NCERT Solutions & Concepts for Basic Geometrical Ideas Ex 4.5 Class 6

What is a Quadrilateral? An Overview

A quadrilateral is a simple closed polygon formed by four line segments. The word "quadrilateral" is made of two parts: 'quad' meaning four, and 'lateral' meaning sides. Every quadrilateral has exactly four vertices (points where sides meet), four sides (the line segments), and four angles formed at the vertices. To understand it fully, imagine drawing four points A, B, C, and D in order, and connecting them. This gives us quadrilateral ABCD.

A very important aspect of quadrilaterals is the relationship between their parts. Elements can be either adjacent (next to each other) or opposite (across from each other). For instance, sides AB and BC are adjacent because they meet at the vertex B. On the other hand, sides AB and CD are opposite because they do not share any vertex. Similarly, angles can be adjacent or opposite. Finally, we have diagonals, which are line segments joining opposite vertices, such as AC and BD. These diagonals always intersect in the interior of a convex quadrilateral.

Step-by-Step: Identifying Parts of a Quadrilateral

1. Step 1: Sketch and Label — Draw any simple four-sided figure on your paper. Label the four vertices in order, moving either clockwise or counter-clockwise (e.g., P, Q, R, S).
2. Step 2: List Vertices and Sides — Write down the 4 corner points as vertices (P, Q, R, S) and the 4 line segments as sides (PQ, QR, RS, SP).
3. Step 3: Find Opposite Pairs — Find pairs that do not touch or share elements. Opposite sides are (PQ, RS) and (QR, SP). Opposite angles are (Angle P, Angle R) and (Angle Q, Angle S).
4. Step 4: Find Adjacent Pairs — Find pairs that are next-door neighbors. Adjacent sides share a vertex (e.g., PQ and QR share vertex Q). Adjacent angles share a common side (e.g., Angle P and Angle Q share side PQ).
5. Step 5: Draw Diagonals — Connect opposite vertices with straight lines inside the shape. These segments, PR and QS, are the diagonals of the quadrilateral.

Key Concepts to Remember

• Naming Rule: Always name a quadrilateral in a continuous cyclic order around its boundary (like ABCD or ADCB). Never skip across (like ACBD).
• Diagonals count: A quadrilateral always has exactly two diagonals.
• Interior Intersections: For any convex quadrilateral, the two diagonals will always intersect each other in the interior region of the shape.
• Vertices: The four corners where the sides meet are called vertices (plural of vertex).

Avoid This Common Exam Mistake!

Students often lose marks in CBSE exams by naming quadrilaterals incorrectly. If you are given vertices K, L, M, and N, write the name of the quadrilateral as KLMN or KNML. If you write KMLN, it is mathematically incorrect because KM and LN are diagonals, not boundary sides. Always draw a quick rough sketch on your exam sheet to verify that your chosen name follows the outer boundary of the shape!

Practice Questions with Solutions

• Q: Draw a rough sketch of a quadrilateral PQRS. Draw its diagonals. Name them. Is the meeting point of the diagonals in the interior or exterior of the quadrilateral? A: Step 1: Draw a four-sided figure and label the vertices consecutively in clockwise order as P, Q, R, and S. Step 2: Identify and draw the diagonals by connecting opposite vertices. Connecting P to R gives diagonal PR, and connecting Q to S gives diagonal QS. Step 3: Mark the intersection point of PR and QS as 'O'. Observe where this point 'O' lies relative to the boundary of the figure. Final answer: The diagonals are PR and QS. Their meeting point (intersection point) lies in the interior of the quadrilateral.
• Q: State two pairs of opposite sides and two pairs of opposite angles for quadrilateral KLMN. A: Step 1: Write down the four sides of KLMN, which are KL, LM, MN, and NK. Step 2: Identify opposite sides (sides that do not share any corner point). The pairs are (KL, MN) and (LM, NK). Step 3: Identify opposite angles (angles that do not share a common arm). The pairs are (Angle K, Angle M) and (Angle L, Angle N). Final answer: Two pairs of opposite sides are (KL, MN) and (LM, NK). Two pairs of opposite angles are (Angle K, Angle M) and (Angle L, Angle N).
• Q: State two pairs of adjacent sides and two pairs of adjacent angles for quadrilateral ABCD. A: Step 1: Identify sides that meet at a common vertex. For example, AB and BC meet at B. BC and CD meet at C. Step 2: Select two pairs of these adjacent sides. Let's write (AB, BC) and (CD, DA). Step 3: Identify angles that share a common side. For example, Angle A and Angle B share side AB, while Angle C and Angle D share side CD. Step 4: Select two pairs of these adjacent angles, which are (Angle A, Angle B) and (Angle C, Angle D). Final answer: Adjacent sides: (AB, BC) and (CD, DA). Adjacent angles: (Angle A, Angle B) and (Angle C, Angle D).
• Q: In a quadrilateral WXYZ, if you draw a line segment connecting W and Y, what is this segment called? Does it lie in the interior of the quadrilateral? A: Step 1: Identify the positions of vertices W and Y. In quadrilateral WXYZ, W and Y are non-adjacent (opposite) vertices. Step 2: A straight line segment joining two non-adjacent vertices of a polygon is defined as a diagonal. Therefore, WY is a diagonal. Step 3: Since WXYZ is a standard convex quadrilateral, the diagonal WY lies entirely inside the boundary. Final answer: The segment WY is called a diagonal, and it lies in the interior of the quadrilateral.

Frequently Asked Questions