basic geo

Contents

1. INTRODUCTION TO GEOMETRY
2. BASIC CONCEPTS
   1. POINT
   2. LINE
   3. LINE SEGMENT
   4. RAY
   5. PLANE
   6. ANGLE
3. IMPORTANT CONCEPTS RELATED TO LINES
   1. PERPENDICULAR LINES
   2. PARALLEL LINES
   3. LINE SEGMENT BISECTOR
   4. PERPENDICULAR BISECTOR
   5. ANGLE BISECTOR
   6. TRANSVERSAL AND PARALLEL LINES
   7. EQUAL INTERCEPT RATIO THEOREM

Basics of Geometry

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INTRODUCTION TO GEOMETRY

Geometry is a branch of mathematics that is concerned with the study of zero, one, two and three dimensional figures and the relationships among them. It focuses on the study of properties and relationships of size, shape, location, direction and orientation of figures like points, lines, planes, angles, polygons, and solids.

It is an important topic for any aptitude test. Geometry can be used to model and represent many mathematical and real life situations; hence, it is used as a tool to test spatial reasoning abilities.

BASIC CONCEPTS

POINT

A point is the smallest unit in a plane. It is characterized only by its position. It does not have length, width, or thickness. A mark of a sharp pencil tip on a piece of paper closely resembles the concept of a point. A point is denoted by a capital letter. In the figure below, A, B, C, D, and E are all points.

LINE

A line is a set of all the points in one dimension. It can be extended in both the directions; hence its length is infinite. A line does not have either width or thickness.

Intersection of two lines is a point.

LINE SEGMENT

A line segment is a part of a line. It has a fixed length, with a defined starting point and an end point. Like a line, it too does not have width or thickness.

Here, AB represents a line segment. Length of a line segment AB is denoted by “AB” or “ℓ(AB)”.

RAY

A ray is a part of a line that has a defined starting point (called end point) and extends upto infinity in one direction. A ray has no fixed length. It does not have any width or thickness.

A ray with end point O and passing through A is given below and is denoted by “ray OA” or

PLANE

A plane is the set of all the points in two dimensions. It does not have any thickness but is indefinitely extended in all directions.

Intersection of two planes is a line which lies in both the planes.

ANGLE

When two rays emerge from a common point, they form an angle. The common point is known as the vertex. Angles are measured in degrees () and radians.

The angle in the following picture is written as AOB or BOA or O.

Measure of AOB is denoted by mAOB. Some characters like , etc. or small letters can also be used to denote the measure of an angle.

An angle that measures more than 0 and less than 90 is known as an acute angle.

BOA in the figure below is an acute angle.

An angle that measures exactly 90 is known as a right angle. The angle shown in the figure below is a right angle.

An angle that measures between 90 and 180 is known as an obtuse angle. QOP in the following figure is an obtuse angle.

An angle that measures exactly 180 is called a straight angle. The angle shown in the figure below is a straight angle.

An angle that measures between 180 and 360 is known as a reflex angle. COA in the figure below is a reflex angle.

REMEMBER

• 0 < Acute Angle < 90

• 90 = Right Angle

• 90 < Obtuse Angle < 180

• 180 = Straight Angle

• 180 < Reflex Angle < 360

1. SUPPLEMENTARY ANGLES

If the measures of two angles add up to 180, then the angles form a pair of supplementary angles.

If A and B are supplementary angles, then

mA + mB = 180.

The supplementary angle of an angle x is equal to (180 x).

2. COMPLEMENTARY ANGLES

If the measures of two angles add up to 90, then both the angles form a pair of complementary angles.

If A and B are complementary angles, then

mA + mB = 90.

The complementary angle of an angle x is equal to (90 x).

Example 1:

Find an angle which is one third of its supplementary angle. Solution: Let the angle be x, then its supplementary angle is 3x. x + 3x = 180 4x = 180 Hence, the required angle x = 45.

Example 2:

Find an angle which is two third of its complementary angle. Solution: Let the angle be 2x, then its complementary angle is 3x. 2x + 3x = 90 5x = 90 x = 90/5 = 18 Hence, the required angle 2x = 36.

