Different types of angles

An Angle can formed when two rays originate from the same endpoint.  The rays making angle are called the arm of the angle. angle can also form by intersecting two lines or line segments. We can see the various type of angle in the fig-1. Angle can be represent by ∠ABC. Where  AB and BC are the arms of the angles and B is the vertex of the angle.

Complementary angle : If the sum of 2 angles are equal to 90°, then that angles are called complementary angles.

Supplementary angles : If the sum of 2 angles are equal to 180°, then that angles are called Supplementary angles.

Adjacent angles: If two angles are adjacent angles then they have a common vertex .a common arm and there non -common arm are the different side of the common arm. In the below figure-2 we can see a adjacent angles.

Here the angle ∠ABD and ∠DBC are adjacent angle. But also note that ∠ ABC is not adjacent angle. More over when there is two adjacent angle, then the sum is always equal to the angle formed by two non-common arms

that is  ∠ABD+∠CBD=∠ABC

Linear pair of Angle

If the non-common arm of an adjacent angle are opposite ray the angles are called linear pair of angle. As shown in the fig3, AB and BC are non common arm and the angle ABD and angle CBD are linear pair of angle.

From the fig3 we can see that ∠ABD+∠CBD=∠ABC (adjacent angle).  And ∠ABC =180° as it is straight angle. So from the above statement we can make the axiom 6.1 as follows

If a ray stand on a line, then the sum of two adjacent angle so formed is 180°

The opposite (converse of each other) of the axiom 6.1 can be state like bellow.

If sum of two adjacent  angle is 180° then the non common arm of the angle form a line.

This two axioms together called linear pair axiom.

Facts about lines / definitions of lines

Lines/(straight line)

Eventually distributed points in any direction can called  line.

Eventually distributed points in particular direction  can called   straight line.

In this section we are discussing about the straight line only.

As shown in the fig1-a A line has no end. It can be extended in either side endlessly. It can denoted by as shone in the fig.

As shown in the figure 1-b a line segment have a start and end point.  It can be denoted as shown in the figure.

A ray have a start point but don’t have an end point.  it can also be denoted by as shown in the figure 1-c.

But in normal case the line, line segment and the ray are denoted by AB only. And the  length (measure) of a line segment is also denoted by AB.

Parallel lines are two lines whose every point have equal perpendicular distance each other. it can be see in the figure 1-d.

Intersetting ling are two lines which a common point.  An intersecting line will also create angles on intersecting point. We can see an example of intersecting line as shown in the fig-e.

Line can also denoted by a single letter like m,n or l.

If three or more points lie on a same line, then this points are called collinear points. Otherwise they are called non collinear points.

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Algebraic identities or equations

  1. (x+y)(x+y) = x²+2xy+y²
  2. (x-y)(x-y) = x²-2xy+y²
  3. (x+y)(x-y) = x²-y²
  4. (x+a)(x+b) = x²+(a+b)x+ab
  5. (x+y+z)² = x²+y²+z²+2xy+2xz+2yz
  6. (x+y)³ = x³+y³+3x²y+3y²x  or  x³+y³+3xy(x+y)
  7. (x-y)³ = x³-y³-3x²y+3y²x  or  x³-y³-3xy(x-y)
  8. x³+y³+z³-3xyz = (x+y+Z)(x²+y²+z²-xy-xz-yz)
  9. x³+y³ = (x+y)(x²-xy+y²)
  10. x³-y³ = (x-y)(x²+xy+y²)
  11. if x+y+z = 0 then x³+y³+z³ = 3xyz

Equations for surface area and volume of geometrical shapes

surface area

Area of rectangle = l x b -length x birth

Area of the circle = πr²

Surface area of cuboid = 2( lb+bh+hl) where l= length, b= breth h = height

Surface area of cube = 6a² where a= edge of the cube

Lateral Surface area of cube = 4a² where a= edge of the cube

Curved surface area of a cylinder = 2πrh where r= radius of the cynder, h= height of the cylinder. 

Total surface area of the cylinder =  2πr(r+h) where r= radius of the cynder, h= height of the cylinder.

Curved surface area of cone = πrl, where r= radius of the cone, l is slant height of cone.

Total surface area of a cone = πr(l+r), where r radius, l= slant height

if we know height and radius of the cone we can find the slant height (l) of the cone by  l= √(r²+h²)

Surface area of a sphere = 4πr², where r is the radius of the sphere.

Curved surface area of a hemisphere (half sphere) = 2πr² 

Total surface area of a  hemisphere (half sphere) = 3πr² 

Volume

volume of cuboid = l x b x h where l= length, b= breth, h= height

Volume of the cube = a³ where a is the edge of the cube

Volume of the cylinder = πr²h where  r= radius, h = height

Volume of the cone = 1/3(πr²h) where r= radius of the cone , h= height of the cone

Volume of the sphere = 4/3(πr³) where r = radius of the sphere