagoras to solve! Parallel Chords Example: Distance between two chords = add/subtract their distances from center. Emoji Legend 🗺️ ⚫ Point 📏 Line/Length ➡️ Ray/Direction ✖️ Intersect 🌟 Concurrent ⬛ Right angle/Square 🔽 Acute/Small 🔼 Obtuse/Large 🔄 Reflex/Alternate 🤝 Complementary 👫 Supplementary 🔁 Equal/Vertically opposite/Similar ✂️ Bisector 〰️ Parallel ➖ Chord/Transversal △ Triangle 🏕️ Area 🧮 Calculation 🔹 Median/Centroid ⬆️ Altitude/Orthocenter 🛑 Polygon 🟦 Rectangle 🔲 Parallelogram 💎 Rhombus 🟫 Trapezium ⚪ Circle 🍕 Sector 🚏 Tangent 🟰 Equal 🟡 Key Point 🏆 Example Use this guide for quick revision and easy understanding of all geometry basics for your exam! Good luck! 🍀12