00\ \0 !:~~'?::,'7C ---- Understanding,Quadrilaterals. Learn and Remember (U) is known as a kite. It has following

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Learn and Remember 1. Parallelogram. A quadrilateral having two pairs of opposite sides equal and parallel is known as a parallelogram. t has following properties : (i) Opposite sides are equal.

1 Learn and Remember 1. Parallelogram. A quadrilateral having two pairs of opposite sides equal and parallel is known as a parallelogram. t has following properties : (i) Opposite sides are equal. (ii) Opposite angles are equal. Understanding,Quadrilaterals (iii) Diagonals bisect one another at mid point. 2. Rhombus. A rhombus which is a kind of parallelogram having all its B sides equal is known as rhombus. t has following properties : (i) All sides are equal. (ii) Diagonals bisect each other at 9 at mid point. (iii) Opposite angles are equal.!:'?::,'7c A -, B 3. Rectangle. A paranelogramndm having adjacent angle 9 each is known as rectangle. t has following properties : K L (i) Opposite sides and opposite angles are equal. (ii) Each adjacent angle is a right angle. (iii) Diagonals are of equal lengths. 4. Square. A rectangle having all four sides o of equal length, is known as a square. t has following properties : (i) All four sides are of equal length. (ii) Adjacent angle are of 9. (iii) Diagonals are equal and they bisect each other at Kite. A quadrilateral with exactly two pairs of equal consecutive sides A c ---- is known as a kite. t has following properties : (i) Diagonals are perpendiculars to one another. (ii) One of the diagonals bisects the other. (iii) One pair of opposite angles of short diagonal is equal while another pair of angles is unequal. 6. Diagonal. A diagonal is a line segment connecting two consecutive vertices of a polygon. t is given in the following figures. D', ;1C <,," -;,'"...,,". JO.,,,,'..... A"/'.J8 s A AR A convex and concave polygons A polygon having pointed portion or raised portion in its exterior part is known as convex and those which have pointed or raised portion in its interior is known as concave polygon. \ \ (i) Convex polygons. <{KG (U) Concave polygons. D B W c UNDERSTANDNG QUADRLATERALS 51

2 52 Classification of polygons We classify the following polygons according of sides. MATHEMATCS-Vii - to the num Number of sides Classification Sample figure 3 Triangle 1 G 4 Quadrilateral D 5 Pentagon 1 6 Hexagon " 7 Heptagon 1 8 Octagon 1 9 Nonagon 1 having nine sides UNDERSTANDNG QUADRLATERALS Match the following: Figures -. Types --- (1) (a) Simple closed curve. o (2) A closed curve that is not simple. (3) ok1 (c) Simple curve that is not closed. (4) (d) Not a simple curve. Figures Types (1) (d) Not a simple curve. (2) (3) (c) (a) (4) K1 Simple curve that is not closed. Simple closed curve. A closed curve that is not simple

3 TEXTBOOK QUESTONS SOLVED 54 MATHEMATcs-V -- (c) Polygon : UNDERSTANDNG QUADRLATERAlS 55 EXERCSE 3.1 (Page-41-42) Q1. Given here are some figures. V*(4) (1) (2) VO* (1) (2) (3) (4) (d) Convex polygon : V(1) (e) Concave polygon: CJDs) * (5) (6) (7) (8) Classify each of them on the basis of the following: (a) Simple curve Simple closed curve (c) Polygon (d) Convex polygon (e) Concave polygon (a) Simple curve: VCJO (1) (2) (5) (6) Simple closed curve: (a) (1) (4) Figures 1, 2, 5, 6 and 7 are simple curve. Figures 1, 2, 5, 6 and 7 are simple closed curve. (c) Figures 1,2 and 4 are polygon. (d) Figure 1 is convex polygon (e) Figures 1 and 4 are concave polygon. Q2. How many diagonals does each of the following have? (a) Aconvexquadrilateral Aregularhexagon (c) A triangle (a) A convex quadrilateral has two diagonals. AC and BD are two diagonals. vcjd A regular hexagon has 9 diagonals as AD, AE, BD, BE andfc. FB, AC, EC, FD. (1) (2) (5) (8) (7)

$56 MATHEMATcs-V UNDERSTANDNG QUADRLATERALS 57 ED 1\ " J..'" ',.',L,...'" \/, -, Fi-----. --+--: C... \ ",'" 't... \,,( AB : 1" -, :"..Jr\ : (C) A triangle has no diagonal. Q3.$

