Parallel Lines PDF

# Parallel Lines ## Let us recall We have heard about parallel lines in class 6. Lines that don't meet, keeping the same distance between them. We have also drawn them with a scale and a set square. Two lines drawn at the same slant to a given line are parallel. We've seen this also. * Two lines...

# Parallel Lines ## Let us recall We have heard about parallel lines in class 6. Lines that don't meet, keeping the same distance between them. We have also drawn them with a scale and a set square. Two lines drawn at the same slant to a given line are parallel. We've seen this also. * Two lines with angles 60°. * Two lines with angles 100°. * A road with parallel lines. We know that a parallelogram is a quadrilateral in which the two pairs of opposite sides are parallel. Can you draw this parallelogram with measures as given? * A parallelogram with a side of 3 cm and a 120 degree angle. ## Lines and angles When a line crosses another line, how many angles are formed between them? If we know one of these, can we calculate the others? * Diagram of two lines with a 50 degree angle This too was seen in class 6. * A line with angles 50° and 130°. What can we say about the relation between four such angles in general? Nothing much to say if the crossing lines are perpendicular. All angles are 90°. * A horizontal line perpendicular to a vertical line. What if one line is a bit tilted? * A horizontal line tilted to a diagonal line. Two small angles and two large angles. Mark the other angles also like this. What is the relation between the angle measures? Try changing the position of D. Don't you see a change in the angles? Does the relation change? * A diagonal line with two lines with a 45 degree angle. What is the relation between them? * The two small angles are of the same measure. * The two large angles are of the same measure. * The sum of a small angle and a large angle is 180°. ## Look at the earlier figure once again: * A blue horizontal line and a diagonal line. * The blue line with two 50 degree angles and two 130 degree angles. What if we draw another line above, parallel to the blue line? * Two parallel blue horizontal lines with 50 degree and 130 degree angles. * Two parallel blue horizontal lines with a diagonal line crossing through them. What is the relation between the eight angles you have marked now? Try changing the position of D. Does the relation change when the angle measures change? Try changing the position of E. What happens when E takes the place of C? ## Now the blue line above makes four angles with the green line. What can you say about them? * Two parallel blue horizontal lines with a diagonal green line crossing through them. Let's look only at the small angle below and the marked angle above. The blue lines are parallel. So these two must be of the same measure. * Two blue parallel lines with a diagonal line making a 50 degree angle. What about the other angles above? * Two blue parallel lines with a diagonal line making a 130 degree angle. Suppose we start with an angle other than 50°. The measures of the other angles will change. But the relation between the angles will be the same. That is, A line intersects two parallel lines at angles of the same measure. * Two parallel lines with a diagonal line. ## A change in angle Suppose there is slight change in the slant of two lines with another line. The two lines won't be parallel. For example, look at this figure: * Two almost parallel lines with different angle measures. The blue lines in the figure appear to be parallel. As there is a difference of 1° in the slants, they will meet when extended sufficiently. We can calculate how much to be extended. You've to extend them by more than a metre for them to meet! ## Matching angles We have seen the relations between the four angles made by two intersecting lines. What can we say about the relation between the eight angles formed when a line cuts two parallel lines? Let's take another look at this figure which we saw earlier: * Two parallel lines with a diagonal line. * Two parallel lines with a diagonal line and two 50 and 130 degree angles. We know the relation between the four angles below. Same is the relation between the four angles above. What if we take an angle from below and an angle from above? If both are small angles, each is 50°. If both are large, each is 130°. If one is small and the other is large; the small one is 50°, the large one 130°; and the sum is 180°. The relations remain the same even if the angles change, right? So we can say this in general: * Of the angles made when two parallel lines are cut by a slanting line, * the small angles are of the same measure. * the large angles are of the same measure. * a small angle and a large angle add up to 