Understanding Quadrilaterals PDF

6 Understanding Quadrilaterals Curve A plane figure formed by joining...

6 Understanding Quadrilaterals Curve A plane figure formed by joining a number of points without lifting a pencil from the paper and without retracing any portion of the drawing other than single points is called a curve. Examples: Curves can be of different types - open, closed, simple closed etc. Region Interior → Region inside the boundary of the closed curve. Exterior → Region outside the boundary of the closed curve. Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals A B Point A lies in the interior region of the curve and point B lies in the exterior region of the curve. Polygons A polygon is a simple closed curve made of many line segments. Examples of curves that are polygons → Examples of curves that are not polygons → B A C (not a closed curve) (BC–not a line segment)  [ 41 ] NCERT Basics: Class 8 Classification Polygons are classified by the number of their sides. 3 sides 4 sides 5 sides 6 sides The polygon having no. Triangle Quadrilateral Pentagon Hexagon of sides 20, 30, 100 and 1000 are called 7 sides 8 sides 9 sides 10 sides..........n sides icosagon, triacontagon, hectogon and chiliagon respectively. Heptagon Octagon Nanagon Decagon....... n-gon Diagonals A diagonal is a line segment connecting two non-consecutive vertices of a polygon. F C E G B A D I H AC and BD EH, HF, FI, IG, - diagonals GE - diagonals Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals  0° < Acute angle < 90° 90° < Obtuse angle < 180°  180° < Reflex angle < 360° Convex and Concave Polygons Convex polygon: In convex polygon, the measure of each interior angle is less than 180°. Convex polygons have all their diagonals in the interior of the polygon. Example: Concave polygon: In concave polygon, the measure of at least one of the interior angles is more than 180°. Concave polygons could have some of their diagonals in the exterior of the polygon. [ 42 ]  Mathematics Example: B P R Q A C T S D Interior BCD > 180° and diagonal BD lies outside the polygon boundary. Interior PQR > 180° and diagonal PR lies outside the polygon boundary. Regular and Irregular polygons A Regular polygon is one which is both equilateral (all sides equal) and equiangular (all angles equal). Equilateral triangle, square → equilateral and equiangular → Regular Rectangle → Equiangular but not equilateral → Irregular Name Regular Irregular Triangle Quadrilateral Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals Pentagon Hexagon 1 Determine whether each figure is a polygon. If it is, classify the polygon and state whether it is regular. If it is not a polygon, explain why? (i) (ii) Explanation (i) The figure has 6 equal sides and 6 equal angles. It is a regular hexagon. (ii) The figure is not a polygon since it has a curved side.  [ 43 ] NCERT Basics: Class 8   Sum of all the three angles of a triangle is 180°.  Angle sum property of a quadrilateral. (a polygon with 4 sides) → Sum of all the four angles of a quadrilateral = 360° Angle sum property Proof : Let ABCD be a quadrilateral. Draw one of its diagonals, AC C B 3 4 2 1 A D Clearly, A = 1 + 2 and C = 3 + 4 In ABC, 2 + 3 + B = 180° (Angle sum property of a triangle) …(i) In ADC, 1 + 4 + D = 180° (Angle sum property of a triangle) …(ii) Adding (i) and (ii), we get 2 + 3 + B + 1 + 4 + D = 180 + 180  1 + 2 + 3 + 4 + B + D = 360  A + C + B + D = 360 1 The three angles of a quadrilateral are 75°, 105° and 85°. Find its fourth angle. Solution Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals Let the fourth angle of the quadrilateral be x. According to the angle sum property of a quadrilateral, 75° + 105° + 85° + x = 360°  265° + x = 360°  x = 360° – 265°  x = 95° Sum of the measure of the exterior angles of a polygon The sum of the exterior angles of any polygon is equal to 360°. Let us prove this for the case of a quadrilateral ABCD. 1 B A 2 4 C D 