Geometry and Mensuration - Important Concepts & Formulas

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Geometry is one of the main areas of Mathematics. Basically it is a study of shapes. The word 'Geo' means 'Earth' and 'Metry' means 'Measure'. So Geo - metry is literally 'Measurement of earth'. Traditional geometrical concepts are called Euclidian Geometry (mainly in two-dimension). Geometry is a sure area in all Placement Test & B-School entrance exams. In this article, you can learn Important Concepts & Formulas of Geometry and Mensuration along with the examples

Here we are considering main two areas of geometry and those are,
  • Plane geometry: Lines, angles, polygons and their properties, Mensuration etc.
  • Solid geometry: This is basically a three dimensional geometry derived in three dimensional spaces. This area is dealing with different kinds of prisms, pyramids, spheres and other various three dimensional figures.

 

The main objectives of this article are
  • Lines and Angles.
  • Properties of angles.
  • Basic properties of triangles.
  • Geometric centre's of a triangle.
  • Similarity and congruency in triangles.
  • Area of a triangle (Various forms)
  • Important theorems and useful results on triangles..
  • Properties of right-angled triangles.
  • Polygons.
  • Quadrilaterals
  • Classifications of quadrilaterals
  • Properties of various quadrilaterals
  • Circles
  • Sectors & Segments
  • Important properties and results related to circles
  • Tangents & Secants

 

Lines and angles

Classification of angles

  • Acute angle:

An angle greater than 0° but less than 90° is called an acute angle.

i.e. 0° < θ < 90°

Eg: 30°, 45°, 89° etc.

  • Right angle:

An angle exactly equal to 90 ° is called a right angle.

i.e. θ = 90°

  • Obtuse angle:

An angle greater than 90° but less than 180° is called an obtuse angle.
i.e. 0° < θ < 180°

Eg: 110°, 175° etc.

  • Straight angle:

An angle exactly equal to 180° is called a straight angle.

  • Reflex angle:

An angle greater than 180° but less than 360° is called a reflex angle.

i.e. 0° < θ < 360°

Eg: 190°, 270° etc.

  • Complete angle:

An angle exactly equal to 360° is called a complete angle.

  • Complementary angles:

If the sum of two angles equal to 90°, then the angles are complementary.

i.e. θ1 + θ2 = 90°

Eg: complementary of 30° is 60°.

Complementary of 70° is 20°.

  • Supplementary angles:

If the sum of two angles equal to 180°, then the angles are supplementary.

i.e. θ1 + θ2 = 180°

Eg: supplementary of 30° is 150°.

Supplementary of 110° is 70°.

  • Adjacent angles:

If two angles have a common hand and a common vertex then they are adjacent.

In diagram, ∠AOB and ∠BOC are adjacent. Therefore∠AOC = ∠AOB + ∠BOC.

Properties of angles

Case I. When two lines intersecting:

Two lines 'l' and 'm' are intersecting, and then there are four different angles formed. Each of the four angles is mutually related in the following manner.

  • Linear pairs:

If two angles are adjacent and supplementary too, then they are called Linear Pairs.

(1, 2), (2, 3), (3, 4) and (4,1) are linear pairs.

i.e. ∠1 + ∠2 = 180°

∠2 + ∠3 = 180° and so on.

  • Vertically opposite angles:

Out of the four angles forms by intersecting two lines, opposite angles are called vertically opposite angles, and they are equal in measure.

i.e. ∠1 = ∠3 and ∠2 = ∠4

  • Angle around a point:

Around the point of intersection of two lines, there are four angles and the sum of these angles equal to 360°.

i.e. ∠1 + ∠2 + ∠3 + ∠4 = 360°.

Case II: A transversal passes through two parallel lines

In diagram lines 'l' and 'm' are parallel and a transversal 't' passes through both the lines, then there are eight different angles formed.

By observation, it is possible to categorize these eight angles as acute and obtuse.

Acute angles: ∠2, ∠4, ∠6 and ∠8.

All acute angles in the diagram are equal in measure.

Obtuse angles: ∠1, ∠3, ∠5 and ∠7.

All obtuse angles in the diagram are equal in measure.

In this diagram, sum of any one of the acute angles and any one of the obtuse angles is equal to 180°.

  • Alternate interior angles: (Z type angles)

For easiness to understand, these are the Z shape angles.

Both the corners of Z are equal in both directions.

i.e. ∠4 = ∠6 and ∠3 =∠5.

Therefore (4, 6) and (3, 5) are alternate interior angles.

  • Corresponding angles (F type angles)

These angles formed as the corners of alphabet 'F' in different directions and corresponding angles are equal in measure.

Therefore;

∠1 = ∠5

∠2 = ∠6

∠3 = ∠7

∠4 = ∠8

  • Co - interior angles (C type angles)

Co-interior formed as the corners of alphabet 'C'. These angles are supplementary.

Therefore;

∠3 + ∠6 = 180° and ∠4 +∠5 = 180°

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