Understanding Quadrilaterals

A polygon is a simple closed curve made up only of line segments. In this chapter you will classify polygons by their number of sides, distinguish convex polygons, in which no part of any diagonal lies outside, from concave ones, and identify regular polygons, which are both equilateral and equiangular. By dividing a polygon into triangles from one vertex, you will discover that the sum of the interior angles of an n-sided polygon is (n - 2) x 180 degrees, and that the sum of the exterior angles of any polygon, taken one at each vertex, is always 360 degrees. The second half studies the kinds of quadrilaterals: trapezium, kite, parallelogram, rhombus, rectangle and square. You will learn and use the properties of a parallelogram, namely that opposite sides and opposite angles are equal, adjacent angles are supplementary and the diagonals bisect each other, and the special properties of the diagonals of a rhombus, a rectangle and a square.

Detailed Worksheet: Understanding Quadrilaterals Section A - Definitions (10 marks) 1. What is a polygon? Distinguish between a convex polygon and a concave polygon. (2 marks) 2. What is a regular polygon? Give one example of a polygon that is equilateral but not regular. (2 marks) 3. Write the angle sum property of a quadrilateral and the formula for the sum of the interior angles of a polygon with n sides. (2 marks) 4. What is the sum of the measures of the exterior angles of any polygon? Does it depend on the number of sides? (2 marks) 5. Define a trapezium, a kite and a parallelogram. (2 marks) Section B - Calculations and Applications (15 marks) 6. (i) Find the sum of the interior angles of a polygon with 7 sides. (ii) Find each interior angle of a regular hexagon. (iii) Three angles of a quadrilateral are 50, 130 and 120 degrees; find the fourth angle. (3 marks) 7. (i) How many sides does a regular polygon have if each exterior angle is 45 degrees? (ii) If each exterior angle is 24 degrees? (iii) Is it possible to have a regular polygon with each exterior angle of 22 degrees? Give a reason. (3 marks) 8. (i) In parallelogram RING, angle R = 70 degrees. Find the other three angles. (ii) Two adjacent angles of a parallelogram are in the ratio 3 : 2. Find all the angles of the parallelogram. (3 marks) 9. The diagonals of parallelogram ABCD meet at O. If OA = 2y + 3 cm and OC = 3y - 1 cm, find y and the length of diagonal AC. Which property did you use? (3 marks) 10. (i) The diagonals of a rhombus are 16 cm and 12 cm. Find its side and perimeter. (ii) The diagonals of rectangle PQRS meet at O. If OP = 6.5 cm, find the length of diagonal QS. (3 marks) Section C - Diagrams (10 marks) 11. Draw a convex pentagon and a concave pentagon. Draw all the diagonals in each and show the diagonal that lies outside the concave pentagon. (4 marks) 12. Draw a hexagon and divide it into triangles by drawing all the diagonals from one vertex. Use your diagram to find the sum of the interior angles of a hexagon. (3 marks) 13. Draw a diagram showing how the parallelogram, rectangle, rhombus and square are related, with a trapezium and a kite placed correctly. Write one property next to each shape. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Show that the sum of the exterior angles of a polygon with n sides is 360 degrees, using the fact that each interior and exterior angle pair at a vertex adds to 180 degrees. Find the number of sides of a regular polygon each of whose interior angles is 165 degrees. Can a regular polygon have each interior angle equal to 100 degrees? (5 marks) 15. State whether true or false, giving a reason: (i) All rectangles are squares. (ii) All rhombuses are parallelograms. (iii) All squares are rhombuses and also rectangles. (iv) All kites are rhombuses. (v) All rhombuses are kites. (5 marks) 16. Explain what kind of parallelogram is obtained when (i) its diagonals are equal (ii) its diagonals are perpendicular (iii) its diagonals are equal and perpendicular. In rectangle PQRS, the diagonals meet at O and angle POQ = 110 degrees. Find angle OPQ, giving reasons. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Give a reason for every angle you calculate and draw all figures neatly with a ruler.