3. VERTICALLY OPPOSITE ANGLES

When two lines intersect each other, we get four angles. Two alternate, opposite angles form a pair of vertically opposite angles. Two such pairs are formed at the intersection of two lines. The angles in each pair of vertically opposite angles are always equal.

In the above figure, a = c, and b = d, as these are vertically opposite angles.

4. ADJACENT ANGLES

When two angles share a common side and a common vertex, we get two adjacent angles. For two angles to be adjacent, no angle should be inside the other.

In the above figure, AOC and BOC are adjacent angles. However, although AOC and AOB share one side and have a common vertex, they are not adjacent angles as one angle is inside the other.

REMEMBER

• If the sum of two adjacent angles is 180 then these angles form a linear pair.

• Angles making a linear pair are supplementary to each other.

Example 3:

Find the measure of ACD forming a linear pair with ACB, if ACB is twice that of ACD. Solution: Let mACD = x mACB = 2x ACD and ACB form a linear pair. mACD + mACB = 180 x + 2x = 180 3x = 180 x = 180/3 = 60 Hence, mACD = 60.

IMPORTANT CONCEPTS RELATED TO LINES

PERPENDICULAR LINES

Two lines intersecting each other at 90 are said to be perpendicular to each other.

PARALLEL LINES

Two lines in the same plane, which never intersect each other, are called parallel lines.

REMEMBER

• Two lines in the same plane that are perpendicular to a given line are parallel to each other.

LINE SEGMENT BISECTOR

A line or a ray or a segment that divides the given line segment into two equal parts is known as the line segment bisector.

PERPENDICULAR BISECTOR

A line segment bisector which makes an angle of 90 with the given segment is known as the perpendicular bisector for the given segment.

In the above figure, ℓ(PA) = ℓ(AQ) and

mPAR = mQAR = 90.

REMEMBER

• Any point on the perpendicular bisector is at an equal distance from both the ends of the given line segment.

REMEMBER

• Distance of a point from a line means the length of the perpendicular drawn from the point to the line. It is the shortest distance of the point from the line.

• Distance between two parallel lines is the perpendicular distance between them.

• Distance between two coplanar but non-parallel lines is always zero, because these lines intersect each other at some point.

ANGLE BISECTOR

A line or a ray (ray BD, in the given figure) which divides the given angle (ABC) into two equal parts (ABD and DBC) is known as an angle bisector.

REMEMBER

• Any point on the angle bisector (ray BD) is at an equal distance from both the arms of the given angle (i.e. from ray BA and from ray BC).

TRANSVERSAL AND PARALLEL LINES

A line that cuts two or more parallel lines is known as a transversal. There are many important properties related to two parallel lines and a transversal. To understand these, refer to the following figure.

Here,

Vertically opposite angles are equal: a = d, b = c, e = h and f = g

Alternate interior angles are equal: c = f and d = e

Alternate exterior angles are equal: a = h and b = g

Corresponding angles are equal: a = e, b = f, c = g and d = h

Interior angles on the same side of the transversal are supplementary: c + e = 180 and d + f = 180

Exterior angles on the same side of the transversal are supplementary: a + g = 180 and b + h = 180

Example 4:

In the above figure, if thrice of a = twice of b, then find the sum of d and h. Solution: From the given condition, it is clear that the ratio of the two angles a and b is 2:3. Let a = 2x b = 3x Since, these two angles form a linear pair, a + b = 180 2x + 3x = 180 5x = 180 x = 36 d = h = a = 2x = 2 36 = 72. ( d and h are corresponding angles and d and a are vertically opposite angles.) Hence, d + h = 72 + 72 = 144

EQUAL INTERCEPT RATIO THEOREM

For a set of three or more parallel lines (L₁, L₂ and L₃ for example), and two or more transversals (T₁ and T₂ for example), the ratio of the lengths of the intercepts of any transversal is equal.

i.e. AB/BC = MN/NO

Example 5:

In the above figure, if ℓ(AB) is twice that of ℓ(BC), ℓ(MN) = 10 units, then find ℓ(MO). Solution: By Equal Intercept Ratio Theorem, AB/BC = MN/NO Let ℓ(BC) = x. ℓ(AB) = 2x 2x/x = 10/ℓ(NO) 2 = 10/ℓ(NO) ℓ(NO) = 5 units Hence, ℓ(MO) = 10 + 5 = 15 units.