Understanding Quadrilaterals

Understanding Quadrilaterals Polygons A simple closed curve made up of only line segments is called a polygon. Classification of polygons Polygons can be classified by the number of sides they have: Diagonals A diagonal is a line segment connecting two non-consecutive vertices of a polygon. Interior and exterior of a closed curve Interior region: The portion of the plane which is inside the curve. Exterior region: The portion of the plane which is outside the curve. Convex and concave polygons A polygon which has all the interior angles less than 180° is called convex polygon. That means all the vertices of the convex polygon point outwards. A polygon which has one or more interior angles greater than 180° is called concave polygon. That means at least one vertex of the concave polygon point inwards. Regular and irregular polygons Regular polygons: A polygon is said to be regular if all the angles are equal and all the sides are equal. Irregular polygons: A polygon is said to be irregular if all the angles are not equal and all the sides are not equal. Angle sum property Theorem: The sum of four angles of a quadrilateral is 360°. Proof: In ∆ABC, ∠1 + ∠2 + ∠3 = 180° …… (1) (From the angle sum property of the triangle) In ∆ACD, ∠4 + ∠5 + ∠6 = 180° ……… (2) (From the angle sum property of the triangle) From (1) and (2) ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360° It can be written as: (∠1 + ∠6) + ∠2 + (∠3 + ∠4) + (∠5) = 360° ⇒ ∠A + ∠B + ∠C + ∠D = 360° Hence Proved. For a regular n-sided polygon, Each interior angle = {[(2n - 4) ×90]/n}° Example: Find the sum of the interior angles of a nonagon? Solution: For a nonagon, the number of sides, n = 9. Sum of interior angles = (2n - 4) × 90° = (2 × 9 - 4) × 90° = 14 × 90° = 1260° Sum of the Measures of the Exterior Angles of a Polygon Figure Figure shows a pentagon. Its external angles are named from ‘a’ to e. The aim is to find the sum of these five angles. It is known that the sum of internal angles of a Pentagon = (5 - 2) × 180° = 3 × 180° = 540° ∴Each interior angle of the pentagon measures 540°/ 5 = 108° The interior and exterior angles form linear pairs and hence are supplementary. ∴Each exterior angle measures 180° - 108° = 72° ∴ Sum of five exterior angles = 5 × 72° = 360° It can be proved that the sum of the exterior angles for any polygon is 360 0. Sum of interior angles of an n sided polygon = (n - 2) 180°. ∴ Measure of each internal angle = ∴ Each exterior angle = ∴ Sum of n exterior angles = = = Conclusion: The sum of interior angles of a polygon is dependent on the number of sides but the sum of the exterior angles is always 360°. Example: One exterior angle of a regular polygon is 60°. How many sides does it have? Solution: Let the number of sides be n. The polygon is regular so one exterior = 360°/n 360°/n = 60° It gives n = 6. So the polygon has 6 sides and it is hexagon. Kinds of Quadrilaterals Based on the nature of the sides or angles of a quadrilateral, it gets special names. Trapezium A trapezium is a quadrilateral in which only one pair of opposite sides is parallel and the other two sides are non-parallel. In the given figure, ABCD is a trapezium in which AB II CD, but AD is not parallel to BC. Kite A kite is a quadrilateral which has two pairs of equal adjacent sides, but unequal opposite sides. In the given figure, ABCD is a kite in which AB = BC, AD = CD, AD ≠ BC and CD ≠ AB. Parallelogram A parallelogram is a quadrilateral in which each pair of opposite sides is parallel. In the given figure, ABCD is a parallelogram in which AB II DC and AD II BC. Elements of a parallelogram • • • • • • There are four sides in a parallelogram that are AB, BC, CD and DA. Four angles are ∠A, ∠B, ∠C and ∠D. There are two pairs of opposite sides that are AB and DC, DA and CB. AB and BC, BC and CD, CD and DA, DA and AB are the adjacent sides. The opposite sides of a parallelogram have equal length. (∠A and ∠B) and (∠B and ∠C) are adjacent angles. They are at the ends of the same side. Property: In a parallelogram, the opposite sides are of equal length. Given: AB || CD and AD || BC To prove: AB = CD, AD = BC Construction: Draw in the diagonal AC. Proof: consider ∆ABC and ∆CDA, AB II CD (given) AD II BC (given) ‫ס‬BAC = ‫ס‬DCA (alternate interior angles) ‫ס‬DAC = ‫ס‬BCA (alternate interior angles) And AC = AC (common side) So, ∆ABC ≅ ∆CDA (by ASA congruent rule) Hence, AB = CD (CPCT) And AD = BC (CPCT) Angles of a parallelogram Property: The adjacent angles in a parallelogram are supplementary. Given: ABCD is a parallelogram. To Prove: ‫ס‬A + ‫ס‬B = ‫ס‬B + ‫ס‬C = ‫ס‬C + ‫ס‬D = ‫ס‬D + ‫ס‬A = 180° Proof: ABCD is a parallelogram ‫ ׵‬AD||BC (Opposite sides of parallelogram are parallel) ֜ ‫ס‬DAB + ‫ס‬ABC = 180° (Sum of consecutive interior angles is supplementary) ‫ס׵‬A + ‫ס‬B = 180° Similarly, ‫ס‬B + ‫ס‬C = 180°, ‫ס‬C + ‫ס‬D = 180° and ‫ס‬D + ‫ס‬A = 180°. Thus, the adjacent angles of a parallelogram are supplementary. Diagonals of a parallelogram Property: The diagonals of a parallelogram bisect each other. Given: ABCD is a parallelogram. To Prove: OB = OD and OA = OC Proof: consider two triangles AOB and COB Then, ∠ (AOB) = ∠ (COD) (vertically opposite angles) Side AB = side CD (since, opposite sides of parallelogram are equal) And, ∠ (ABO) = ∠ (CDO) (since, AB parallel to CD and transversal BD intersects them So, ∠ (ABO) = ∠ (CDO) [alternate angles] Therefore, triangles AOB and COB are congruent (By AAS rule) Therefore OB = OD (By CPCT) And OA = OC (By CPCT) Therefore, diagonals of a parallelogram bisect each other. Some Special Parallelograms Rhombus • A rhombus is a quadrilateral with all sides having equal length. • The opposite sides of a rhombus are of equal length i.e., AB = DC, AD = BC. • The opposite angles are also equal i.e., ∠A = ∠C, ∠B = ∠D so it has all the properties of a parallelogram and of a kite. Property: Diagonals of a Rhombus bisect each other at right angles In the given rhombus, ABCD, AC and BD are the diagonals that meet at point O. It is observed that AO = CO, BO = DO and ∠DOC = ∠COB = ∠BOA = ∠AOD = 90°. Hence, Diagonals of a Rhombus bisect each other at right angles. A rectangle A quadrilateral with opposite sides equal that are AB = DC, AD = BC and all the four angles equal to right angle that are ∠A = ∠B = ∠C = ∠D = 90° is called rectangle. Property: Diagonals of a rectangle are equal and bisect each other. In the given rectangle ABCD, It can be observed that AO = CO, BO = DO, And AC = BD. So that diagonals of a rectangle are equal and bisect each other. A square A quadrilateral with all its four sides equal that are AB = BC = CD = DA and each of its four angles a right angle that are ∠A = ∠B = ∠C = ∠D = 90° is called a square. Property: The diagonals of a square are perpendicular bisectors of each other. Diagonals of a square are equal i.e. AC = BD, and perpendicular to each other i.e. AC ⊥ BD and bisects each other.”