The Triangle and its Properties

A triangle is a closed figure made of three line segments. A median of a triangle joins a vertex to the midpoint of the opposite side, and an altitude is the perpendicular from a vertex to the opposite side or its extension. Every triangle has three medians, which meet at one point, and three altitudes. The angle sum property states that the three interior angles of a triangle always add up to 180 degrees. The exterior angle property states that an exterior angle is equal to the sum of its two interior opposite angles. You also study special triangles. An equilateral triangle has all sides equal and each angle 60 degrees, while an isosceles triangle has two equal sides and the angles opposite them are equal. The sum of the lengths of any two sides of a triangle is always greater than the third side. In a right-angled triangle, the Pythagoras property holds: the square on the hypotenuse equals the sum of the squares on the other two sides, as in a triangle with sides 3, 4 and 5 cm.

Detailed Worksheet: The Triangle and its Properties Section A - Definitions (10 marks) 1. Define a median and an altitude of a triangle. How many of each does a triangle have? (2 marks) 2. State the angle sum property of a triangle. (2 marks) 3. State the exterior angle property of a triangle with a labelled example. (2 marks) 4. What are the properties of the sides and angles of an isosceles triangle and an equilateral triangle? (2 marks) 5. State the Pythagoras property. In a right triangle, which side is the hypotenuse? (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the unknown angle in a triangle with angles (i) 50 degrees and 60 degrees (ii) 30 degrees and 90 degrees (iii) x, x and 70 degrees. (3 marks) 7. An exterior angle of a triangle is 110 degrees and one of its interior opposite angles is 45 degrees. Find the other interior opposite angle and the third angle of the triangle. (3 marks) 8. Is it possible to have a triangle with sides (i) 2 cm, 3 cm, 5 cm (ii) 3 cm, 6 cm, 7 cm (iii) 6 cm, 3 cm, 2 cm? Justify each. (3 marks) 9. Find the unknown side in a right-angled triangle: (i) legs 6 cm and 8 cm (ii) hypotenuse 25 cm and one leg 24 cm (iii) legs 5 cm and 12 cm. (3 marks) 10. A 15 m long ladder reaches a window 12 m high from the ground on placing it against a wall. Find the distance of the foot of the ladder from the wall. (3 marks) Section C - Diagrams (10 marks) 11. Draw a triangle ABC and construct (i) the median from A to BC (ii) the altitude from A to BC. Draw an obtuse triangle and show an altitude lying outside it. (4 marks) 12. Draw a triangle PQR with side QR extended to S. Mark the exterior angle PRS and label the interior opposite angles. If angle P = 40 degrees and angle Q = 65 degrees, find angle PRS. (3 marks) 13. Draw a right-angled triangle with legs 3 cm and 4 cm and draw squares on all three sides. Use the squares to show the Pythagoras property. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Answer with reasons: (i) Can a triangle have two right angles? (ii) Can a triangle have all angles less than 60 degrees? (iii) Can a triangle have all angles greater than 60 degrees? (iv) Can a triangle have two obtuse angles? (v) Can an exterior angle be smaller than an interior opposite angle? (5 marks) 15. The angles of a triangle are in the ratio 2 : 3 : 4. (i) Find each angle. (ii) What type of triangle is it by angles? (iii) Find the exterior angle at the vertex with the smallest angle. (iv) Verify the exterior angle property for it. (5 marks) 16. A tree is broken at a height of 5 m from the ground and its top touches the ground at a distance of 12 m from the base. (i) Draw a rough figure. (ii) Find the length of the broken part. (iii) Find the original height of the tree. (iv) Which property did you use? (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw a rough figure for every problem and write the property used.