Parallel and Intersecting Lines — Question 6

Draw a line RS through N, which is parallel to line LM, and line TU through O, which is parallel to line PQ. ∠LMN = ∠MNS [Alternate Angles] Therefore, w° = 56° Given, ∠MNO = 96° w° + x° = 96° x° = 96° – 40° = 56° Now, RS is parallel to TU, and NO is a transversal. So, ∠SNO = ∠NOT [Alternate Angles] Therefore, y° = x° = 56° Now TU is parallel to PQ, and OP is a transversal. So, ∠TOP = ∠OPQ [Alternate Angles] z° = 52° [Given ∠OPQ = 52°] Thus, a° = y° + z° = 56° + 52° = 108° Parallel Illusions NCERT In-Text Questions (Page 125) There do not seem to be any parallel lines here. Or, are there? What causes these illusions? Solution: (a) At first glance, this image may appear to be a confusing mix of lines going in all directions, giving the impression that nothing is straight or parallel. However, if we take a closer look, we can see that the vertical lines are perfectly straight, evenly spaced, and are parallel. In contrast, the other lines in the image fan out like spokes on a wheel. These lines are not parallel; they are slanted and converge at a central point. Due to their orientation and the way they intersect with the vertical lines, our brains can become misled. This phenomenon is known as an optical illusion. It occurs because the slanted lines create a sensation that everything is angled or distorted. The focal point in the centre draws our attention and makes it difficult to concentrate on the vertical lines. (b) This pattern appears to be filled with tilted or zigzagging lines, and the black shapes create a confusing background. However, if we look closely at the white spaces in between, we can see that the horizontal white lines are parallel. So why do they not seem that way? The bold, slanted black shapes visually interrupt the lines, causing our eyes to perceive them as slanting or shifting. This phenomenon is known as an optical illusion—our brain interprets the shapes around the lines, leading us to see something that isn’t there. (c) When you first look at this picture, it appears that nothing is parallel. The lines seem bent, the shape appears to curve inward, and everything feels like it’s being pulled toward the centre. However, the two horizontal lines at the top and bottom of the image are parallel! This is a classic optical illusion. It occurs because of the many diagonal lines radiating from a central point, resembling the spokes of a wheel. These radiating lines distort our perception, leading our brains to interpret the space as curved due to the way the lines fan out from the centre, creating a sense of depth. As a result, the ends of the horizontal lines seem to bend, even though they are perfectly straight. This visual trick deceives our eyes into believing that the horizontal lines are curving inward, but if we measure them or place a ruler along them, we can see that they are straight and parallel.