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## 📐 **Geometry Notes**
### **1. Basics of Geometry**
- **Point:** A location (no size).
- **Line:** Infinite set of points in a straight path.
- **Plane:** Flat surface extending in all directions.
- **Segment:** Part of a line with two endpoints.
- **Ray:** A line with one endpoint that extends infinitely in one direction.
- **Angle:** Formed by two rays with a common endpoint (vertex).
**Types of Angles:**
- Acute (< 90°), Right (= 90°), Obtuse (> 90°), Straight (= 180°)
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### **2. Logic and Proof**
- **Conditional Statement:** If p, then q.
- **Converse, Inverse, Contrapositive:** Variations of conditionals.
- **Deductive Reasoning:** Using facts, definitions, logic.
- **Two-column Proof:** Step-by-step logical argument.
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### **3. Parallel and Perpendicular Lines**
- **Parallel Lines:** Never intersect, same slope.
- **Perpendicular Lines:** Intersect at 90°, slopes are negative reciprocals.
- **Angle Relationships:**
- Corresponding, Alternate Interior, Alternate Exterior, Consecutive Interior.
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### **4. Triangles**
- **Triangle Sum Theorem:** All angles = 180°
- **Types by sides:** Equilateral, Isosceles, Scalene.
- **Types by angles:** Acute, Right, Obtuse.
- **Pythagorean Theorem:** \( a^2 + b^2 = c^2 \) (right triangles).
- **Congruence Postulates:** SSS, SAS, ASA, AAS, HL.
- **Triangle Inequality Theorem:** Sum of any two sides > third side.
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### **5. Polygons**
- **Sum of Interior Angles:** \( (n - 2) \times 180^\circ \)
- **Regular Polygon:** All sides and angles equal.
- **Exterior Angles:** Always sum to \( 360^\circ \).
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### **6. Quadrilaterals**
- **Types:**
- Parallelogram, Rectangle, Rhombus, Square, Trapezoid.
- **Properties:**
- Opposite sides/angles equal (parallelograms).
- Diagonals bisect each other (parallelograms).
- Diagonals are congruent (rectangles).
- Diagonals are perpendicular (rhombus).
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### **7. Circles**
- **Parts:**
- Radius, Diameter, Chord, Tangent, Secant, Arc.
- **Formulas:**
- Circumference: \( C = 2\pi r \)
- Area: \( A = \pi r^2 \)
- **Central Angle = Arc**
- **Inscribed Angle = ½ Arc**
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### **8. Transformations**
- **Types:**
- Translation (slide), Reflection (flip), Rotation (turn), Dilation (resize).
- **Congruent Figures:** Same shape and size.
- **Similar Figures:** Same shape, proportional size.
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### **9. Area and Volume**
- **Area Formulas:**
- Rectangle: \( A = l \times w \)
- Triangle: \( A = \frac{1}{2} b h \)
- Trapezoid: \( A = \frac{1}{2} (b_1 + b_2) h \)
- Circle: \( A = \pi r^2 \)
- **Surface Area / Volume (3D):**
- Prism Volume: \( V = B \times h \)
- Cylinder: \( V = \pi r^2 h \)
- Pyramid: \( V = \frac{1}{3} B h \)
- Sphere: \( V = \frac{4}{3} \pi r^3 \)
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### **10. Coordinate Geometry**
- **Distance Formula:**
\( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
- **Midpoint Formula:**
\( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
- **Slope Formula:**
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
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