4 56 MATHEMATcs-V UNDERSTANDNG QUADRLATERALS 57 ED 1\ " J..'" ',.',L,...'" \/, -, Fi : C... \ ",'" 't... \,,( AB : 1" -, :"..Jr\ : (C) A triangle has no diagonal. Q3. What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!) LetABCD is a convex quadrilateral, then we draw a diagonal AC which divides the quadrilateral in two triangles. LA + LB + LC + LD = L1 + L6 + L5 + L4 + L3 + L2 = (L1 + L2 + L3)+ (L4 + L5 + L6) (By angle sum property of a triangle). = + o = 36 c Hence, the sum of measures of the triangles of a convex quadrilateral /-:::'J)O.J...J- B is 36. A Yes, if quadrilateral is not convex then, this property will also be applied. Let ABCD is a non-convex quadrilateral and join BD. Which also divides the quadrilateral in two triangles. nmbd, A L1 + L2 + L3 =...(i) (By angle sum property of triangle) nbdc, L4 + L5 + L6 =...(ii) (By angle sum property of a triangle) Adding equation, (z) and (ii), L1 + L2 + L3 + L4 + L5 + L6 = + 18 L1 + L2 + (L3 + L4) + L5 + L6 = 36 LA + LB + LC + LD = 36. Proved. C o -Q4. Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.) 6 [ "... ''\,... \, -... " ' \ Figure ' ' \. " \,, Side Angle 1 x 18 2 x 18 3 x 18 4x 18 sum = (3-2) = (4-2) = (5-2) = (6-2) x 18 x 18 x 18 x 18 What can you say about the angle sum of a convex polygon with number of sides? (a) 7 8 (e) 1 (d) n (a) When n = 7 Then, angle sum of a polygon = (n - 2) x = (7-2) x 18 = 5 x = 9. When n = 8 Then, angle sum of a polygon = (n - 2) x = (8-2) x = 6 x = 18. (c) When n = 1 Then, angle sum of a polygon = (n - 2) x = (1-2) x (d) = 8 x = 144. When n = n sides Then, angles sum of a polygon = (n - 2) x =(n - 2). (Where n is number of sides.) Q5. What is a regular polygon? State the name of a regular polygon of (i) 3 sides (ii) 4 sides (iii) 6 sides A regular polygon : A polygon having all sides of equal length and the interior angles of equal size is known as regular polygon. (i) 3 sides Polygon having three sides is called triangle.

58 Q6. (ii) 4 sides Polygon having four sides is called a quadrilateral. (iii) 6 sides A polygon having six sides is called hexagon. Find the angle measure x in the following figures.

5 58 Q6. (ii) 4 sides Polygon having four sides is called a quadrilateral. (iii) 6 sides A polygon having six sides is called hexagon. Find the angle measure x in the following figures. (a) (c) (a) We know that sum of angles of a quadrilateral is 36. Now, x = 36 (d) 3 + x = x = 36-3 (a) or x = 6. We know that sum of angles of a quadrilateral is 36. Now, x = x = 36 x = or x = 14 (c) First base interior angle = - 7 = 11 Second base interior angle = - 6 = 12 There are five sides, n = 5 (c) UNDERSTANDNG Q7. QUADRLATERALS Angle sum of a polygon = (n - 2) x = (5-2) x 1 = 3 x 18 = 54. Sum of angles of irregular pentagon = 54. Then, 3 + x x = x = 54 2x = x = 28 or x = = 14 Hence, each unknown angle is 14. (d) Angle sum of a regular pentagon = (n - 2) x = (5-2) x = 3 x = 54 Since, sum of angles of regular pentagon = 54 (From part (c)) So, x + x + x + x + x = 54 5x = 54 x = = 18 Hence, each interior angle is (a) Findx+y+z Findx+y+z+w (a) m = 18 (Angle sum property of triangle) 12 + m = 18, m = - 12 m = 6 Since, sum of linear pair is also x = (i) z z + 3 = (ii) y + 6 = (iii) Adding equations (i), (ii) and (iii) 9 + x + z y + 6 = + + (d) 59 3 y

6 x + y + z + w x + y +z + x +y +z x +y +z = 54 = 54 - = 36. Sum of angles of a quadrilateral = 36 6 + 8 + 12 + n = 36 26 + n = 36 n = 36-26 n=1 Since, sum of linear pair is also.

