Class 8 – Geometry

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Geometry
  • Review of geometry
  • Congruence of triangles
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Geometry
  • Quadrilaterals
  • Polygons
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Geometry
  • Inequalities of a triangle
  • Circles,Arcs and Chords
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Geometry
  • Areas (Parallellogram and triangle)
  • Pythagoras Theorem
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Point: A point is represented by a dot. It has no dimension like length, breadth or thickness. It has only position. Points are denoted by capital letters A, B, C, D etc.

Line: A geometrical line is a set of points that extends endlessly in both the directions ie., a line has no end points. It has only length. The arrow heads show that the line goes on endlessly in both directions.

Lines are denoted by small letters ,…… or capital letters as , ,….

Line-segment: A line segment is a line which has end points. In the below figure, the part of the line between the points. ‘A’ and ‘B’ including ‘A’ and ‘B’ is a line-segment.

The line-segment AB is represented as  or segment AB.

http://images.clipartpanda.com/bloc-clipart-note-md.pngThe shortest distance between two points is called line segment.

Plane: A plane is a set of points. It is a flat surface with length and breadth. A geometrical plane extends endlessly in all the directions. Small letters are used to denote a plane.

Surface of a sheet of a paper, surface of a wall, surface of a table,….

Coplanar Points: The points that belong to the same plane are called coplanar points.

Coplanar Lines: The lines that lie in the same plane are called coplanar lines.

 

Incidence Properties: The relation between a point and line is called an incidence property.

Distance between Two Points: The distance between two points A and B is the length of the line-segment joining them.

The distance between A and B is denoted by d(A, B) or AB.

Midpoint: Given a line segment AB, a point M is said to be the midpoint of AB, if M is an interior point of AB, such that AM = MB.

Perpendicular Bisector: Aline ‘l’ passing through the midpoint ‘M’ of a line segment  and perpendicular to AB is called the perpendicular bisector of the line-segment .

Collinear Points: If three or more points lie on a straight line, then those points are called collinear points.

A, B, C, D, E are collinear.

Non-collinear: The points which do not lie on the straight line are called non-collinear points.

G, H do no lie on the straight line ‘l’.

Hence, they are non-collinear points.

http://images.clipartpanda.com/bloc-clipart-note-md.pngA point ‘C’ is said to line between ‘A’ and ‘B’ if

a) A, B, C are collinear and

b) AB = AC + BC

 

Ray: Let ‘l’ be aline and A, B be two distinct points on ‘l’.

A ray is a part of line ‘l’ which has one end point as ‘A’ and contains the point ‘B’. A ray is denoted by the symbol.

Properties of straight lines:

i)    Ona single line ‘l’ there exists infinite number of points.

ii)  An infinite number of lines pass through a single points p (concurrent lines).

iii) Only a single line passes through two distinct points ‘A’ and ‘B’.

iv) Given two different lines ‘l’ and ‘m’ there exists only one point ‘P’. Which lies on both the lines ‘l’ and ‘m’.

i.e., the two lines intersect at P.

http://images.clipartpanda.com/bloc-clipart-note-md.pngTwo distinct lines cannot have more than one point in common.

Angle: An angle is the union of two rays with a common initial point.

When a ray AB rotates in the plane about the point ‘A’ and takes new position ‘AC’, an angle BAC is formed. Symbol of angle is ∠.

The angle formed by the two rays and  is denoted by ∠BAC or ∠CAB or ∠A

The two rays  and  are called the arms and the common initial point ‘A’ is called the vertex.

Bisector of an angle: Aline which divides an angle into two equal parts is called the bisector of the angle.

The line OP is called the bisector of ∠AOB

Types of Angles:

i)    Zero angle: An angle in said to be a zero angle which two rays coin side and the measure is 00.

ii)  Acute angle: An angle which is more than 0° and less than 900 is called an acute angle.

                          

iii) Right angle: An angle which is exactly 900is called a right angle.

                                   

iv)  Obtuse angle: An angle which is greater than 900 and less than 1800 is called an obtuse angle.

v)   Straight angle: An angle which is exactly 1800 is called a straight angle.

vi)  Reflex angle: An angle which is greater than 1800 and less than 3600 is called reflex angle.

vii)  Complete angle: An angle which is exactly 3600is called a complete angle.

