NCERT Solutions Class 6 Maths Chapter 5 Exercise 5.7: Understanding Quadrilaterals
Welcome back to your YoLearn AI Math sketchbook! Today, we are exploring Chapter 5, Exercise 5.7 of CBSE Class 6 Mathematics. In this exercise, we will unlock the secrets of four-sided closed figures called quadrilaterals. Think of them as a big family of shapes with special properties. You will learn to recognize rectangles, squares, parallelograms, rhombuses, and trapeziums. More importantly, we will understand how they are related to each other—like why a square is actually a special type of rectangle! Mastering this topic will build a solid foundation for your geometry studies in higher classes. Let's dive in with easy-to-follow explanations, visual comparisons, and step-by-step solved questions so you can score perfect marks in your school exams.
Understanding the Family of Quadrilaterals
A quadrilateral is any simple closed figure made of four line segments. Depending on the properties of their sides, angles, and diagonals, they are classified into different types. Imagine them as a family tree! At the top is the general Quadrilateral. Below it, we have the Parallelogram (which has opposite sides parallel and equal) and the Trapezium (which has only one pair of parallel sides). Within the parallelogram family, we find special members like the Rectangle (four right angles), the Rhombus (four equal sides), and the Square, which is the ultimate VIP shape because it combines all properties: it has four equal sides, four right angles, and opposite sides parallel! Understanding these connections is the secret to solving Exercise 5.7 easily.
Comparing Quadrilaterals & Their Special Rules
| Aspect | Details |
|---|---|
| Parallelogram | Think of it as a tilted rectangle. Opposite sides never meet! |
| Rectangle | A perfectly upright box where all corners form an L-shape. |
| Rhombus | Think of a diamond shape or a tilted square. |
| Square | The VIP shape! It is a rectangle, a rhombus, and a parallelogram all at once. |
| Trapezium | Looks like a table top. One pair of opposite sides will eventually cross if extended. |
Crucial Relationships and Common Confusion
- A square is a special rectangle: A rectangle has opposite sides equal and all angles equal to 90 degrees. A square fits this definition perfectly because its opposite sides are equal (since all four sides are equal anyway) and its angles are 90 degrees.
- A square is also a special rhombus: A rhombus has all four sides equal. A square also has all four sides equal. The only difference is that a square has all its angles equal to 90 degrees.
- A parallelogram is NOT always a rectangle: A rectangle must have 90-degree angles. While all rectangles have opposite sides parallel (making them parallelograms), normal parallelograms do not need to have 90-degree angles.
- A trapezium is NOT a parallelogram: A trapezium only has one pair of parallel sides, whereas a parallelogram must have two pairs of parallel sides.
Practice Questions with Solutions
- Q: State whether the following statement is True or False: 'Every square is a rhombus.' A: Step 1: Recall the definition of a rhombus. A rhombus is a quadrilateral with all four sides of equal length. Step 2: Compare this with the properties of a square. A square is a quadrilateral that has all four sides of equal length and all four angles equal to 90 degrees. Step 3: Since all four sides of a square are always equal, a square satisfies all conditions of being a rhombus. Final answer: True. Every square is a rhombus.
- Q: Give reason for: 'A rectangle can be thought of as a special parallelogram.' A: Step 1: Recall the definition of a parallelogram. A parallelogram is a quadrilateral whose opposite sides are parallel and equal in length. Step 2: Recall the definition of a rectangle. A rectangle is a quadrilateral whose opposite sides are parallel, opposite sides are equal, and all four angles are 90 degrees. Step 3: Since a rectangle has opposite sides parallel and equal, it already satisfies all properties of a parallelogram. It just has an extra property of having all 90-degree angles. Final answer: Hence, a rectangle is a special parallelogram with each angle equal to 90 degrees.
- Q: State whether the following statement is True or False: 'The opposite sides of a trapezium are parallel.' A: Step 1: Recall the definition of a trapezium. A trapezium is a quadrilateral in which only one pair of opposite sides is parallel. Step 2: Look at the other pair of opposite sides in a trapezium. They are non-parallel and will intersect if extended. Step 3: Therefore, it is incorrect to say all opposite sides of a trapezium are parallel. Final answer: False. Only one pair of opposite sides of a trapezium is parallel.
- Q: Why is a square also classified as a quadrilateral? A: Step 1: Recall the definition of a quadrilateral. Any closed polygon with exactly four sides is called a quadrilateral. Step 2: Count the number of sides of a square. A square is formed by joining four line segments, so it has exactly four sides. Step 3: Since any four-sided closed figure is a quadrilateral, a square fits this category. Final answer: A square is classified as a quadrilateral because it is a closed shape made up of exactly four line segments.
Frequently Asked Questions
What is the main difference between a rectangle and a square?
A square has all four sides equal in length, whereas a rectangle only requires opposite sides to be equal in length. Both shapes have four 90-degree angles.
Is a trapezium a parallelogram?
No, a trapezium is not a parallelogram. A trapezium has only one pair of parallel sides, while a parallelogram must have two pairs of parallel sides.
Why is a square considered a regular quadrilateral?
A regular polygon must have all sides equal and all angles equal. A square has four equal sides and four equal angles of 90 degrees, satisfying both conditions.