180°. If the intersecting line is perpendicular to one of the parallel lines, it would be perpendicular to the other line too, and all angles would be right angles. ## Now, look at this figure: * Two parallel lines with a diagonal line. * A 45 degree angle and a 45 degree angle labeled. The top and bottom lines are parallel. What is the measure of angle above? To see all the angles clearly, let's suppose the lines are extended. * Two parallel lines with a diagonal line and two 45 degree angles labelled. Look at the two small angles formed when the slanting line cuts the top and bottom parallel lines. These are the angles in the first figure. So they are of the same measure. ## Position and angle We may group the angles formed when two parallel lines are cut by another line based on their position. The figures below show angle pairs in the same position. * Two horizontal parallel lines with a diagonal line. * Different areas of the line labeled with top right, bottom right, top left, bottom left. Angles in each such pair are called corresponding angles. The angles in each pair measure the same. ## Opposite positions The figures below show the angles in opposite positions when two parallel lines are cut by another line. That is, the angle above is also 45°. * Two parallel lines with a diagonal line. * Different areas of the line labeled as above. What if the figure looks like this? * A diagonal line with an angle of 40 degrees. This is the first figure with the angle slightly changed and turned a bit, isn't it? What is the measure of the other angle? Now look at this figure: * Two vertical parallel lines with a diagonal line. * Some angles labeled as 30 degrees. The vertical lines are parallel. But there is no line cutting them. How about drawing another vertical line? * Two vertical parallel lines with two diagonal lines. * Some angles labeled as 30 degrees. Now the, middle angle is in two parts. We can find the left part. * Two vertical parallel lines with two diagonal lines. * Two angles labeled as 30 degrees. What about the right part? Let's try different measures for this angle: * Two vertical parallel lines with two diagonal lines. * Two angles labeled as 30 degrees. * An angle labeled as 40. What should be this angle to get a nice figure? * Two vertical parallel lines with two diagonal lines. * Two angles labeled as 30 degrees and one label as 60 degrees. ## In and Out The figures below show the interior and exterior angle pairs when two parallel lines are cut by another line. * Two horizontal parallel lines with a diagonal line. * The different areas of the lines are labeled: inner left, inner right, outer left, outer right. The first two pairs are called co-interior angles and the last two pairs co-exterior angles. The sum of the angles in each such pair is 180°. ## Let's draw a parallelogram. Draw two lines AB and AC. Through B draw a line parallel to AC and through C draw a line parallel to AB. The point of intersection of these lines is D. Draw parallelogram ABDC using Polygon tool. We can see all angles if we click in the parallelogram using Angle tool. * A parallelogram with some of the angles labeled. What is the relation between these angles? Try changing the position of C. Do the angles change? And the relation between them? The 55° angle and the angle above it form a pair of small angle and large angle. So their sum is 180°. This means the top angle = 180° – 55° = 125°. Now look at the angle to the right of the marked angle. To calculate this, look at the angles made by the left and right parallel sides with the bottom line. * A parallelogram with some of the angles labeled as 55 degrees and 125 degrees. The 55° angle and the angle on its right are a small angle and a large angle of these angles. So, this angle also is 125° as calculated earlier. Can't you find the fourth angle, like this? ## Now try these problems: 1. Draw the parallelogram below with the given measures. * A parallelogram with a 60 degree angle, a 3 cm side, and a 5 cm side. Calculate the other three angles. 2. The top and bottom blue lines in the figure are parallel. Find the angle between the green lines. * Two parallel blue lines and two diagonal green lines with a 50 degree, 40 degree angle. 3. In the figure, the pair of lines slanted to the left are parallel; and also the pair of lines slanted to the right. * Two parallel lines with a 40 degree angle. Draw this figure: 4. The left and right sides of the large triangle are parallel to the left and right sides of the small triangle. * Two triangles. Calculate the other two angles of the large triangle and all angles of the small triangle. 