 

 

Pair of angles:

i)    Complementary Angles: If the sum of the measures of two angles is equal to 900, then the angles are called be complementary angles.
x + ∠y = 900
∴ ∠x and ∠y are complementary angles.

ii)  Supplementary angles: If the sum of the measures of two angles is equal to 1800, then the angles are called be supplementary angles,
∠x + ∠y =1800

∴ ∠xand ∠y are supplementary.

Adjacent angles: Angles having the same vertex and a common side and which lie on the opposite sides of the common side are called adjacent angles.

 with common vertex O and common side OB are adjacent angles.

Linear Pair of Angles: Two adjacent angles are said to form a linear pair of angles if they lie on the same straight line. The two non-common arms are opposite rays.

The sum of two adjacent angles of a linear pair of angles is 180 °, hence they are also supplementary.

Congruent Angles: Two angles are said to be congruent if they have the same measure.

Vertically opposite angles: If two lines AB and CD intersect at a point ‘O’, then the pair of angles ∠AOC and ∠BODis said to be a pair of vertically opposite angles. Also ∠AOD and  ∠BOC form another pair of vertically opposite angles.

                                            

Angles forming a pair of vertically opposite angles are congruent (or equal)

Angle Addition Axiom: ∠ABC =∠PBC +∠PBA

http://images.clipartpanda.com/bloc-clipart-note-md.pngThe sum of the measures of the angles formed around a point is 3600.

 

Properties of angles:

i)    The adjacent angles formed when one striaght line stands over another are together equal to two right angles ie., 1800.

ii)  If two adjacent angles are supplementary then their outer arms are on a straight line.

iii) If two straight lines cut one another, the four angles so formed are together equal to four right angles ie., 3600

iv)  If two straight lines cut one another then, the vertically opposite angles are equal.

v)   When a number of straight lines meet at a point the sum of all angles so formed at that point is equal to four right angles ie., 3600.

Pair of lines:

Parallel lines: Two lines in a plane which do not intersect when they are produced endlesly in the directions are called parallel lines.In the given figures, some examples of parallel are given, which are written as:

i)    lm

ii)  PQ║RS

iii) AB║CD

Intersecting lines: Two lines are said to be intersecting lines if they have a common point. Consider the letter Y made up of line segements and the grill-door of a window. These are examples of intersecting lines.

For example: In the figure given alongside. AC and BE intersect at P, AC and BC intersect at C, AC and EC intersect at C.

Transversal: A straight line which intersect two or more given lines in a plane at different distinct points is called a transveral to the given lines.

In figure (i), the tranversal p cuts three parallel lines l, m and at points X, Y and Z respectively.

In figure (ii), the tranversal q cuts two non-parallel lines AB and CD at points E and F respectively.

Angles made by a transveral: When a traversal cuts two lines (parallel or non-parallel), eight angles are formed. In the adjoining figure, l and m are two lines and transversal t intersecting l and m at P and Q respectively.

These eight angles formed are marked 1 to 8. These angles so formed are given some distinguishing names.

i)    Exterior angles: The angles that do not contain PQ as one of their arms are called exterior angles. In the adjoining figure ∠1, ∠2, ∠7 and ∠8 are exterior angles.

ii)  Interior angles: The angles that contain PQ as one of their arms are called interior angles. In the adjoining figure, ∠3, ∠4, ∠5 and ∠6 are interior angles.

Alternate angles:
Alternate Exterior angles: The two pairs ∠1, ∠7 and ∠2, ∠8 are called alternate exterior angles.

Alternate Interior angles: The two  pairs ∠3, ∠5 and ∠4, ∠6 are called alternate interior angles or simply alternate angles.

iii) Corresponding angles: There are four pairs of corresponding angles.

In the given figures, the respective pairs are ∠1, ∠5; ∠2, ∠6; ∠3, ∠7 and ∠4, ∠8.

iv)  (i) Co-interior (or consecutive interior or conjoined or allied) angles: The two pairs ∠3, ∠6 and ∠4, ∠5 are called co-interior angles.

