Try to give a few more examples and non-examples for a polygon. Draw a rough figure of a polygon and identify its sides and vertices. 3.2.1 Classification of polygons We classify polygons according to the number of sides (or vertices) they have. Number of sides or vertices Classification Sample figure 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon # # # n n-gon 3.2.2 Diagonals A diagonal is a line segment connecting two non-consecutive vertices of a polygon (Fig 3.1). Can you name the diagonals in each of the above figures? (Fig 3.1) Is PQ a diagonal? What about LN ? You already know what we mean by interior and exterior of a closed curve (Fig 3.2). Interior Fig 3.2 Exterior The interior has a boundary. Does the exterior have a boundary? Discuss with your friends. 3.2.3 Convex and concave polygons Here are some convex polygons and some concave polygons. (Fig 3.3) Fig 3.3 Can you find how these types of polygons differ from one another? Polygons that are convex have no portions of their diagonals in their exteriors. Is this true with concave polygons? Study the figures given. Then try to describe in your own words what we mean by a convex polygon and what we mean by a concave polygon. Give two rough sketches of each kind. In our work in this class, we will be dealing with convex polygons only. 3.2.4 Regular and irregular polygons A regular polygon is both ‘equiangular’ and ‘equilateral’. For example, a square has sides of equal length and angles of equal measure. Hence it is a regular polygon. A rectangle is equiangular but not equilateral. Is a rectangle a regular polygon? Is an equilateral triangle a regular polygon? Why? Regular polygons Polygons that are not regular [Note: Use of or indicates segments of equal length]. In the previous classes, have you come across any quadrilateral that is equilateral but not equiangular? Recall the quadrilateral shapes you saw in earlier classes – Rectangle, Square, Rhombus etc. Is there a triangle that is equilateral but not equiangular? 3.2.5 Angle sum property Do you remember the angle-sum property of a triangle? The sum of the measures of the three angles of a triangle is 180°. Recall the methods by which we tried to visualise this fact. We now extend these ideas to a quadrilateral. (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 polygon (e) Concave polygon 2. How many diagonals does each of the following have? (a) A convex quadrilateral (b) A regular hexagon (c) Atriangle 3. 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!) 4. Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.) Figure Side 3 4 5 6 Angle sum 180º 2 × 180° = (4 – 2) × 180° 3 × 180° = (5 – 2) × 180° 4 × 180° = (6 – 2) × 180° What can you say about the angle sum of a convex polygon with number of sides? (a) 7 (b)8 (c)10 (d) n 5. What is a regular polygon? State the name of a regular polygon of (i) 3 sides (ii) 4 sides (iii) 6 sides 6. Find the angle measure x in the following figures. (a) (b) (c) (d) (a) Find x + y + z (b) Find x + y + z + w 3.3 Sum of the Measures of the Exterior Angles of a Polygon On many occasions a knowledge of exterior angles may throw light on the nature of interior angles and sides. Take a regular hexagon Fig 3.10. 1. What is the sum of the measures of its exterior angles x, y, z, p, q, r? 2. Is x = y = z = p = q = r? Why? 3. What is the measure of each? (i) exterior angle (ii) interior angle 4. Repeat this activity for the cases of (i) a regular octagon (ii) a regular 20-gon Example 2: Find the number of sides of a regular polygon whose each exterior angle has a measure of 45°. Solution:Total measure of all exterior angles = 360° Measure of each exterior angle = 45° 360 Therefore, the number of exterior angles = = 845The polygon has 8 sides. EXERCISE 3.2 1. Find x in the following figures. (a) (b) 2. Find the measure of each exterior angle of a regular polygon of (i) 9 sides (ii) 15 sides 3. How many sides does a regular polygon have if the measure of an exterior angle is 24°? 4. How many sides does a regular polygon have if each of its interior angles is 165°? 5. (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? 6. (a) What is the minimum interior angle possible for a regular polygon? Why? (b) What is the maximum exterior angle possible for a regular polygon? 3.4 Kinds of Quadrilaterals Based on the nature of the sides or angles of a quadrilateral, it gets special names. 3.4.1 Trapezium Trapezium is a quadrilateral with a pair of parallel sides. These are trapeziums These are not trapeziums Study the above figures and discuss with your friends why some of them are trapeziums while some are not. (Note: The arrow marks indicate parallel lines). 