Congruence of Triangles

Two figures are congruent if they have exactly the same shape and size, so that one can be placed over the other to cover it exactly. Examples around us include two identical stamps, coins of the same value and keys made from the same mould. In this chapter you will learn that two line segments are congruent if they have the same length, and two angles are congruent if they have the same measure. For triangles, congruence depends on a correct correspondence of vertices: if triangle ABC is congruent to triangle PQR, then A corresponds to P, B to Q and C to R, and the corresponding sides and angles are equal. You will study the four criteria for congruence of triangles: SSS (three sides equal), SAS (two sides and the included angle equal), ASA (two angles and the included side equal) and RHS (in right-angled triangles, the hypotenuse and one side equal). You will also see why AAA is not a criterion, since triangles with equal angles can have different sizes.

Detailed Worksheet: Congruence of Triangles Section A - Definitions (10 marks) 1. What are congruent figures? Give two examples from daily life. (2 marks) 2. When are two line segments congruent? When are two angles congruent? (2 marks) 3. What is meant by correspondence of vertices in congruent triangles? (2 marks) 4. State the SSS and SAS congruence criteria. (2 marks) 5. State the ASA and RHS congruence criteria. (2 marks) Section B - Calculations and Applications (15 marks) 6. Triangle ABC is congruent to triangle FED. (i) Write all the pairs of corresponding vertices. (ii) Write the pairs of corresponding sides. (iii) If angle A = 50 degrees and angle B = 60 degrees, find angle F, angle E and angle D. (3 marks) 7. In triangles PQR and XYZ, PQ = 4 cm, QR = 5 cm, angle Q = 40 degrees, XY = 4 cm, YZ = 5 cm and angle Y = 40 degrees. (i) Are the triangles congruent? (ii) Name the criterion. (iii) Write the congruence correctly and find XZ if PR = 3.2 cm. (3 marks) 8. In right triangles ABC and PQR, angle B = angle Q = 90 degrees, AC = PR = 13 cm and BC = QR = 5 cm. (i) Name the criterion for congruence. (ii) Write the congruence statement. (iii) Find AB if PQ = 12 cm. (3 marks) 9. In triangle DEF and triangle LMN, angle D = 60 degrees, angle E = 70 degrees, DE = 6 cm, angle L = 60 degrees, angle M = 70 degrees and LM = 6 cm. (i) Are they congruent? By which criterion? (ii) Find angle F and angle N. (iii) Which side of triangle LMN equals EF? (3 marks) 10. Which congruence criterion would you use in each case? (i) All three sides of one triangle equal the three sides of another. (ii) Two sides and the angle between them are equal. (iii) Two angles and the side between them are equal. Give one reason why AAA is not a congruence criterion. (3 marks) Section C - Diagrams (10 marks) 11. Draw two triangles ABC and PQR with AB = PQ = 5 cm, BC = QR = 6 cm and CA = RP = 7 cm. Mark equal sides with the same marks, write the congruence statement and name the criterion. (4 marks) 12. Draw a figure in which AB and CD are two line segments bisecting each other at O. Show that triangle AOC is congruent to triangle BOD and name the criterion. (3 marks) 13. Draw an isosceles triangle ABC with AB = AC and a line AD from A to the midpoint D of BC. Show that triangle ABD is congruent to triangle ACD, and state the criterion. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. In quadrilateral ABCD, AB = AD and AC bisects angle A. (i) Draw the figure. (ii) Show that triangle ABC is congruent to triangle ADC. (iii) Name the criterion. (iv) What can you say about BC and CD? (5 marks) 15. Two triangles have all three angles equal: 50, 60 and 70 degrees. One has sides 3 cm, 3.5 cm and 4 cm and the other has sides 6 cm, 7 cm and 8 cm. (i) Are they congruent? (ii) Why not? (iii) What does this show about AAA? (iv) What do such triangles have in common? (5 marks) 16. To measure the width of a river, a student uses triangles. She stands at A opposite a tree T on the far bank, walks along the bank to B, places a stick at the midpoint M, and then walks from B at right angles until she sees M and T in a line from point C. (i) Draw the figure. (ii) Which two triangles are congruent? (iii) By which criterion? (iv) Which length on her side equals the river's width AT? (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Write congruence statements with vertices in the correct order and mark equal parts in every figure.