Class 9 Mathematic Euclid's Geometry All Formula & Concept Notes | Formula Sheet

Introduction to Euclid's Geometry

Euclid's Geometry forms the basis of plane geometry as we know it today. In this chapter, we explore the foundational concepts and logical structure that Euclid introduced, which have become the standard in geometry. Below is a detailed overview of the formulas and concepts.

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1. Axioms and Postulates

Axioms (or Common Notions)

Axioms are universally accepted truths, which do not require proof. Some of the basic axioms include:

1. Things that are equal to the same thing are equal to one another.

   • Example: If $a = b$ and $b = c$, then $a = c$.

2. If equals are added to equals, the wholes are equal.

   • Example: If $a = b$ and $c = d$, then $a + c = b + d$.

3. If equals are subtracted from equals, the remainders are equal.

   • Example: If $a = b$ and $c = d$, then $a - c = b - d$.

4. Things that coincide with one another are equal to one another.

   • Example: Two figures that perfectly overlap are identical.

5. The whole is greater than the part.

   • Example: $a > b$ if $b$ is a part of $a$.

6. Things which are double of the same things are equal to one another.

   • Example: If $a = b$, then $2a = 2b$.

7. Things which are halves of the same things are equal to one another.

   • Example: If $a = b$, then $\frac{a}{2} = \frac{b}{2}$.

Postulates

Postulates are specific geometrical assumptions accepted without proof. Euclid's five postulates are:

1. A straight line may be drawn from any one point to any other point.

   • A line segment can be drawn between any two points.

2. A terminated line can be produced indefinitely.

   • A line segment can be extended in both directions without limit.

3. A circle can be drawn with any center and radius.

   • Given any point as the center and any length as the radius, a circle can be constructed.

4. All right angles are equal to one another.

   • Any angle measuring 90 degrees is congruent to another right angle.

5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two lines, if produced indefinitely, meet on that side.

   • This is known as the parallel postulate and forms the basis for parallel lines in Euclidean geometry.

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2. Definitions

Euclid defined various geometric terms, which are essential for understanding his geometry.

1. Point: A point has no dimensions, only position.
2. Line: A line has length but no breadth.
3. Straight Line: A line that lies evenly with the points on itself.
4. Surface: A surface has length and breadth only.
5. Plane Surface: A flat surface that lies evenly with the straight lines on it.
6. Angle: The inclination of two lines to each other in a plane which meet.
   • Right Angle: An angle of 90°.
   • Acute Angle: An angle less than 90°.
   • Obtuse Angle: An angle greater than 90°.

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3. Theorems and Proofs

Euclid’s geometry relies heavily on logical deductions from the axioms and postulates. Some important theorems include:

1. Theorem 1:

   • If two lines intersect, then the vertically opposite angles are equal.
   • Proof Outline:
     1. Assume two intersecting lines $AB$ and $CD$.
     2. Angles $\angle AOC$ and $\angle BOD$ are vertically opposite.
     3. By axiom, $\angle AOC + \angle BOC = 180^\circ$ and $\angle BOD + \angle BOC = 180^\circ$.
     4. Subtracting $\angle BOC$ from both sides, $\angle AOC = \angle BOD$.

2. Theorem 2:

   • The sum of the angles of a triangle is equal to 180 degrees.
   • Proof Outline:
     1. Draw a triangle $ABC$ and extend line $BC$.
     2. Show that $\angle ABC + \angle BCA + \angle CAB = 180^\circ$ using parallel postulate and properties of angles.

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4. Parallel Lines and Angles

Parallel lines and the angles formed when a transversal cuts them are key concepts:

1. Corresponding Angles:

   • When a transversal intersects two parallel lines, the corresponding angles are equal.
   • Formula: $\angle 1 = \angle 2$.

2. Alternate Interior Angles:

   • Alternate interior angles are equal when a transversal cuts two parallel lines.
   • Formula: $\angle 3 = \angle 4$.

3. Sum of Interior Angles on the Same Side:

   • The sum of the interior angles on the same side of the transversal is 180°.
   • Formula: $\angle 5 + \angle 6 = 180^\circ$.

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5. Important Concepts

1. Congruence of Triangles:

   • Two triangles are congruent if:
     1. SAS (Side-Angle-Side): Two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle.
     2. ASA (Angle-Side-Angle): Two angles and the included side of one triangle are equal to two angles and the included side of another triangle.
     3. SSS (Side-Side-Side): Three sides of one triangle are equal to three sides of another triangle.

2. Parallel Lines:

   • Criteria: Two lines are parallel if:
     1. Corresponding angles are equal.
     2. Alternate interior angles are equal.
     3. Interior angles on the same side sum up to 180°.

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6. Euclid's Elements

• Euclid compiled his geometric work into a series of books known as "The Elements".
• The first book covers plane geometry, including the concepts, axioms, and postulates we've discussed.
• The rest of the books explore more advanced topics like number theory, solid geometry, and the theory of proportions.

Conclusion

Euclid's Geometry is foundational in understanding the principles of plane geometry, providing a logical structure through axioms, postulates, definitions, and theorems. Mastery of these concepts is essential for higher-level mathematics and geometrical reasoning.