 Hello and welcome to the session. In this session, first we will discuss parallel lines and a transversal. We know that a line which intersects two or more lines at distinct points is called a transversal. Now here we have taken two lines M and N. This line L intersects these two lines at two distinct points where these two points be P and Q. Then we say that this line L is a transversal. Now angles 1, 2, 7 and 8, these are the exterior angles. Then 3, 4, 5 and 6 are the interior angles. Now let's find out the corresponding angles in this figure. Angle 1 and angle 5. Then angle 2 and angle 6. Angle 4 and angle 8. Also angle 3 and angle 7. These are the corresponding angles. Then alternate interior angles are given by angle 4 and angle 6, angle 3 and angle 5. Next we have alternate exterior angles. These are given by angle 1 and angle 7, angle 2 and angle 8. Then interior angles on same side of the transversal are given by angle 4 and angle 5. Then angle 3 and angle 6. These are the interior angles on the same side of a transversal. We can also call them consecutive interior angles. We can also call them co-interior angles. We have some results related to the parallel lines and transversal. Like if a transversal intersects two lines, then we have each pair of corresponding angles is equal. Like suppose we have that line M is parallel to the line N and this L is the transversal. Then each pair of corresponding angles would be equal. That is we have angle 1 is equal to angle 5, angle 2 is equal to angle 6, angle 4 is equal to angle 8 and angle 3 is equal to angle 7. And this is also called corresponding angles axiom. Also we have that if a transversal intersects two parallel lines, then each pair of alternate interior angles is equal. That is for the given parallel lines M and N we have angle 4 is equal to angle 6 and angle 3 is equal to angle 5. That is pair of alternate interior angles are equal. Also if a transversal intersects two parallel lines, then we say that each pair of interior angles, the same side of the transversal is supplementary. That is in this case angle 4 plus angle 5 is equal to 180 degrees. That is angle 4 and angle 5 are supplementary and angle 3 plus angle 6 is also equal to 180 degrees. Now if a transversal intersects such that either any one pair corresponding angles is equal one pair of alternate interior angles is equal or any one pair of interior angles on the same side of the transversal is supplementary, then we say the lines are parallel. We have two lines M and N intersected by a transversal L. Now if we have that any one pair of corresponding angles is equal. That is angle 1 equal to angle 5 or angle 2 equal to angle 6 or angle 4 equal to angle 8 or angle 3 equal to angle 7. Now if we have any of these four possibilities then we can say that the line M is parallel to the line N. Also if we get that any one pair of alternate interior angles is equal that is angle 4 equal to angle 6 or angle 3 equal to angle 5 then also we can say that line M is parallel to the line N. And if we get any one pair of interior angles on the same side of the transversal is supplementary that is angle 4 plus angle 5 equal to 180 degrees or angle 3 plus angle 6 is equal to 180 degrees. That is if we get any of these two possibilities then also we can say that line M is parallel to the line N. We have an important theorem related to the lines parallel to the same line which says that lines which are parallel to the same line are parallel to each other. Suppose that this line M is given to be parallel to the line L and also it is given that line N is parallel to the line L. Then from this theorem we can conclude that line M is parallel to line N. In this figure it is given that line AB is parallel to the line CD and the transversality cuts AB at point E and transversality cuts CD at point F. If suppose that we are given angle 1 is equal to 70 degrees let's find the measure of each of the remaining marked angles. Now angle 1 and angle 3 form a pair of vertically opposite angles so angle 1 would be equal to angle 3 equal to 70 degrees. Then angle 1 and angle 5 are the corresponding angles so angle 5 would be equal to angle 1 that is 70 degrees again. Now angle 5 and angle 7 are vertically opposite angles so angle 7 is equal to angle 5 that is 70 degrees. Angle 1 and angle 2 form a linear pair so angle 1 plus angle 2 is equal to 180 degrees and from here we have angle 2 is equal to 180 degrees minus angle 1 that is 70 degrees and that is 110 degrees. Now angle 4 and angle 2 form vertically opposite angles so angle 4 is equal to angle 2 that is 110 degrees. Angle 2 and angle 6 are a pair of corresponding angles so angle 2 is equal to angle 6 so we get angle 6 is equal to angle 2 that is 110 degrees. Angle 6 and angle 8 are vertically opposite angles, so angle 6 is equal to angle 8, that is angle 8 is equal to 110 degrees again, so when we are given angle 1 we have found out angle 2, angle 3, angle 4, angle 5, angle 6, angle 7 and angle 8. Now we discuss angles and property of a triangle, now the theorem says that the sum of the angles of a triangle is 180 degrees, this is the angles and property of a triangle. We have another theorem according to which we have, if a side of a triangle is produced then the exterior angle so formed is equal to the sum of the two interior opposite angles. Now suppose that this ABC is a triangle, now we produce this side BC like this, this angle ACD is the exterior angle formed after producing the side BC of the triangle, according to this theorem we have that this exterior angle that is angle ACD is equal to the sum of the two interior opposite angles, that is it is equal to angle CAB plus angle ABC and it is obvious from this result that an exterior angle of a triangle is greater than either of its interior opposite angles, consider this figure, this is a triangle ABC we are given BD is the perpendicular to AC, angle EAC is 40 degrees, angle DBC is 50 degrees, we need to find X and Y. Now let's consider triangle BDC, in this we have angle BCD plus angle CDB plus angle DBC is equal to 180 degrees by the angles and property of a triangle, now BCD is X plus angle CDB is 90 degrees since we have BD is perpendicular to AC plus DVC is 50 degrees, this is equal to 180 degrees, so from here we get X is equal to 40 degrees, for the triangle AEC angle Y is the exterior angle, so this Y is equal to the sum of the interior opposite angles that is equal to X plus 40 degrees, now we have found out that X is equal to 40 degrees, so Y is equal to 40 degrees plus 40 degrees, thus we get Y equal to 80 degrees, this completes the session, hope you have understood the concept of parallel lines at a transversal and the angles and property of a triangle.