3 1, 2, 3 and 4 represents the four exterior angles of the quadrilateral ABCD. As we can see in the given figure, 1 and BAD form a linear pair, [ 44 ]  Mathematics  1 + BAD = 180° …(1) Similarly,2 + ABC = 180° …(2) 3 + BCD = 180° …(3) 4 + ADC = 180° …(4) Adding (1), (2), (3) and (4), 1 + BAD + 2 + ABC + 3 + BCD + 4 + ADC = 180 + 180 + 180 + 180  1 + 2 + 3 + 4 + ABC+BCD+ADC+BAD = 720°  1 + 2 + 3 + 4 + 360° = 720° [Angle sum property of a quadrilateral]  1 + 2 + 3 + 4 = 360° Hence, proved. This property holds true for all the polygons. You can try to prove it yourself for a pentagon or a hexagon. 2 Find the number of sides of a regular polygon whose each exterior angle has a measure of 60°. Solution The measure of all the exterior angles = 360° Measure of each exterior angle = 60° 360  Number of sides of the polygon = =6 60 Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals Thus, the polygon has 6 sides.  Exterior angle of quadrilateral Incorrect Correct a x b w d y c z  [ 45 ] NCERT Basics: Class 8 Quadrilaterals A polygon with four sides and vertices is called a quadrilateral. Before learning the different types of quadrilaterals, we must know some important terms related to quadrilaterals. Vertex: A, B, C and D are vertices of the quadrilateral ABCD. B A D C Sides: AB, BC, CD and DA are the sides of the quadrilateral ABCD. Angles: A, B, C and D are the angles of the quadrilateral ABCD. Adjacent sides: AB & BC (Common vertex B), BC & CD (Common vertex C), CD & DA (Common vertex D) and DA & AB (common vertex A) are the four pairs of adjacent sides in the quadrilateral ABCD. Opposite sides: AB & DC and BC & AD are the two pairs of opposite sides of the quadrilateral ABCD. Adjacent angles: A & B (common side AB), B & C (common side BC), C & D (common side CD) and D & A (common side AD) are the four pairs of adjacent angles in the quadrilateral ABCD. Opposite Angles: A & C and B & D are the two pairs of opposite angles in the quadrilateral ABCD. Any four-sided closed shape is a quadrilateral, but the sides have to be straight and it   Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals has to be 2-dimensional (2-D). Different kinds of Quadrilaterals Trapezium A quadrilateral in which a pair of opposite sides are parallel is called a trapezium. B A D C AB   CD  ABCD is a trapezium. Other examples: (Parallel sides donated by arrows) [ 46 ]  Mathematics If the non-parallel sides of a trapezium are equal, then it is called an isosceles trapezium. A B D C Properties of Trapezium In a trapezium, the interior angles on the same side of each of the non-parallel sides are supplementary.  A + D = 180° (as AB||CD and AD is a transversal). A B D C A and D are interior angles on the same side of the transversal) Similarly, B + C = 180° Kite Kite is a special type of Quadrilateral in which one pair of adjacent sides are equal to each other and the other pair of adjacent sides are equal to each other. B A C D In the kite ABCD, AB = BC and CD = AD Properties of Kite Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals B A C D (i) The diagonals are perpendicular to one another. (ii) One of the diagonals bisects the other. In the figure BD bisects AC. (iii) One pair of opposite angles (the ones that are between the sides of unequal length) are equal in size. In the figure, A = C but B  D. Parallelogram A parallelogram is a quadrilateral in which both the pairs of opposite sides are parallel. Quadrilateral ABCD is a parallelogram where AB  CD and BC  AD. A B D C  [ 47 ] NCERT Basics: Class 8 Properties of Parallelogram (i) The opposite sides of a parallelogram are equal in length. (ii) The opposite angles in a parallelogram are equal. (iii) The adjacent angles in a parallelogram are supplementary. (iv) The diagonals of a parallelogram bisect each other. Proof: Consider a parallelogram ABCD. Draw any diagonal, say AC. Looking at the angles, 1 = 4. (as AB   CD and AC is a transversal 1 and 