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Chapter Analysis
Intermediate15 pages • EnglishQuick Summary
This chapter on quadrilaterals in Class 9 Mathematics explores the properties and types of quadrilaterals, with a focus on parallelograms. It delves into the characteristics like congruency of triangles formed by diagonals, equal opposite sides and angles, and bisecting diagonals. The chapter provides theorems and examples to demonstrate various properties of quadrilaterals and their relevance in understanding shapes such as rectangles, rhombuses, and squares.
Key Topics
- •Properties of parallelograms
- •Theorems related to diagonals
- •Mid-point theorem in triangles
- •Types of quadrilaterals
- •Congruency in triangles
- •Properties of rhombuses and squares
Learning Objectives
- ✓Understand andapply the properties of a parallelogram
- ✓Prove theorems related to quadrilateral diagonals
- ✓Demonstrate mid-point theorems in triangles
- ✓Identify different types of quadrilaterals
- ✓Apply congruency concepts in triangle proof
Questions in Chapter
If the diagonals of a parallelogram are equal, then show that it is a rectangle.
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Show that the diagonals of a square are equal and bisect each other at right angles.
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Diagonal AC of a parallelogram ABCD bisects ∠A. Show that it bisects ∠C also and ABCD is a rhombus.
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ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that ABCD is a square.
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In a parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ. Show that APCQ is a parallelogram.
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ABCD is a trapezium in which AB || CD and AD = BC. Show that diagonal AC = diagonal BD.
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Additional Practice Questions
Prove that the opposite angles of a parallelogram are equal.
mediumAnswer: In a parallelogram ABCD, opposite angles are equal due to the property of parallel lines that alternate interior angles are equal. This can be proven using the properties of transversals and by considering alternate interior angles formed between the sides and a transversal.
Demonstrate that in any quadrilateral, if one pair of opposite sides is both parallel and equal, then it is a parallelogram.
hardAnswer: Given a quadrilateral ABCD, if AB || CD and AB = CD, then triangle properties and parallel line theorems can be used to show that the opposite angles and sides conform to a parallelogram's properties, such as equal opposite sides and angles, thus proving ABCD is a parallelogram.
Show that the sum of the interior angles in a quadrilateral is 360°.
easyAnswer: A quadrilateral can be divided into two triangles. Since the sum of interior angles in a triangle is 180°, the total for two triangles (a quadrilateral) is 360°.
Prove that the diagonals of a rectangle are equal.
mediumAnswer: In a rectangle, the diagonals bisect each other and overlap to form congruent triangles. Using the properties of congruent triangles and equal opposite sides, we can demonstrate that the diagonals are equal.
Establish that the area of a parallelogram is the base multiplied by the height.
mediumAnswer: Using the theorem that the area of a rectangle is the product of its length and width, if you draw a parallelogram inside a rectangle so that it has the same base and height, the area is found using the rectangle’s area minus another similar parallelogram.