6 6 x + y + z + w x + y +z + x +y +z x +y +z = 54 = 54 - = 36. Sum of angles of a quadrilateral = n = n = 36 n = n=1 Since, sum of linear pair is also. w + 1 = x + 12 = y + 8 = 18 z + 6 = Adding equations, (i), (ii), (iii) and (iv) x + y + z + w + 36 = 72 x + y + z + W = or x + y + z + W = 36. EXERCSE 3.2 (Page - 44) Ql. Find x in the following figures. (a) (a) m = MAHEMATCS-V... (i)... (ii)...(iii)...(iv) (Linear pair of angles) m = -125 = 55 (Linear pair of angles) and n = n = = 55 Exterior angle Xo = sum of opposite interior angles or Xo = = 11 -UNDERSTANDNG QUADRLATERALS Or Since, sum of exterior angles of a triangle = Xo = Xo = 36 Xo = = 11. Sum of angles of a pentagon = (n - 2) x =(5-2) x = 3 x = 54 By linear pairs of angles L1 + 9 = (i) L2 + 6 = (ii) L3 + 9 = (iii) L4 + 7 = (iv) L5 + x = (v) Adding equations (i), (ii), (iii), (iv) and (u) x + (L1 + L2 + L3 + L4 + L5) + 31 = 9 x = 9 x + 85 = 9 x = 9-85 = 5. Q2. Find the measure of each exterior angle of a regular polygon of (i) 9 sides (ii) 15 sides. (i) Sum of angles of a regular polygon = (n - 2) x (9-2) x = 7 x = 126 h i. _ Sum of interior angles Eac mtenor ang e - Nurnber f'sid S es = 14 Each exterior angle = - 14 = 4 Or 126 =-9-. _Sum of exterior angles _ 36 _ E xtenor ang e - N urn be r f sid S es 9 (ii) Sum of exterior angles of a regular polygon = 36 61

62 MATHEMATcs-V Q3. Q4. Exteri h. 15 d Sum of exterior angles enor ang e avmg si es = -------"'-- Number of sides = 36 = 24.

7 62 MATHEMATcs-V Q3. Q4. Exteri h. 15 d Sum of exterior angles enor ang e avmg si es = "'-- Number of sides = 36 = How many sides does a regular polygon have, if the measure of an exterior angle is 24? Let number of sides be n The measures of all exterior angles = 36 N b f sid Sum of exterior angles urn er Sl es =. Each extenor angle 36 n = -- = Hence, the regular polygon has 15 sides. How many sides does a regular polygon have if each of its interior angles is 165? Let the number of sides be n. Exterior angle = -165 = 15. Sum of exterior angles of a regular polygon = 36. N b f sid Sum of exterior angles urn er Sl es = = Each exterior angle 36 n= =24 Hence, the regular polygon has 24 sides. Q5. (a) s it possible to have a regular polygon with of each exterior angle as 22? Can it be an interior angle of a regular polygon? Why? (a) No; (since, 22 is not a divisor of 36). No; (because each exterior angle is - 22 = 158, which is not a divisor of 36 ). Q6. (a) What is the minimum interior angle possible for a regular polygon? Why? What is the maximum exterior angle possible for a regular polygon? (a) The equilateral triangle being a regular polygon of3 sides has the least measure of an interior angle = 6. '::RSTANDNGQUADRLATERALS Since, angle-sum of a triangle =. x + x + x = 3x = EXERCSE 3.3 (Page-SO-52) Ql. Given a parallelogramabcd. Complete each statement along with the definition or property used. m AD =. (ii) LDCB = (iii) OC = (iv) mldab + mlcda =. D\_,>?<:: B A ' (i) AD = BC (Since, opposite sides of a parallelogram are equal.) (ii) LDCB = LDAB (Opposite angles are equal in gm) (iii) OC = OA (Since diagonals of a parallelogram bisect each other) (iv) mldab + mlcda = (..' The adjacent angles in a parallelogram are supplementary) Q2. Consider the following parallelograms. Find the values of the unknowns x, y, %. DC A (i) <l> (iii) B x = -- =6 3 By (a), we can see that the greatest exterior angle is _ 6 = 12. (ii) (iv) 63

