 Hello and welcome to the session. In this session we will distinguish between different types of curves and we will solve problems involving angle measurement. First of all let us discuss different types of curves. First of all let us discuss simple curve. Now a simple curve does not intersect itself except if the starting point and the finishing point are same. Now let us draw a curve in a pattern of a wave. Now here we see that this curve starts from one point and moves continuously and this curve does not intersect anywhere. So it is a simple curve. Now let us draw another curve. Now this curve intersects at one point with itself. So it is not a simple curve. Now let us discuss what is a closed curve. Now a closed curve is that curve whose starting point and finishing point coincide. It can be drawn by starting and stopping at the same point. Now a circle is an example of a closed curve. Here we start from one point and draw the circle and we stop at the same point. So it is a closed curve. Now let us discuss what are polygons. Now polygons are simple closed curves with sides that are only line segments. Now see here we have drawn two polygons one having three sides and other having six sides and see here all the sides are line segments. Also these polygons are closed and simple because they have same starting and finishing point and the curves do not intersect itself at any point. Now let us discuss convex curves. Now convex curves are simple and closed curves such that the segment joining any two points in the integer of the curve is completely contained inside the curve. Now see these closed and simple curves out of which one is a polygon. Now let us join any two points on these curves on this circle. Let us join these two points and here on this polygon let us join these two points and here we see that line segment joining these two points lies entirely within the curve. So both of these figures are convex figures. Now let us discuss concave curve. Now concave curves are simple and closed curves but are not convex. The line segment joining any two points in the integer of the curve does not lie completely inside the curve. Now let us see these two figures. Let us join any two points on these two curves. Here let us join these two points on this curve and on the second curve. Let us join these two points. Here we see that the line segment joining these two points does not lie completely within the curve. Part of the line segment lies outside the curve. So both these figures are concave figures. Now let us solve problems involving angle measurement. Now let us recall few definitions used in angles. First is supplementary angles. Now any two angles whose sum of measure is 180 degrees are called supplementary angles. In this figure you can see angle A and angle B are supplementary angles because angle A plus angle B is equal to 180 degrees. Now let us discuss complementary angles. Now any two angles whose sum of measure is 90 degrees are called complementary angles. Now in this diagram we can see that angle A and angle B are complementary angles because angle A plus angle B is equal to 90 degrees. Now let us discuss an example. Here we have to find the complement of angle 53 degrees. Now let us start with the solution. Let angle A is equal to 53 degrees and we have to find angle B such that angle A plus angle B is equal to 90 degrees. That is angle A and angle B are complementary angles. So this applies 53 degrees plus angle B is equal to 90 degrees which implies angle B is equal to 90 degrees minus 53 degrees and this gives angle B is equal to 37 degrees. So complement of angle measuring 53 degrees is equal to 37 degrees. Now let us discuss angles of a regular polygon. Now a regular polygon is a polygon whose length of all sides is same. And measure of all angles is also same and these are the interior angles of the polygon. Now sum of interior angles of a regular polygon with n sides is given by n minus 2 whole into 180 degrees. And measure of each interior angle is given by 1 upon n into n minus 2 whole into 180 degrees. For example suppose we have a regular polygon with n sides. So its sum of n by interior angles is equal to n minus 2 whole into 180 degrees. Now here n is equal to 8. So sum of 5 interior angles will be equal to 8 minus 2 whole into 180 degrees which is equal to 6 into 180 degrees. That is equal to 1080 degrees. Now measure of each interior angle is equal to 1 upon n into n minus 2 whole into 180 degrees. So this will be equal to 1 upon 8. Now we have already found the value of n minus 2 whole into 180 degrees. And that is equal to 1080 degrees. So putting this value here and solving this is equal to 135 degrees. So measure of each interior angle of a regular polygon with n sides is equal to 135 degrees. So in this session we have discussed how to distinguish between different types of curves and how to solve problems involving angle measurement. And this completes our session that we all have enjoyed the session.