NCERT Solutions for Class 6 Maths Chapter 5 Exercise 5.5

Welcome to your step-by-step guide for understanding elementary shapes ex 5 5 class 6 ncert. In this exercise, we dive into the fascinating concept of perpendicular lines. When two lines intersect such that the angle between them is a right angle (90°), they are called perpendicular lines. Understanding this relationship is a vital building block in geometry, helping you identify structures in the real world—from the corners of your textbook to the perpendicular grids of city streets. By mastering this topic, you will learn to spot perpendicularity in daily objects, determine precise angle measures at points of intersection, explore the geometry box's set-squares, and understand the concept of a perpendicular bisector. Let YoLearn AI Tutor guide you through these simple visual steps to make geometry easy, fun, and highly intuitive!

Understanding Perpendicular Lines

In geometry, lines can relate to each other in different ways. They can be parallel, meaning they never meet, or they can intersect at a point. When two intersecting lines form a perfect right angle (90°) at their point of intersection, they are called perpendicular lines. We write this using the special symbol '⊥'. For example, if line AB is perpendicular to line CD, we write AB ⊥ CD.

You see perpendicular lines everywhere in daily life! Look at the corner of your notebook, the adjacent edges of a table, or the letter 'L'. In all these cases, two straight edges meet to form a precise 90° angle. Understanding this concept helps you analyze shapes like rectangles and squares, which are built entirely on perpendicular relationships. Let's explore how we identify, measure, and draw these lines with ease.

Key Geometric Concepts

Perpendicular Lines
Two intersecting lines or line segments that form an angle of exactly 90° at their point of intersection.
Perpendicular Bisector
A perpendicular line or segment that intersects another line segment at its exact midpoint, dividing it into two equal halves.
Set-squares
Triangular drawing instruments found in a geometry box used to construct and verify perpendicular and parallel lines.

How to Identify Perpendicular Lines and Bisectors

  1. Observe the Point of Intersection — Look at the point where the two lines or line segments meet. If they do not meet or run parallel (like railway tracks), they cannot be perpendicular.
  2. Measure or Verify the Angle — Use a protractor or the corner of a square piece of paper (which is a perfect 90° angle) to check the angle. If the angle is exactly 90°, the lines are perpendicular.
  3. Check for Bisecting Properties — To determine if a perpendicular line is also a 'bisector', measure the segments on both sides of the intersection point. If the two segments are equal in length, then the line bisects the segment.

Pro Tip: Avoid the Parallel Track Trap

A common mistake in exams is confusing perpendicular lines with parallel lines. Remember: Perpendicular lines must intersect at a 90° angle. Parallel lines (like railway tracks) never intersect, meaning they have a constant distance between them and can never be perpendicular. Always double-check if the lines meet before declaring them perpendicular!

Practice Questions with Solutions

  • Q: Which of the following are models for perpendicular lines? (a) The adjacent edges of a postcard. (b) The lines of a notebook page. (c) The letter 'V'. (d) The letter 'T'. A: Step 1: Let us analyze each option based on geometric definitions. Step 2: (a) The adjacent edges of a postcard meet at a corner, forming a perfect right angle (90°). So, they are perpendicular. Step 3: (b) The lines of a notebook page run parallel to each other and never meet. So, they are not perpendicular. Step 4: (c) The lines forming the letter 'V' meet at an acute angle, which is much less than 90°. So, they are not perpendicular. Step 5: (d) In the letter 'T', the horizontal line meets the vertical line at a 90° angle. So, they are perpendicular. Final answer: Options (a) and (d) are models for perpendicular lines.
  • Q: Let line segment AB be perpendicular to segment CD. Let them intersect at point O. What is the measure of angle AOC? A: Step 1: Identify the given geometrical relationship. We are told that segment AB is perpendicular to segment CD (AB ⊥ CD). Step 2: Recall the definition of perpendicularity. When two lines are perpendicular, all four angles formed at the intersection point O measure exactly 90°. Step 3: Therefore, the angle AOC must be a right angle. Final answer: The measure of angle AOC is 90°.
  • Q: Look at your geometry box set-squares. What are the angle measures of the two set-squares? Do they share any common angle measure? A: Step 1: Examine the first set-square. It is a scalene right-angled triangle with angles measuring 30°, 60°, and 90°. Step 2: Examine the second set-square. It is an isosceles right-angled triangle with angles measuring 45°, 45°, and 90°. Step 3: Compare the angles of both set-squares: {30°, 60°, 90°} and {45°, 45°, 90°}. Step 4: Identify the common angle in both sets, which is 90°. Final answer: The angles of the set-squares are 30°-60°-90° and 45°-45°-90°. They share a common angle of 90°.
  • Q: On a straight line segment AH, points are marked in order: A, B, C, D, E, F, G, H at equal intervals of 1 cm each. A perpendicular line m passes through point E. Answer the following: (a) Is CE = EG? (b) Does line m bisect segment DF? A: Step 1: First, calculate the lengths using the equal 1 cm intervals. Step 2: For (a), the distance from C to E is E - C = 2 units (D and E). Since intervals are equal, CE = 2 cm. The distance from E to G is G - E = 2 units (F and G), so EG = 2 cm. Since CE = EG = 2 cm, the statement is true. Step 3: For (b), the segment is DF. The point E lies exactly halfway between D and F because DE = 1 cm and EF = 1 cm. Since line m passes perpendicular through E, it cuts DF into two equal parts. Final answer: (a) Yes, CE = EG. (b) Yes, line m bisects segment DF.

Frequently Asked Questions

What does perpendicular mean in simple words?

Perpendicular means that two lines or segments cross each other to make a perfect corner, forming an angle of exactly 90 degrees.

What is the difference between perpendicular lines and a perpendicular bisector?

While perpendicular lines simply meet at a 90-degree angle, a perpendicular bisector does this while also dividing a line segment into two perfectly equal halves.

Where do we see perpendicular lines in our classroom?

You can see them in the corners of the blackboard, the edges of doors and windows, and where the walls meet the floor.

How do we denote perpendicular lines symbolically?

We use the upside-down 'T' symbol (⊥). For example, XY ⊥ PQ means line XY is perpendicular to line PQ.