8 64 (v) MATHEMATCS-V Note. For getting correct answer, read 3 = 3 in (Fig. (iii». (i) LB + LC = Dx C (The adjacent angles in a parallelogram are supplementary.) z 1 => 1 + x = A B => x = - 1 => x = 8, z = x = 8 opposite angle of 1/ y = 1 (opposite angle of 1/ gm are equal). (ii) x + 5 = (The adjacent angles in a parallelogram are supplementary.) => x = - 5 => x = 13 => z = x = 13 gm) (Corresponding angles) "" => y = z = 13 => x =y = 13 (Alternate angles) (Opposite angles of a parallelogram are equal) (Vertically opposite angles) (Angle sum property of a triangle) => y = => y + 12 = => => (iii) x = 9 => y + x + 3 = (iv) y = - 12 = 6 z = y = 6 (Alternate z. angles) -- UNDERSTANDNG QUADRLATERAlS Q3. => z = 8 (Corresponding angles) => x + 8 = (The adjacent angles in a parallelogram aresupplementary.) => x = - 8 = 1 y = 8 (Opposite angles are equal in 1/gm.) (v) y = 112 (Opposite angles of parallelogram are equal.) =>4 + y + x = (Angle-sum property of a triangle.) => x = 18 => x = => x = => x = 28 => z = x = 28 (Alternate angles). Can a quadrilateral ABCD be a parallelogram, if (i) LD + LB =? (ii) AB = DC = 8 em, AD = 4 em and BC = 4.4 em? (iii) LA = 7 and LC = 65? (i) LD + LB =. t can be, but here, it needs not to be? (ii) (iii) No. n this case. One pair of opposite sides are equal and another pair of opposite sides are unequal. So, it is not a parallelogram. No; LA :f. LC (Since opposite angles are equal in gm) Here opposite angles are not equal in quadrilateral ABCD. So, it is not a parallelogram. A DC 8crn B C 65 4:1 }4= A C 8crn B

$\ \ ' ---- LHOP = - 7 = 11. LE = LHOP UNDERSTANDNG QUADRU\TERALS (Angles of linear pair) (Opposite angles of a paralellogram are equal) x = 11 (Alternate angles) LPHE = LHPO Er 71 P Q5. Q6. Q7.$

9 67 66 Q4. Draw a rough figure of a quadrilateral that is no parallelogram but has exactly two opposite angles equal measure. ABCD is a quadrilateral in which opposite angles LA = L 11. \ \ ' ---- LHOP = - 7 = 11. LE = LHOP UNDERSTANDNG QUADRU\TERALS (Angles of linear pair) (Opposite angles of a paralellogram are equal) x = 11 (Alternate angles) LPHE = LHPO Er 71 P Q5. Q6. Q7.. A \ '> B So, it is a kite. The measure of two adjacent angles of a parallelogr are in the ratio 3 : 2. Find the measure of each of t angles of the parallelogram. Let two adjacent angles be 3x and 2x. Since the adjacent angles in a parallelogram a supplementary. 3x + 2x = 5x = x = = 36 5 One angle = 3 x 36 = 18 Another angle = 2 x 36 = 72. Two adjacent angles of a parallelogram have equ measure. Find the measure of each of the angles of t parallelogram. Let each adjacent angle be x. Since sum of two adjacent angles in gmare supplementa x + x =. 2x = x = -- =9 2 Hence, each adjacent angle is 9. Ans. The adjacent figure HOPE is a parallelogram. angle measures x, y and z. State the properties to find them. LHOP + 7 = 18 Find tb you u Q8. H Y = 4 LEHO = La = z = 7 z = 7 _4 Hence, Z = 3, x = 11, y = 4, Z = 3. Ans. o (Corresponding angles) The ON following figures GUNS and RUNS are parallelogram. Find x and y. (Lengths are in em). 26 (i) (ii) (i) N S",,2 1--";-' "\l.> - " 3X >' <> _,.l- ;"""" -,.> G 3)'- 1 u R ' -- - u n gtnguns GS = UN (Opposite sides ofllgtnare equal.) 3x = x = 3=6cm GU = SN (Opposite sides ofllgtnare equal) and 3y -1 = 26 3y = y = y = -=9cm 3 Hence, x = 6 cm, Y = 9 ern, Ans.

10 68 Q9. (ii) n gmruns MATHEMATCS-V y + 7 = 2...(i) (The diagonals of a parallelogram bisect each other.) x + y = 16...(ii) From equation (z), y+7=2 Y = 2-7 = 13 Putting y = 13 in equation (ii), x + 13 = 16 x = = 3 Hence, x = 3 em and y = 13 em. Ans. K E 8 U, 7 ' \ 12 X, ( V 1\,,, 7 R J C L n the above figure both RSK and CLUE are parallelograms. Find the value of x. nllgmrsk R LRS = LRS = Zm = Zm. = and LEC = m +n +LEC= 6 + n + 7 = 13 + n = K E 8, 12 «x, ( V 1\ u --J o n \ 12m' 7 C LK 12 (Opposite angles of a parallelogram are equal.) '(Sum oflinear pairs) - 12 = 6 LL = 7 (Corresponding angles) 18 (Angles sum property of triangle) n = - 13 n = 5 L UNDERSTANDNG QUADRLATERALS Q. QU. x = n = 5 (Vertically opposite angles) x = 5. Explain how this figure is a trapezium. Which of its two sides are parallel? K UN M 1 8 LM + LL = = (Sum of interior opposite angles is ) un Since, sum of co-interior angles is. Lines NM and KL are parallel. L M 1 Hence, KLMN is a trapezium K L Find mlc in figure if AB DC, LB + LC = Since DC AB 12 + mlc = \ J C mlc = - 12 = 6. Q12. Find the measure of LP and LS if SP RQ in given figure. (f you find mlr, is there more than one method to find mlp?) LP + LQ = 18 c... Sum of co-interior angles is ) LP + 13 = LP = - 13 = 5 LR 9 (given) LS + 9 = LS = - 9 = 9 each c... AB line ) \ JC 81 R p 69 Q

