Triangles

This chapter studies the congruence of triangles in depth. Two figures are congruent if they have exactly the same shape and size, and two triangles are congruent if their corresponding sides and angles are equal. You will learn to write congruence correctly, matching corresponding vertices, and use the abbreviation CPCT, corresponding parts of congruent triangles, to deduce equal parts. You will study the criteria for congruence: SAS, which is taken as an axiom; ASA, which is proved from it; AAS; SSS; and RHS for right triangles, where the hypotenuse and one side are equal. You will also learn why SSA and AAA are not valid criteria. Using these criteria, you will prove important properties of isosceles triangles: the angles opposite to equal sides of a triangle are equal, and conversely, the sides opposite to equal angles are equal. You will see that each angle of an equilateral triangle is 60 degrees, that the bisector of the vertical angle of an isosceles triangle bisects the base at right angles, and apply congruence to prove results in many geometric figures.

Detailed Worksheet: Triangles Section A - Definitions (10 marks) 1. What are congruent figures? When are two triangles congruent? (2 marks) 2. State the SAS congruence rule. Why is it taken as an axiom? (2 marks) 3. State the ASA and AAS congruence rules. (2 marks) 4. State the SSS and RHS congruence rules. (2 marks) 5. What does CPCT stand for? How is it used? (2 marks) Section B - Calculations and Applications (15 marks) 6. In triangle ABC, AB = AC and angle B = 70 degrees. Find angles C and A. (3 marks) 7. In an isosceles triangle, the vertex angle is 30 degrees more than each base angle. Find all the angles. (3 marks) 8. Triangle PQR is congruent to triangle XYZ. If PQ = 5 cm, QR = 7 cm, angle Q = 60 degrees and angle R = 50 degrees, find XY, YZ, angle Y, angle Z and angle X. (3 marks) 9. For each pair, state whether the triangles are congruent and name the rule, or say why not: (i) AB = PQ, BC = QR, angle B = angle Q (ii) angle A = angle P, angle B = angle Q, AB = PQ (iii) all three angles equal in both triangles. (3 marks) 10. In triangle ABC, AB = AC and the exterior angle at C is 115 degrees. Find all the angles of the triangle. (3 marks) Section C - Diagrams (10 marks) 11. Draw a figure in which AB and CD bisect each other at O. Prove that triangle AOC is congruent to triangle BOD and that AC is parallel to BD. (4 marks) 12. Draw an isosceles triangle ABC with AB = AC and AD the bisector of angle A meeting BC at D. Prove that BD = DC and AD is perpendicular to BC. (3 marks) 13. Draw a figure in which line l bisects angle A, and B is a point on l. BP and BQ are perpendiculars from B to the arms of angle A. Prove that BP = BQ. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Prove that the angles opposite to equal sides of a triangle are equal. State and prove its converse. (5 marks) 15. ABC is an isosceles triangle with AB = AC. BE and CF are the altitudes to AC and AB respectively. Prove that BE = CF. (5 marks) 16. ABC and DBC are two isosceles triangles on the same base BC with vertices A and D on the same side of BC. If AD is extended to meet BC at P, prove that triangle ABD is congruent to triangle ACD, triangle ABP is congruent to triangle ACP, and AP bisects angle A and BC. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw neat figures, mark equal parts and state the congruence rule used in every proof.