5. A triangle is drawn inside a parallelogram. * A triangle inside a parallelogram. Calculate the angles of the triangle. ## Triangle sum Look at this figure: * Two parallel lines with a 60 degree angle. The top and bottom lines are parallel. So, can you calculate the angle at the top? * Two parallel lines with a 140 degree angle. If this angle is drawn less than 140°, then the two lines will meet. Let's decrease it by 60°. * Two parallel lines with a 140 degree angle and an 80 degree angle. Now we have a triangle. What are the angles in this triangle? The left angle is 40°. The top angle is 140° − 60° = 80°. What about the third angle? It is one of the small angles which the new slanted line makes with the parallel lines. * A triangle with 40 degrees and 80 degrees. Its measure is the same as that of the small angle which this line makes with the top line. Isn't the top small angle 60°? So the bottom small angle is also 60°. * A triangle with 40 degrees, 80 degrees, and two 60 degree angle. Thus the 60° which we took away from the top reappears at the bottom as an angle of the triangle. The sum of this angle and the top angle of the triangle 80° + 60° = 140°. Now look at this figure: * A triangle with a 50 degree angle. Can you calculate the sum of the other two angles of this triangle? In the first problem, the right side of the triangle was got by drawing a slanted line instead of the parallel line. Let's think in the reverse. A parallel line instead of the slanted right side. What is the angle which this parallel line makes with the left side? * A triangle with a 50 degree angle and a 130 degree angle. When we drew the triangle, this angle split into two. One part is the top angle of the triangle. What about the other part? * A triangle with a 50 degree angle and a 130 degree angle. That is, one part of the 130° angle is the top angle of the triangle and the other part is the angle on the right in the triangle. So, the sum of these two angles of the triangle is 130°. How do we state in general what we've learnt from this problem? If we subtract the measure of one angle of a triangle from 180°, we get the sum of the other two angles. For example, if one angle of a triangle is 60°, the sum of the other two angles: 180° -60° = 120° What about the sum of the three angles of the triangle? This is true for any triangle. The sum of all angles of a triangle is 180°. ## Now try this problem: One angle of a right triangle is 40°. What is the measure of the angle other than the right angle? Let's think this way. The sum of the angles other than the right angle is 180° - 90° = 90° One of them is 40°. Then the other angle is 90°-40° = 50° We can think in another way as well. The sum of the three angles is 180°. The sum of two of them 90° + 40° = 130° So the third angle 180° - 130° = 50° ## Another problem: One angle of a triangle is 72°. The other two angles are of equal measure. What are their measures? What is the sum of the other two angles? 180°-72° = 108° Since the other two angles are equal, each is half the sum, isn't it? So each is 108°/2 = 54° ## Now try these problems: 1. Draw the triangle with the given measures. * A triangle with a 40 degree angle, a 60 degree angle, and a 5 cm side. 2. The figure shows a triangle drawn in a rectangle. * A triangle inside a rectangle with a 40 degree angle, a 25 degree angle, and a 60 degree angle. Calculate the angles of the triangle. 3. The top and bottom lines in the figure are parallel. * Two parallel lines with a diagonal line and a triangle. Calculate the third angle of the bottom triangle and all angles of the top triangle. 4. The left and right sides of the large triangle are parallel to the left and right sides of the small triangle. * Two triangles. Calculate the other two angles of the large triangle and all angles of the small triangle. 5. A triangle is drawn inside a parallelogram. * A triangle inside a parallelogram. Calculate the angles of the triangle. ## Draw two parallel lines and mark a point on each. Mark a third point in between them. Draw lines joining the points on the lines with the third point. Mark the angles which these lines make with the parallel lines. Also mark the angle between the lines. * Two parallel lines with a diagonal line and different angles labeled. What is the relation between the angles of the triangle? Try changing the positions of the points. ## Draw a triangle and mark a point on one of the sides. Draw a line through this point parallel to another side of the triangle. Mark the point where this line meets the third side. Draw the small triangle with one corner of the triangle and the points on the sides as vertices. Mark the angles of the first triangle and the small triangle. What is the relation between these angles? Try changing the corners of the triangle. * A triangle with a diagonal line going through it creating a smaller triangle, some angles labeled.