4 are alternate interior angles.) Similarly, 2 = 3 A 1 B 2 3 D 4 C In triangles ADC and ABC, 1 = 4 AC = AC (common side), 2 = 3 So, ADC  CBA (ASA congruence rule)  CD = AB and AD = BC Hence proved. (ii) The opposite angles in a parallelogram are equal. Proof: We have proved above that ADC  CBA A B D C Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals  B = D Similarly, we can draw the other diagonal BD and prove DAB ≅ BCD  A = C Hence proved. (iii) The adjacent angles in a parallelogram are supplementary. Proof : AB   CD and AD is a transversal, A and D are interior angles on the same side of the transversal.  A + D = 180° A B D C Similarly, B + C = 180° C + D = 180° D + A = 180° [ 48 ]  Mathematics (iv) The diagonals of a parallelogram bisect each other. Proof: Let we draw the two diagonals AC and BD (meeting at O) of the parallelogram ABCD. In triangles AOB and COD, A 1 3 B O 4 2 D C Alternate interior angle with AB||CD and AC–transversal. 1 = 2 [alternate interior angle] 3 = 4 [alternate interior angle] CD = AB [opposite sides of parallelogram are equal] So, AOB  COD (ASA congruency)  AO = CO and OB = OD Hence proved. 2 Draw a parallelogram. Label the congruent angles. Explanation B 1 C 2 A D Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals ABCD is a parallelogram with ABCD and BCAD. We know, the opposite angles of a parallelogram are equal.  A =C and B = D The 2 set of congruent angles 1 Complete the statements about parallelogram ABCD. Justify your answer. D C G A B (i) DAB  ? (ii) ABD  ? (iii) AB   ? (iv) BG = ?  [ 49 ] NCERT Basics: Class 8 3 In the given parallelogram PQRS, OS = 5 cm and PR is 7 cm more than QS. Find OP. P Q O S R Solution OS = 5 cm  OQ = 5 cm (Diagonals of a parallelogram bisect each other.)  QS = 10 cm Given that PR = QS + 7 = 10 + 7 = 17 cm 1 1  OP = × PR = × 17 = 8.5 cm. 2 2 4 Find the value of x in each quadrilateral. (i) Kite ABCD B A 90° x C Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals 60° D (ii) Trapezium EFGH E H x F 50° G (iii) Parallelogram IJKL K J 150° L x I [ 50 ]  Mathematics Solution (i) We know, in a kite, the opposite angles that are between the sides of unequal length are equal. Here, AB = BC and AD = CD So, A = C (A - between AB and AD, C - between BC and CD)  C = A = x = 90° (ii) We know, in a trapezium, the interior angles on the same side of each of the non- parallel sides are supplementary. Here, EH  FG  x + 50° = 180°  x = 130° (iii) We know, in a parallelogram, opposite angles are equal. So, I = K  x = 30°. Some special parallelograms Rhombus B C A D A rhombus is a parallelogram with all sides equal. ABCD is a rhombus where AB  CD and BC  AD. Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals Also, AB = BC = CD = DA. Rhombus has all the properties of a parallelogram. Besides these, a rhombus has some other properties also. Properties B C A D (i) All sides are equal (Definition of Rhombus) AB = BC = CD = DA (ii) Opposite angles are equal (Property of parallelogram) A = C B = D (iii) Diagonals bisect each other at right angles.  [ 51 ] NCERT Basics: Class 8 Proof : ABCD is a rhombus, therefore, it is a parallelogram.  Its diagonals bisect each other.  OA = OC and OB = OD The diagonals of a rectangle need not be Now, in triangles AOB and COB perpendicular to each B C other. O A D AO = CO OB = OB (Common side) AB = BC (All sides of Rhombus are equal)  AOB  COB (SSS congruency)  AOB = COB We know, AOB and COB form a linear pair, hence, AOB + COB = 180°  AOB = COB = 90° Hence proved. Similarly, AOD = COD = 90° All parallelograms are not rhombus, but all rhombuses are parallelogram.   Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals 1 (i) DAB = DCB (ii) ABD = BDC (iii) AB || CD (iv) BG = DG 2 Which property is not a characteristic of Rhombus? (1) Opposite sides parallel (2) 4 Right angles (3) 4 congruent sides (4) Opposite sides congruent 5 ABCD is a Rhombus in which AC and BD are diagonals intersecting at O. If OAB = 55°, find CDO. [ 52 ]  Mathematics Solution The diagonals of a rhombus bisect each other at right angles.  