Practical Geometry

Practical geometry is about drawing accurate figures using only a ruler and a compass, with a protractor where angles must be measured. In this chapter you first learn to draw a line parallel to a given line through a point outside it. The method uses the idea of equal alternate angles: you draw a transversal through the point, copy the angle it makes with the given line at the point, and the new line drawn through the point is parallel to the given line. The main part of the chapter is the construction of triangles from given measurements. A unique triangle can be drawn when you know three sides (SSS), two sides and the included angle (SAS), two angles and the included side (ASA), or the hypotenuse and one side of a right-angled triangle (RHS). Before constructing, you draw a rough sketch and mark the given parts. You also check whether the sides can form a triangle at all, since the sum of any two sides of a triangle must be greater than the third side.

Detailed Worksheet: Practical Geometry Section A - Definitions (10 marks) 1. Name the instruments in a geometry box needed for constructions and state the use of the compass. (2 marks) 2. What is meant by the included angle between two sides? Give an example using triangle PQR. (2 marks) 3. Write the full forms of SSS, SAS, ASA and RHS criteria for constructing a triangle. (2 marks) 4. Why should we draw a rough sketch before constructing a triangle? (2 marks) 5. State the property of the sides of a triangle that must be checked before constructing it from three sides. (2 marks) Section B - Calculations and Applications (15 marks) 6. Check whether a triangle can be constructed with sides (i) 3 cm, 4 cm, 5 cm (ii) 2 cm, 3 cm, 6 cm (iii) 5 cm, 5 cm, 10 cm. Give reasons. (3 marks) 7. In triangle ABC, angle A = 60 degrees and angle B = 70 degrees. Find angle C. Can a triangle be constructed with angles 90 degrees, 60 degrees and 40 degrees? Justify. (3 marks) 8. For each set of measurements, state which criterion you would use: (i) PQ = 5 cm, QR = 6 cm, angle Q = 50 degrees (ii) AB = 6 cm, angle A = 40 degrees, angle B = 60 degrees (iii) LM = 5 cm, MN = 4 cm, LN = 6 cm (iv) hypotenuse 6 cm, one side 4 cm, angle 90 degrees. (3 marks) 9. Can you construct a triangle with angle A = 110 degrees, angle B = 80 degrees and AB = 5 cm? Explain. Also, can you construct a unique triangle given three angles 50, 60 and 70 degrees? Explain. (3 marks) 10. In an isosceles triangle, the equal sides are 6 cm each and the angle between them is 40 degrees. Find the other two angles and state which criterion will be used to construct it. (3 marks) Section C - Diagrams (10 marks) 11. Draw a line l and a point A not on it. Construct a line through A parallel to l using a ruler and compass. Write the steps of construction. (4 marks) 12. Construct triangle XYZ with XY = 4.5 cm, YZ = 5 cm and ZX = 6 cm. (3 marks) 13. Construct triangle PQR with PQ = 3 cm, QR = 5.5 cm and angle PQR = 60 degrees. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Construct triangle ABC with BC = 7.5 cm, angle B = 30 degrees and angle C = 100 degrees. (i) Write the steps. (ii) Measure AB and AC. (iii) Find angle A by calculation and check by measuring. (iv) Which criterion was used? (5 marks) 15. Construct a right-angled triangle LMN with angle M = 90 degrees, hypotenuse LN = 5 cm and MN = 3 cm. (i) Write the steps. (ii) Measure LM. (iii) Verify that LN x LN = LM x LM + MN x MN using your measurement. (iv) Name the criterion. (5 marks) 16. Construct an equilateral triangle of side 5 cm. (i) Measure each angle. (ii) Through one vertex, draw a line parallel to the opposite side. (iii) Which pairs of alternate angles appear in your figure? (iv) Explain why the line you drew is parallel. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Use only a ruler and compass where asked, keep construction arcs visible, and draw a rough sketch first.