3.4.2 Kite Kite is a special type of a quadrilateral. The sides with the same markings in each figure are equal. For example AB = AD and BC = CD. These are kites These are not kites Study these figures and try to describe what a kite is. Observe that (i) A kite has 4 sides (It is a quadrilateral). (ii) There are exactly two distinct consecutive pairs of sides of equal length. 3.4.3 Parallelogram A parallelogram is a quadrilateral. As the name suggests, it has something to do with parallel lines. LM & ON QP &SR AB & ED LO & MN QS & PR BC & FE These are parallelograms These are not parallelograms Study these figures and try to describe in your own words what we mean by a parallelogram. Share your observations with your friends. AB & DC AB & CD AD & BC Looking at the angles, ∠1 = ∠2 and ∠3 = ∠4 (Why?) Since in triangles ABC and ADC, ∠1 = ∠2, ∠3 = ∠4 and AC is common, so, by ASA congruency condition, Δ ABC ≅Δ CDA (How is ASA used here?) This gives AB = DC and BC = AD. Example 3: Find the perimeter of the parallelogram PQRS (Fig 3.22). Solution: In a parallelogram, the opposite sides have same length. Therefore, PQ = SR = 12 cm and QR = PS = 7 cm So, Perimeter = PQ + QR + RS + SP = 12 cm + 7 cm + 12 cm + 7 cm = 38 cm Fig 3.223.4.5 Angles of a parallelogram We studied a property of parallelograms concerning the (opposite) sides. What can we say about the angles? Does this tell you anything about the measures of the angles A and C? Examine the same for angles B and D. State your findings. Property: The opposite angles of a parallelogram are of equal measure. Fig 3.24 Studying Δ ABC and ΔADC (Fig 3.25) separately, will help you to see that by ASA congruency condition, Δ ABC ≅Δ CDA (How?) Fig 3.25 This shows that ∠B and ∠D have same measure. In the same way you can get m∠A = m ∠C. Example 4: In Fig 3.26, BEST is a parallelogram. Find the values x, y and z. Solution: S is opposite to B. So, x = 100° (opposite angles property) y = 100° (measure of angle corresponding to ∠x) z = 80° (since ∠y, ∠z is a linear pair) We now turn our attention to adjacent angles of a parallelogram. In parallelogram ABCD, (Fig 3.27). ∠A and ∠D are supplementary since DC & AB and with transversal DA , these two angles are interior opposite. ∠A and ∠B are also supplementary. Can you Fig 3.27 say ‘why’? AD & BC and BA is a transversal, making ∠A and ∠B interior opposite. Identify two more pairs of supplementary angles from the figure. Property: The adjacent angles in a parallelogram are supplementary. Example 5: In a parallelogram RING, (Fig 3.28) if m∠R = 70°, find all the other angles. Solution: Given m∠R = 70° Then m∠N = 70° because ∠R and ∠N are opposite angles of a parallelogram. Since ∠R and ∠I are supplementary, Fig 3.28 m∠I = 180° – 70° = 110° Also, m∠G = 110° since ∠G is opposite to ∠I Thus, m∠R = m∠N = 70° and m∠I = m∠G = 110° 2. Consider the following parallelograms. Find the values of the unknowns x, y, z. (i) (ii) (iii) (iv) (v) 3. Can a quadrilateral ABCD be a parallelogram if (i) ∠D + ∠B = 180°? (ii) AB = DC = 8cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure. 5. The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram. 6. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram. 7. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them. 8. The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm) (i) (ii) In the above figure both RISK and CLUE are parallelograms. Find the value of x. 10. Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig 3.32) 11. Find m∠C in Fig 3.33 if AB & DC . 12. Find the measure of ∠P and ∠S if SP & RQ in Fig 3.34. (If you findm∠R, is there more than one method to findm∠P?) Fig 3.34 3.5 Some Special Parallelograms 3.5.1 Rhombus We obtain a Rhombus (which, you will see, is a parallelogram) as a special case of kite (which is not a a parallelogram).Note that the sides of rhombus are all of same length; this is not the case with the kite. A rhombus is a quadrilateral with sides of equal length. Since the opposite sides of a rhombus have the same length, it is also a parallelogram. So, a rhombus has all the properties of a parallelogram and also that of a kite. Try to list them out. You can then verify your list with the check list summarised in the book elsewhere. Kite Rhombus The most useful property of a rhombus is that of its diagonals. Property: The diagonals of a rhombus are perpendicular bisectors of one another. Here is an outline justifying this property using logical steps. ABCD is a rhombus (Fig 3.35). Therefore