7 71 MATHEMATiCS-V S TANDNGQUADRLATERALS Or Yes, angle sum property of quadrilateral. LS + LR + LQ + LP = 36 9 + 9 + 13 + LP = 36 31 + LP = 36 LP = 36-31 LP = 5. Ans. EXERCSE 3.4 (Page-55) Q1.

11 7 71 MATHEMATiCS-V S TANDNGQUADRLATERALS Or Yes, angle sum property of quadrilateral. LS + LR + LQ + LP = LP = LP = 36 LP = LP = 5. Ans. EXERCSE 3.4 (Page-55) Q1. State whether True or False. (a) All rectangles are squares. All rhombuses are parallelograms. (e) All squares are rhombuses and also rectangles. (d) All squares are not parallelograms. (e) All kites are rhombuses. (n All rhombuses are kites. (g) All parallelograms are trapeziums. (h) All squares are trapeziums. (a) False. All rectangles are squares. Since, squares have all sides equal. True. All rhombuses are parallelograms. Since, in rhombus, opposite angles are equal and diagonals intersect at mid points. (c) True. All squares are rhombus and also rectangles. Since, squares have the same property of rhombus, but not a rectangle. Cd) False. All squares are not parallelograms. Since, all squares have the same property of parallelograms. (e) False. All kites are not rhombuses. Since, all kites do not have equal sides. if) Yes, all rhombus are kites, since, they have equal sides and diagonals bisect each other. (g) True. All parallelograms are trapezium because trapezium has only two parallel sides. (h) True. All squares are trapeziums. Since, squares have also two parallel sides. Q2. dentify all the quc,drilaterals that have. (a) four sides of equal length. four right angles. 8 1 (a) Rhombus and square have sides of equal length. or- Square and rectangle have four right angles. Q3. Explain how a square is (i) a quadrilateral. (ii) a parallelogram. (iii) a rhombus. (iv) a rectangle. 81. (i) A square is a quadrilateral, if it has four unequal length of sides. (ii) A square is a parallelogram, since it contains both pairs of opposite sides equal. (iii) A square is already a rhombus. Since, it has four equal sides and diagonals bisect at 9 to each other. (iv) A square is a parallelogram, since having each adjacent angle a right angle, and opposite sides are equal. Q4. Name the quadrilaterals whose diagonals (i) bisect each other. (ii) are perpendicular bisectors of each other. (iii) are equal. (i) f diagonals of a quadrilateral bisect each other then it is a rhombus, parallelogram, rectangle or square. (ii) f diagonals of a quadrilateral are perpendicular bisector of each other, then it is a rhombus or square. (iii) f diagonals are equal, then it is a square or rectangle. Q5. Explain why a rectangle is a convex quadrilateral. Arectangle is a convexquadrilateral since its vertex are raised and both of its diagonals lie in its interior. Q6. ABC is a right-angled triangle and is the mid point of the side opposite to the right angle. Explain why is equidistant from A, Band C. (The dotted lines are drawn additionally to help you.) A" ::;"1D o > B V C Since, two right triangles make a rectangle where is equidistant point from A, B, C and D because is the midpoint of the two diagonals of a rectangle. Since AC and BD are equal diagonals and intersect at mid point. So, is the equidistant from A, B, C and D. DO

Class VIII Chapter 3 Understanding Quadrilaterals Maths. Exercise 3.1

Class VIII Chapter 3 Understanding Quadrilaterals Maths. Exercise 3.1 Question 1: Given here are some figures. Exercise 3.1 (1) (2) (3) (4) (5) (6) (7) (8) Classify each of them on the basis of the following. (a) Simple curve (b) Simple closed curve (c) Polygon (d) Convex