AOB = 90° A 55° B O D C Applying angle sum property in AOB, 55° + 90° + OBA = 180°  OBA = 180° – 145° = 35° We know that opposite sides of Rhombus are parallel to each other, OBA = CDO (AB   CD and BD is transversal)  OBA = CDO (Alternate angles)  OBA = CDO = 35° Rectangle A parallelogram which has all its angles as right angles is called a rectangle. MNOP is a rectangle. M N P O M = N = O = P = 90° The rectangle has all the properties of a parallelogram and a few more too. Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals Properties (i) Opposite sides are parallel and equal (Property of parallelogram) MN  OP, MN = OP MP  NO, MP = NO (ii) All angles are equal and are right angles (Definition of rectangle). (iii) Diagonals bisect each other (Property of parallelogram). (iv) Diagonals are equal. Proof : In rectangle MNOP, let us draw its diagonals NP and OM. M N P O In triangles POM and OPN, PO = OP (common side)  [ 53 ] NCERT Basics: Class 8 MPO = NOP = 90° PM = ON (opposite sides of rectangle)  POM  OPN (SAS congruency)  OM = PN Hence, proved. Diagonals are equal and bisect each other.  MX = XO = NX = XP M N X P O 2 (2) 4 Right angles 3 Determine whether each statement is sometimes, always or never true. (i) A quadrilateral is a trapezium. (ii) A parallelogram is a rectangle. 6 Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals In the given rectangle ABCD, the diagonals AC and BD meet at O. Find x, if OA = 5x – 1 and OD = 4x + 4. D C O A B Solution OA is half the diagonal AC and OD is half the diagonal BD. Since, the diagonals of a rectangle are equal, their halves must also be equal. Hence, OA = OD  5x – 1 = 4x + 4  5x – 4x = 4 + 1  x = 5 [ 54 ]  Mathematics Square A parallelogram with all sides and all angles equal is a square. B C A D ABCD is a square. It has AB = BC = CD = DA and A = B = C = D = 90° Properties (i) Opposite sides are parallel (property of parallelogram) (ii) All sides are equal (property of rhombus) (iii) All angles are right angles (equal) (property of rectangle) Definition (iv) Diagonals are equal (property of rectangle) of square (iv) Diagonals bisect each other at right angles. (property of rhombus) D C O A B A square is the only AC = BD regular quadrilateral. AO = OC = OD = OB BOA = BOC = COD = DOA = 90° Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals  If all sides of quadrilateral are equal it is a rhombus always and sometimes a square.  If all angles of quadrilateral are equal, it is a rectangle always and sometimes a square. 3 Classify the quadrilateral using the name that best describes it. (A) (B) (C) (D) (E) (F) (G) (H) (I)  [ 55 ] NCERT Basics: Class 8 Explanation (A) Since, all sides and all angles are equal. Hence, it is a square. (B) Since, a pair of opposite sides are parallel. Hence, it is a trapezium. (C) Since, one pair of adjacent sides are equal to each other and the other pair of adjacent sides are equal to each other. Hence, it is a kite. (D) Since, all sides are equal in this quadrilateral. Hence, it is a rhombus. (E) Since, adjacent angles are of 90°, opposite sides are parallel. Hence, it is a trapezium. (F) Since, in this quadrilateral both the pairs of opposite sides are equal in length. Hence, it is a parallelogram. (G) Since, opposite sides are equal and all angles are of 90°. Hence, it is a rectangle. (H) Since, it has 4 sides are 4 vertices and has no other special property. Hence, it is a quadrilateral. (I) Since, opposite sides are equal and parallel. Hence, it is a parallelogram. 3 (i) Sometime true (ii) Sometime true Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals  [ 56 ]  Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals  Mathematics [ 57 ] [ 58 ] NCERT Basics: Class 8 Publishing\PNCF\2024-25\Print Module\SET-1\NCERT\Mathematics\8th\M-2\6. Understanding Quadrilaterals