it is a parallelogram too. Since diagonals bisect each other, OA = OC and OB = OD. We have to show that m∠AOD = m∠COD = 90° It can be seen that by SAS congruency criterion Fig 3.35 Δ AOD ≅Δ COD Since AO = CO (Why?)Therefore, m ∠AOD = m ∠COD AD = CD (Why?)Since ∠AOD and ∠COD are a linear pair, OD = OD m ∠AOD = m ∠COD = 90° Example 7: RICE is a rhombus (Fig 3.36). Find x, y, z. Justify your findings. Solution: x =OE y =OR z = side of the rhombus = OI (diagonals bisect) = OC (diagonals bisect) = 13 (all sides are equal ) Fig 3.36=5 =12 3.5.2 A rectangle A rectangle is a parallelogram with equal angles (Fig 3.37). What is the full meaning of this definition? Discuss with your friends. If the rectangle is to be equiangular, what could be the measure of each angle? Fig 3.37 Let the measure of each angle be x°. Then 4x° = 360° (Why)? Therefore, x° = 90° Thus each angle of a rectangle is a right angle. So, a rectangle is a parallelogram in which every angle is a right angle. Being a parallelogram, the rectangle has opposite sides of equal length and its diagonals bisect each other. In a parallelogram, the diagonals can be of different lengths. (Check this); but surprisingly the rectangle (being a special case) has diagonals of equal length. Property: The diagonals of a rectangle are of equal length. This is easy to justify. If ABCD is a rectangle (Fig 3.38), then looking at triangles ABC and ABD separately [(Fig 3.39) and (Fig 3.40) respectively], we have ΔABC ≅ΔABD This is because AB = AB (Common) BC = AD (Why?) m ∠A = m ∠B = 90° (Why?) The congruency follows by SAS criterion. Thus AC =BD and in a rectangle the diagonals, besides being equal in length bisect each other (Why?) Example 8: RENT is a rectangle (Fig 3.41). Its diagonals meet at O. Find x, if OR = 2x + 4 and OT = 3x + 1. Solution: OT is half of the diagonal TE , OR is half of the diagonal RN . Diagonals are equal here. (Why?) So, their halves are also equal. Therefore 3x + 1 =2x + 4 or x =3 3.5.3 A square Fig 3.41A square is a rectangle with equal sides. This means a square has all the properties of a rectangle with an additional requirement that all the sides have equal length. The square, like the rectangle, has diagonals of equal length. BELT is a square, BE = EL = LT = TB In a rectangle, there is no requirement ∠B, ∠E, ∠L, ∠T are right angles. for the diagonals to be perpendicular to BL = ET and BL ⊥ ET .one another, (Check this). OB = OL and OE = OT. In a square the diagonals. (i) bisect one another (square being a parallelogram) (ii) are of equal length (square being a rectangle) and (iii) are perpendicular to one another. Hence, we get the following property. Property: The diagonals of a square are perpendicular bisectors of each other. (d) All squares are not parallelograms. (h) All squares are trapeziums. 2. Identify all the quadrilaterals that have. (a) four sides of equal length (b) four right angles 3. Explain how a square is. (i) a quadrilateral (ii) a parallelogram (iii) a rhombus (iv) a rectangle 4. Name the quadrilaterals whose diagonals. (i) bisect each other (ii) are perpendicular bisectors of each other (iii) are equal 5. Explain why a rectangle is a convex quadrilateral. 6. ABC is a right-angled triangle and O is the mid point of the side opposite to the right angle. Explain why O is equidistant from A, B and C. (The dotted lines are drawn additionally to help you). THINK, DISCUSS AND WRITE 1. A mason has made a concrete slab. He needs it to be rectangular. In what different ways can he make sure that it is rectangular? 2. A square was defined as a rectangle with all sides equal. Can we define it as rhombus with equal angles? Explore this idea. 3. Can a trapezium have all angles equal? Can it have all sides equal? Explain. WHAT HAVE WE DISCUSSED? Quadrilateral Properties Parallelogram: A quadrilateral with each pair of opposite sides parallel. (1) Opposite sides are equal. (2) Opposite angles are equal. (3) Diagonals bisect one another. Rhombus: A parallelogram with sides of equal length. (1) All the properties of a parallelogram. (2) Diagonals are perpendicular to each other. Rectangle: A parallelogram with a right angle. (1) All the properties of a parallelogram. (2) Each of the angles is a right angle. (3) Diagonals are equal. Square: A rectangle with sides of equal length. All the properties of a parallelogram, rhombus and a rectangle. Kite: A quadrilateral with exactly two pairs of equal consecutive sides (1) The diagonals are perpendicular to one another (2) One of the diagonals bisects the other. (3) In the figure m∠B = m∠D but m∠A ≠ m∠C.