Quadrilaterals

This chapter studies the properties of quadrilaterals, especially parallelograms, and proves them using congruence of triangles. You will revise that the sum of the angles of a quadrilateral is 360 degrees and recall special types such as the trapezium, parallelogram, rectangle, rhombus, square and kite. You will prove that a diagonal of a parallelogram divides it into two congruent triangles, that in a parallelogram opposite sides are equal, opposite angles are equal and the diagonals bisect each other, and you will prove the converse of each result to identify a parallelogram. You will also prove that a quadrilateral is a parallelogram if one pair of opposite sides is equal and parallel. You will learn special properties of the diagonals: they are equal in a rectangle, perpendicular in a rhombus, and both equal and perpendicular in a square. Finally, you will study the Mid-point Theorem, which states that the line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it, along with its converse, and apply it to solve problems.

Detailed Worksheet: Quadrilaterals Section A - Definitions (10 marks) 1. Define a parallelogram and a trapezium. (2 marks) 2. What is a rhombus? How is it different from a square? (2 marks) 3. State the angle sum property of a quadrilateral. (2 marks) 4. State the Mid-point Theorem and its converse. (2 marks) 5. State two properties of the diagonals of a rectangle and two of a rhombus. (2 marks) Section B - Calculations and Applications (15 marks) 6. The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles. (3 marks) 7. In a parallelogram ABCD, angle A = (3x + 15) degrees and angle C = (5x - 25) degrees. Find x and all the angles of the parallelogram. (3 marks) 8. The diagonals of a rhombus are 16 cm and 12 cm. Find the length of each side of the rhombus. (3 marks) 9. In triangle ABC, D and E are the mid-points of AB and AC. If BC = 14 cm, AB = 10 cm and AC = 12 cm, find DE and the perimeter of triangle ADE. (3 marks) 10. Two adjacent angles of a parallelogram are in the ratio 2 : 3. Find all its angles. (3 marks) Section C - Diagrams (10 marks) 11. Draw a parallelogram ABCD with diagonal AC. Prove that triangle ABC is congruent to triangle CDA and hence that opposite sides are equal. (4 marks) 12. Draw a triangle and join the mid-points of its three sides. Show that the four triangles formed are congruent. (3 marks) 13. Draw a rhombus with its diagonals and mark the right angles at their intersection. Explain why the diagonals are perpendicular. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Prove that the diagonals of a parallelogram bisect each other. Prove also that if the diagonals of a quadrilateral bisect each other, it is a parallelogram. (5 marks) 15. ABCD is a quadrilateral and P, Q, R and S are the mid-points of AB, BC, CD and DA. Prove that PQRS is a parallelogram. If ABCD is a rhombus, show that PQRS is a rectangle. (5 marks) 16. Prove that if the diagonals of a parallelogram are equal, it is a rectangle. If the diagonals of a quadrilateral are equal and bisect each other at right angles, prove that it is a square. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw neat figures and give reasons for every step of each proof.