"तरुण के स्वप्न" पाठ किसके उद्बोधन का अंश है?

सुभाषचंद्र बोस

"उनके स्वप्न के उत्तराधिकारी आज हम हैं" में "हम" किसके लिए है?

देश के तरुण/युवा वर्ग के लिए

नेताजी ने किस प्रकार के राष्ट्र का स्वप्न देखा था?

सर्वांगीण स्वाधीन और संपन्न राष्ट्र

नेताजी के आदर्श समाज में नारी का स्थान कैसा होगा?

पुरुष के समान अधिकार और भागीदारी

"आलसी तथा अकर्मण्य के लिए कोई स्थान नहीं रहेगा" — क्यों कहा गया?

क्योंकि नेताजी श्रम और कर्म की मर्यादा वाले समाज की कल्पना करते थे

What is true of the diagonals of a rectangle?

They are equal in length and bisect each other.

How do the diagonals of a square differ from those of a general rectangle?

They additionally intersect at \(90^\circ\) and bisect the angles of the square.

Is every rectangle a parallelogram?

Yes, because a rectangle has both pairs of opposite sides parallel.

At what angle do the diagonals of a rhombus intersect?

At \(90^\circ\); they are perpendicular bisectors of each other.

Can a quadrilateral have three right angles and a fourth angle that is not \(90^\circ\)?

No, because the four angles must total \(360^\circ\), forcing the fourth to also be \(90^\circ\).

If a quadrilateral has all four angles equal, what is it?

A rectangle, since each angle must equal \(360^\circ \div 4 = 90^\circ\).

What relationship do adjacent angles of a parallelogram have?

They are supplementary, adding to \(180^\circ\).

Is every kite a rhombus?

No; a rhombus has all four sides equal, while a kite only needs two adjacent pairs equal.

Which property defines an isosceles trapezium?

Its non-parallel sides are equal, which makes the base angles equal.

Quadrilaterals

Q: What is the sum of all angles in any quadrilateral?

Always \(360^\circ\), since a diagonal splits it into two triangles of \(180^\circ\) each.

Class 8 Mathematics · Ganita Prakash Part 1 · Chapter 4

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CBSE Class 8 · Mathematics

7 chapters: square roots to rule of three

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Exponent laws, Ch 2
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CBSE Class 8 · Mathematics

Percentages to area: 7 chapters of rules

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Summary

This chapter studies four-sided figures and proves their properties through geometric reasoning rather than measurement alone. Starting from the Carpenter’s Problem of joining two strips to make a rectangle, it deduces that the diagonals of a rectangle are equal and bisect each other, and that a quadrilateral with all four angles \(90^\circ\) must be a rectangle. A square is a special rectangle with all sides equal; its diagonals are equal, bisect each other at \(90^\circ\), and bisect its angles. A central result is that the angle sum of any quadrilateral is \(360^\circ\), found by splitting it into two triangles. Relaxing conditions builds a family of quadrilaterals: a parallelogram has both pairs of opposite sides parallel (opposite sides and angles equal, diagonals bisecting each other); a rhombus has all sides equal, with diagonals that bisect each other at right angles and bisect the angles; a kite has two adjacent pairs of equal sides; and a trapezium has at least one pair of parallel sides, with the isosceles trapezium having equal base angles. Venn diagrams capture how squares, rectangles, rhombuses and parallelograms are nested.

Rectangles and squaresDiagonal properties and congruenceAngle sum of a quadrilateralParallelograms and rhombusesKites and trapeziums

Key terms

Rectangle

A quadrilateral in which all four angles are \(90^\circ\); its opposite sides are equal and parallel.

Square

A quadrilateral with all angles \(90^\circ\) and all sides equal; a special rectangle and special rhombus.

Parallelogram

A quadrilateral whose opposite sides are parallel; opposite sides and angles are equal and diagonals bisect each other.

Rhombus

A quadrilateral with all four sides equal; its diagonals bisect each other at \(90^\circ\) and bisect its angles.

Kite

A quadrilateral with two distinct adjacent pairs of equal sides; one diagonal bisects the other at right angles.

Trapezium

A quadrilateral with at least one pair of parallel sides; an isosceles trapezium has equal non-parallel sides.

Extra questions & answers

Try one from this chapter

What is the sum of all angles in any quadrilateral?

Flashcards & interactive practice