Basic Geometrical Ideas Ex 4.2 Class 6 NCERT Solutions

Understanding Curves: Open vs. Closed

In geometry, a 'curve' is any drawing you can make without lifting your pencil from the paper. It doesn't have to be curvy; even a straight line is a type of curve! The most important way to classify them is by checking if they are 'open' or 'closed'.

Open Curve: Think of an open gate. You can walk in and out. An open curve is similar. Its starting point and ending point are different. Examples include the letter 'U', the number '2', or a simple wavy line.

Closed Curve: Think of a fenced-in park. To get out, you have to jump over the fence! A closed curve has no endpoints; it starts and ends at the same point, enclosing an area. A circle, a square, or any shape that forms a complete loop is a closed curve. A special type of closed curve is a 'simple closed curve', which does not cross itself. A figure '8' is a closed curve, but it is not simple because it crosses itself in the middle.

Meet the Polygons and Angles

Polygon

A polygon is a simple closed figure made up entirely of straight line segments. Triangles, squares, and pentagons are all examples of polygons.

Sides, Vertices, and Diagonals

The line segments that form a polygon are called its 'sides'. The point where two sides meet is a 'vertex'. A 'diagonal' is a line segment that connects two non-adjacent vertices.

Angle

An angle is formed when two rays share a common endpoint (vertex). It has an 'interior' (the space inside the rays) and an 'exterior' (the space outside the rays).

Triangle

A polygon with three sides, three vertices, and three angles.

Quadrilateral

A polygon with four sides, four vertices, and four angles.

Let's Investigate a Quadrilateral

1. Step 1: Draw and Name — First, let's draw a four-sided polygon. We label the corners, or vertices, with capital letters like P, Q, R, and S.
2. Step 2: Identify the Sides — The four line segments that make the shape are its sides. For our quadrilateral PQRS, the sides are PQ, QR, RS, and SP.
3. Step 3: Identify the Angles — An angle is formed at each vertex. The four angles are ∠P, ∠Q, ∠R, and ∠S. We can also name them using three letters, like ∠SPQ for angle P.
4. Step 4: Find Opposite and Adjacent Sides — Sides across from each other are 'opposite sides' (e.g., PQ and SR). Sides that share a vertex are 'adjacent sides' (e.g., PQ and QR).
5. Step 5: Find Opposite and Adjacent Angles — Angles across from each other are 'opposite angles' (e.g., ∠P and ∠R). Angles next to each other are 'adjacent angles' (e.g., ∠P and ∠Q).

Important Tip for Your Exam

A very common question asks if a circle is a polygon. Remember this forever: A circle is a closed curve, but it is NOT a polygon. Why? Because a polygon must be made of straight line segments only. A circle has a continuous curve, not straight sides. Don't lose marks on this simple concept!

Practice Questions with Solutions

• Q: Classify the following figures as open or closed: (i) A circle (ii) A line segment (iii) A triangle (iv) A curved line that does not meet at its ends. A: Step 1: A figure is closed if its starting and ending points are the same, forming an enclosed region. Otherwise, it's open. Step 2: (i) A circle is a continuous line that meets at its ends, enclosing a region. (ii) A line segment has distinct start and end points. (iii) A triangle is formed by three line segments that meet at their ends, enclosing a region. (iv) A curved line that does not meet has distinct start and end points. Final answer: (i) Closed, (ii) Open, (iii) Closed, (iv) Open.
• Q: Which of the following are polygons? Give a reason for your answer. (i) A figure made of two line segments and a curve. (ii) A square. (iii) A circle. A: Step 1: Recall the definition of a polygon: A polygon is a simple closed figure made up entirely of line segments. Step 2: (i) This figure contains a curve, so it is not made entirely of line segments. (ii) A square is a simple closed figure made up of four line segments. (iii) A circle is a closed figure but is made of a curve, not line segments. Final answer: Only (ii) A square is a polygon because it is a simple closed figure made entirely of line segments.
• Q: Draw a quadrilateral ABCD. Name its sides, vertices, and diagonals. A: Step 1: Draw a four-sided closed figure and label its corners as A, B, C, and D in order. Step 2: Sides are the line segments forming the figure. Vertices are the points where the sides meet. Diagonals are line segments connecting non-adjacent vertices. Step 3: From the drawing, identify the parts. Final answer: Sides: AB, BC, CD, DA. Vertices: A, B, C, D. Diagonals: AC, BD.
• Q: Consider a triangle PQR. Mark a point X in its interior, a point Y on its boundary, and a point Z in its exterior. A: Step 1: Draw a triangle and label its vertices as P, Q, R. Step 2: A point in the interior lies inside the figure. A point on the boundary lies exactly on the lines forming the figure. A point in the exterior lies outside the figure. Step 3: Place point X anywhere inside the triangle PQR. Place point Y on any of the line segments PQ, QR, or RP. Place point Z anywhere outside the triangle PQR. Final answer: Point X is marked inside the triangle, Point Y is marked on one of its sides, and Point Z is marked outside the triangle.

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