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## MATHEMATICS

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**(a+b)3**(a +b)2 MATHEMATICS**Children,**• Have you come across figures of the following types?**Do you know the name of these shapes?.**• “…geometric shapes are the alphabet in which the book of nature is written." -Galileo**What is a Quadrilateral?**• A quadrilateral is a two-dimensional figure formed by connecting four segments endpoint to endpoint with each segment intersecting exactly two others. It also has four sides and four angles.**What is a Quadrilateral?**• Look around you, do you see shapes that could be termed a quadrilateral? • Take about two minutes to draw five different shapes that you think could be classified as a quadrilateral.**QUADRILATERALS**A quadrilateral is a polygon having four sides. D C A quadrilateral has four sides. A A quadrilateral has four angles. A quadrilateral has two diagonals B The sum of four angles of a quadrilateral is 3600**Classification of Quadrilaterals**• A concave quadrilateral contains at least one angle that is >180o. • A quadrilateral is convex if each vertex lies in the interior of the opposite angle.**Types of Quadrilaterals**• Trapezoid • Parallelogram • Square • Rectangle • Kite, Dart • Rhombus • Rectangle**SPECIAL TYPES OF QUADRILATERALS**Quadrilaterals having some special properties are known as special types of Quadrilaterals**Are They Quadrilaterals?**A C B • Examples of shapes or curves that are not classified as quadrilaterals. They do not follow the definition of a quadrilateral. Can you give a reason why?**TRAPEZIUM**Trapezium is a quadrilateral with two parallel sides or bases, generally of unequal length D C A B AB║CD ABCD is a trapezium**PARALLELOGRAM**D C B A AB║CD BC║ AD ABCD is a parallelogram**PARALLELOGRAM HAS SOME SPECIAL PROPERTIES**• DRAW A PARALLELOGRAM ABCD.MEASURE ITS SIDES • LENGTH OF AB= LENGTH OF CD • LENGTH OF AD= LENGTH OF BC • OR • WE CAN SAY THAT OPPOSITE SIDES OF • A PARALLELOGRAM ARE EQUAL. D C A B**DIAGONALS OF A PARALLELOGRAM BISECT EACH OTHER**• In the figure ABCD is a parallelogram. AC and BD are diagonals. • They intersect at O. AO=OC and BO=OD A B O D C**ACTIVITY**• MEASURE THE FOUR ANGLES OF THE PARALLELOGRAM. WHAT DO YOU FIND? • IS THERE ANY RELATION BETWEEN THE OPPOSITE ANGLES?**PROPERTIES OF A PARALLELOGRAM**• Opposite sides are equal. Opposite angles are equal . The diagonals bisect each other. Click to know progress**Self assessment**• Q 1) PQRS is a parallelogram. If PQ= 5cm and PR= 6cm find RS and QR. 5cm Q P 6cm R S**Q 2) In parallelogram ABCD If angle A=1100 find angle C.**A B D C**Q 3) In PQRS Parallelogram PR=8cm and QS=10cm.Find PO and**QO. S P o Q R Go to answer**Q 1)RS=5cm ;QR=6cm**• Q 2)Angle C =1100 • Q 3)PO=4 cm ; QO=5cm**Special types of parallelograms**• Parallelograms which are quadrilaterals are further classified in to • RHOMBUS • RECTANGLE • SQUARE**PROPERTIES OF A RHOMBUS**All sides are equal. Opposite angles are equal . The diagonals bisect each other at right angles.**RECTANGLE**A rectangle is a IIgm with one angle a right angle. D C A B**PROPERTIES OF A RECTANGLE**• Opposite sides are equal. • All angles are right angles. The diagonals bisect each other. • The diagonals are equal.**SQUARE**A square is a IIgm with a pair of adjacent sides equal and one angle 900 D C A B**PROPERTIES OF A SQUARE**• All sides are equal. • All angles are right angles. • The diagonals bisect each other. • The diagonals are equal.**ASSIGNMENT**• STATE TRUE OR FALSE • 1) In a square adjacent sides are equal. • 2) In a rectangle the diagonals bisect each other. • 3)Square is a rhombus in which diagonals are equal. • 4)The diagonals of a parallelogram are always equal. • 5)A square is a rectangle**SUMMARY**• Quadrilateral is a plane figure, (that is, a shape drawn on a flat surface) bounded by four straight lines called its sides. It has four corners, called vertices, and the sum of their internal angles is 360°.Trapezium and parallelograms are special types of quadrilaterals. In this parallelograms are further classified. • If the angles are all right angles, the parallelogram is called a rectangle. If all four sides are equal the parallelogram is a rhombus. A square is both a rectangle and a rhombus, so all its angles are right angles and all its sides are equal. A rectangle, a rhombus, and a square are all examples of parallelogram in which opposite sides are parallel.**Find x in the following figures.**Ans. We know that the sum of all exterior angles of any polygon is 360º. (a) 125° + 125° + x = 360° 250° + x = 360° x= 110° (b) 60° + 90° + 70° + x + 90° = 360° 310° + x = 360° x= 50°**Qn . Find the measure of each exterior angle of a regular**polygon of (i) 9 sides (ii) 15 sides Ans. (i) Sum of all exterior angles of the given polygon = 360º Each exterior angle of a regular polygon has the same measure. Thus, measure of each exterior angle of a regular polygon of 9 sides = (ii) Sum of all exterior angles of the given polygon = 360º Each exterior angle of a regular polygon has the same measure. Thus, measure of each exterior angle of a regular polygon of 15 sides =**Qn. How many sides does a regular polygon have if the**measure of an exterior angle is 24°? Ans.Sum of all exterior angles of the given polygon = 360º Measure of each exterior angle = 24º Thus, number of sides of the regular polygon Qn.(a) Is it possible to have a regular polygon with measure of each exterior angle as 22°? (b) Can it be an interior angle of a regular polygon? Why? Ans. The sum of all exterior angles of all polygons is 360º. Also, in a regular polygon, each exterior angle is of the same measure. Hence, if 360º is a perfect multiple of the given exterior angle, then the given polygon will be possible. (a) Exterior angle = 22° 360º is not a perfect multiple of 22º. Hence, such polygon is not possible. (b) Interior angle = 22° Exterior angle = 180° − 22° = 158° Such a polygon is not possible as 360° is not a perfect multiple of 158°.**(a) What is the minimum interior angle possible for a**regular polygon? (b) What is the maximum exterior angel possible for a regular polygon? Ans . Consider a regular polygon having the lowest possible number of sides (i.e., an equilateral triangle). The exterior angle of this triangle will be the maximum exterior angle possible for any regular polygon. Exterior angle of an equilateral triangle Hence, maximum possible measure of exterior angle for any polygon is 120º. Also, we know that an exterior angle and an interior angle are always in a linear pair. Hence, minimum interior angle = 180º − 120° = 60º