(ii) Exterior Allied Angles: The two pairs ∠1, ∠8 and ∠2, ∠7 are called exterior allied angles.

Example: Name the pairs of angles in given figures:

Solution: In fig (i): ∠1 and∠2 are correspoding angles.

In fig (ii): ∠3 and ∠4 are alternate interior angles.

In fig (iii): ∠5 and ∠6 form a pair of co-interior angles.

Transversal of parallel lines: Let a transveral ‘t’ intersect two parallel lines l and m at points P and Q repectively. The angles formed are marked 1 to 8 as shown. Then,

i)    Corresponding angles are equal
i.e., ∠1 = ∠5                     ∠3 = ∠7
∠2 = ∠6                     ∠4 = ∠8

ii)  Alternate interior angles are equal
i.e., ∠3 = ∠5,                    ∠4 = ∠6

iii) Alternate exterior angles are equal
i.e., ∠1 = ∠7,                    ∠2 = ∠8

iv)  Co-interior angles are supplementary
i.e., ∠3 + ∠6 =1800 and ∠4 + ∠5 = 1800

v)   Co-exterior angles are supplementary
i.e., ∠1 + ∠8 =1800 and ∠2 + ∠7 = 1800

Thus, we conclude that, if two lines are intersected by a transversal and are parallel, if any one of the following is true:

i)    Pair of corresponding angles are equal.

ii)  Pair of alternate (interior or exterior) angles are equal.

iii) Sum of interior angles or exterior angles on the same side of the transversal are supplementary.

http://images.clipartpanda.com/bloc-clipart-note-md.png

In short: ∠1 = ∠3 = ∠5 = ∠7 and∠2 = ∠4 = ∠6 = ∠8

 

Example: State the property that is used in each of the following statements:

Solution:

(i)  Corresponding angles property.        (ii) Alternate interior angles property.

(ii) Interior angles on the same side of the transversal are supplementary.

Example: Find the value of x in each of following figures if .

(i)                                 (ii)                                  (iii)

                     

Solution:

(i)  Sincel m and t is transversal
x= (1800 – 1100) = 700

(ii) x +2x= 1800
(The sum of the interior angles on the same side of the tranversal is 1800)
⇒ 3x= 1800  ⇒  x = 600

(iii)       Sincel m and tis transversal
∴ ∠x =1200   [corresponding angles]

Triangles

Name

Properties

Equilateral triangle

i)     Always acute angled.

ii)   All 3 sides are equal.

iii) 3 angles are equal and each angle is  600.

iv)  Incenter, circumcentre, orthocentre and centriod coincide.

v)    Point of intersection of altitudes, medians and angular bisectors is same.

Isosceles triangle

i)     2 sides equal.

ii)   2 angles equal. (which are opposite to equal sides)

iii) 2 medians, 2 altitudes equal.

iv)  Internal bisectors of 2 angles are equal.

v)    Bisector of vertical angle bisects the base and perpendicular to the base.

vi)  May be acute, obtuse or right angled triangle.

Isosceles right angled triangle

i)     2 sides are equal.

ii)   Angle included by the equal sides is 900.

iii) Other two angles are each 450.

 Right angled triangle

i)     One angle is 900.

ii)   Sum of the other two is 900.

iii) Side opposite to 900 is hypotenuse and is the greatest side.

iv)  Median to the hypotenuse is half of the hypotenuse.

v)    Of the two acute angles, if one is 300. The smallest sides is half of the greatest side or the side opposite to 300 is half of hypotensue.

http://www.aids.harvard.edu/img/news/spotlight/newsletter_layout/logo.gif

i)    Linear pair axiom: If a ray stands on a straight line, then the sum of the adjacent angles so formed, is 1800. Thus, ∠AOB + ∠COB =1800, where AOC is a straight line.

ii)   Converse of (i): If two adjacent angles are supplementary, then the non -common arms of the angles are in a straight line.

Thus, if two adjacent angles ∠AOC and ∠BOC with common arm OC are such that
∠AOC + ∠BOC =1800, then OA and OB are in the same straight line i.e., AOB is a straight line.

iii) Angles at a point: The sum of all the angles at a point is 3600.

In the given figure, we have
∠1 +  ∠2 + ∠3  + ∠4 = 3600