Angles in Grade IV

Hello friends, in today's session we are going to learn about angles/angles related to fractions and parallel/perpendicular lines.

Let's start with angles.

An angle (read this for more information) can be defined as the figure formed when two rays or line segments which are called as the sides come and intersect at a common point called as vertex.

For example in the image given below the angle is <ABC or < CAB. We can also write it as <B which is the name of the vertex.

                                                                         

We should notice that in the above angle representation the vertex B is always written in the middle.

Now we come to the different types of angles. There are three basic types of angles, Acute angles, Right angles and obtuse angles.

An acute angle in the one which measures less than 90° and looks like

                                                                   

The obtuse angle is the one which measures more than 90° and has the following kind of shape

                                                          

And the right angle is the one which measures exactly 90°, and looks like

                                                                           

The other two types which are not used commonly are the Straight angle and the reflex angle.

Straight angles are the one which measure exactly 180°, and its figure looks like a straight line.

The reflex angle is the one which measures more than 180° but less than 360°.

Now we proceed to the pairs of angles.

Complementary angles, the two angles which add up to 90° are called as the complementary angles.

                                                             

<ABD + < DBC = 90°. so it is a complementary angle.

Supplementary angle, the two angles which add upto 180° are called as the supplementary angles.

                                             

< ABD + < DBC = 180°. so these are the supplementary angles.

Since now we have the basic knowledge about an angle and its types, so now we will proceed to the measurement of angle. The size of the angle depends on the opening between the two sides. The angle is generally measured in Degrees and is represented by the symbol °. We take a circle and then divide it into 360 equal part. Each part is known as an angle. So now we can say that the circle is 360° angle. To make our work easier we take the vertex at the center of the circle, keep one side horizontal and rotate the other side like the radius of the circle and measure the angle.

The angle is usually measured with the help of a protractor, each mark on the protractor is 1 degree ° and each thick mark corresponds to 10°.

                                                   

In this protractor we can see that the <ABC = 60°. so with the help of protractor we can easily measure any of the angle till 180°.

Let's take few examples like

                                                           

                          = approximately 180°.

                                                         

                              = approx. 55°

                                                      

                          = approx. 12° .

Now we are going to study some topics that are used to perform operations on the angles.

Congruent angles are those angles which have same angle measurement,it similar to the line which are congruent when they have same length.

Bisectors, these are the lines that divide the angles in two equal parts.

Exterior and the interior angles, let's understand this with the help of a triangle.

The angle which lies inside the triangle is known as the interior angle and the angle formed by extending one of the lines of a triangle is known as the exterior angle.

                                            

Now we take a look at the theorems,

The first one says that the sum of all the interior angles of the triangle is always equal to 180°, it doesn't matter what type of triangle it is.

                                                               

In this the sum of the interiors angle is 45° + 90° + 45° = 180° .

The second theorem states that the measure of the exterior angle of the triangle is always equal to the sum of the interior two opposite angles.

                                                                  

In this figure <RAB= <ABC+ < BCA. Where < RAB is the exterior angle and the angles ABC and BCA are interior opposite angles.

The third theorem states that if the transversal intersects two lines in such a way that the corresponding angles become congruent, then the lines are parallel and vice versa.

                                                          

So this is all we need to perform the basic operations on angles, now we will proceed to the second topic that is angles related to fractions.

We take a whole pie as 360°, now if the pie is to be distributed among two students equally then we divide the pie in two parts by cutting it in the middle. This now reduces our angle to 180°. So if we take 1 for 360° then ½ would be for 180°. similarly if the pie is now need to be divided into 4 equal parts then each part would be having 90° angle. So this implies for ¼ the angle is 90. similarly the values for different math fractions would be

1= 360° (obtuse angle)

½ = 180° (straight angle)

1/3 = 120° (obtuse angle)

¼ = 90° (right angle)

1/5 = 75° (acute angle)

1/6 = 60° (acute angle)

1/8 = 45° (acute angle)

1/9 = 40° (acute angle)

1/10 = 36° (acute angle)

and so on.

So if different fractions are given then we can easily find the angle for them and solve our problems.

This is all the knowledge you would need to solve the general problems that comes for the angles.

Now we come to our second topic of the day which is Parallel and perpendicular lines.

If we have two line segments in a plane then there are two possibilities that either the lines would intersect each other or would not.

If the line segments are intersecting then there is a possibility of perpendicular lines, but if the segments do not intersect then there is a possibility that the two lines could be parallel.

Parallel lines, the lines which do not intersect each other at any point in the plane if extend to infinite length then these lines are called as the parallel lines. The symbol for parallel lines is .

the above diagram shows the parallel lines.

Definition of Perpendicular Lines : Perpendicular lines are those line which intersects in a plane at 90° are known as the perpendicular lines. It is generally represented by a symbol .

The above diagrams given show us what the two perpendicular lines are.

Slope is the measure of the angle from the horizontal at which the line is situated in our plane.

Since the two parallel lines have same angle, so their slope is also equal or we can say that the lines with equal slope are parallel.

The slope of perpendicular lines is bit difficult to understand, if the slope of one line is assumed x then the slope of the line perpendicular to it would be the negative reciprocal that is -1/x. Slope of a line is generally represented by m.

Now how to write the equation of parallel and perpendicular lines.

Let the equation of one line be 2x – 3y = 9 and it passes through a point (4 , -1).

Now we have to find the equation of the line parallel and perpendicular to it.

Let's first find the equation of the line parallel to it. If the lines are parallel then the slope remains the same. Therefore the constants with x and y remains the same.

So we assume the equation as 2x – 3y = a, where a is any constant. Now since the line passes through (4 , -1) so we put this values in our assumed equation and find the value for a.

2 x 4 – 3 x -1 = a

11 = a

so our equation for the line parallel to 2 x – 3 y = 9 is 2 x – 3 y = 11.

Now for the line perpendicular to it. As we discussed above for the perpendicular lines the slope becomes negative reciprocal, so the coefficients of x and y will interchange with the sign before y will reverse.

So the general equation for the line perpendicular to it will be 3 x + 2 y = b.

The value of b could be easily found by putting ( 4 , -1) in it.

3 x 4 + 2 x -1 = b

10 = b.

so the equation of the line perpendicular to 2 x – 3 y = 11 is 3 x + 2 y = 10.

There are other methods also to find the equation of the parallel and the perpendicular lines,

The first one is point and slope form.

y – y₁ = m(x – x₁)

where m is the slope and ( y₁ , x₁) is the given point.

The second is the two point form.

where ( x₁,y₁) and (x₂ , y₂ ) are the two give points.

By these two methods we can easily find the equation of the line parallel and perpendicular to one line.

So it seems like now you have all the knowledge you need to solve the problems on the line segments and the angles.

In upcoming posts we will discuss about Symmetry of Figures in Grade IV and Steps in problem solving. Visit our website for information on 2010 CBSE board papers of political science