Mathematics • Year 7 • Unit 3 • Lesson 14
Introduction to Congruent Figures
Two triangles are congruent when they are the same shape AND the same size. The four tests — SSS, SAS, AAS, RHS — give you a recipe for proving it. Spot which test applies, list the equal parts, write the congruence statement.
1. I do — fully worked example
This is the standard Year 7 layout: list the equal parts, name the test, write the congruence statement.
Problem. In △ABC and △DEF: AB = DE = 7 cm, BC = EF = 5 cm, and ∠B = ∠E = 60°. Prove the triangles are congruent. State the test used.
Step 1 — List the equal parts in order.
AB = DE (side, 7 cm)
∠B = ∠E (angle, 60°)
BC = EF (side, 5 cm)
Reason: write Side, Angle, Side in the order they sit around the triangle.
Step 2 — Check the angle's position.
∠B sits BETWEEN sides AB and BC → it is the INCLUDED angle.
Reason: SAS only works if the angle is the one between the two given sides.
Step 3 — Name the test.
Two sides + included angle equal → SAS.
Step 4 — Write the congruence statement.
∴ △ABC ≡ △DEF (SAS)
Answer: △ABC ≡ △DEF (SAS).
2. We do — fill in the missing steps
An AAS problem. Fill in the blanks. 4 marks
Problem. In △PQR and △XYZ: ∠P = ∠X = 50°, ∠Q = ∠Y = 70°, PQ = XY = 8 cm. Prove the triangles are congruent.
Step 1 — List the equal parts:
∠P = ∠X (____°), ∠Q = ∠Y (____°), PQ = XY (____ cm).
Step 2 — Count what type of parts are equal: _____ angles + _____ side.
Step 3 — Name the test: ______ (one of SSS, SAS, AAS, RHS).
Step 4 — Write the congruence statement:
∴ △PQR ≡ △________ (______)
Step 5 — Match the vertices: P ↔ ____, Q ↔ ____, R ↔ ____.
3. You do — independent practice
For each pair: name the test (SSS / SAS / AAS / RHS) and write the congruence statement matching vertices correctly.
Foundation — identify the test
3.1 In △ABC and △DEF: AB = DE = 5 cm, BC = EF = 6 cm, AC = DF = 7 cm. Which test? Write the congruence statement. 1 mark
3.2 In △LMN and △RST: LM = RS = 4 cm, ∠M = ∠S = 90°, MN = ST = 3 cm. Which test? Write the congruence statement. 1 mark
3.3 Two right-angled triangles △KLM and △NOP: right angles at L and O. Hypotenuses KM = NP = 13 cm. LM = OP = 5 cm. Which test? Write the congruence statement. 1 mark
3.4 In △UVW and △XYZ: ∠U = ∠X = 40°, ∠V = ∠Y = 75°, UV = XY = 6 cm. Which test? Write the congruence statement. 1 mark
Standard — list and justify
3.5 In △ABC and △PQR: AB = PQ = 9 cm, BC = QR = 12 cm, ∠B = ∠Q = 50°. List the three equal parts in order, identify the included angle, name the test, and write the congruence statement. 2 marks
3.6 Triangles △DEF and △LMN have DE = LM = 7 cm, ∠D = ∠L = 35°, ∠F = ∠N = 75°. (a) Show the third angles are also equal. (b) Which test? (c) Write the congruence statement. 2 marks
Extension — spot the test that DOESN'T work
3.7 In △ABC and △DEF: AB = DE = 8 cm, ∠B = ∠E = 50°, AC = DF = 6 cm. (Note: the angle is NOT between AB and AC — it's at B, between AB and BC.) Explain why SSA (side-side-angle, not included) is NOT a valid congruence test, and what extra information would make it valid. 3 marks
3.8 Two triangles have all three pairs of angles equal: ∠A = ∠D = 45°, ∠B = ∠E = 60°, ∠C = ∠F = 75°. Are they necessarily congruent? Explain. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (△PQR ≡ △XYZ)
Step 1: ∠P = ∠X (50°), ∠Q = ∠Y (70°), PQ = XY (8 cm).
Step 2: two angles + one side.
Step 3: AAS.
Step 4: ∴ △PQR ≡ △XYZ (AAS).
Step 5: P ↔ X, Q ↔ Y, R ↔ Z.
3.1 — Three sides match
Three pairs of sides equal → SSS. ∴ △ABC ≡ △DEF (SSS).
3.2 — Two sides + included angle
∠M is between LM and MN → included. Two sides + included angle → SAS. ∴ △LMN ≡ △RST (SAS).
3.3 — Right angle, hypotenuse, side
Right angles equal (90°), hypotenuses equal (13 cm), one other side equal (5 cm) → RHS. ∴ △KLM ≡ △NOP (RHS).
3.4 — Two angles + corresponding side
Two angles + side opposite/adjacent to them → AAS. ∴ △UVW ≡ △XYZ (AAS).
3.5 — SAS list and statement
AB = PQ (9 cm, side), ∠B = ∠Q (50°, angle), BC = QR (12 cm, side). ∠B is between AB and BC → INCLUDED. Test: SAS. ∴ △ABC ≡ △PQR (SAS).
3.6 — Show third angle equal then AAS
(a) Angles in a triangle sum to 180°. In △DEF: ∠E = 180 − 35 − 75 = 70°. In △LMN: ∠M = 180 − 35 − 75 = 70°. So ∠E = ∠M = 70°. ✓
(b) Now we have ∠D = ∠L, ∠F = ∠N, and DE = LM. Two angles + a matching side → AAS.
(c) ∴ △DEF ≡ △LMN (AAS).
3.7 — SSA is not a valid test
The given sides (8 cm and 6 cm) and the angle 50° at B form an SSA arrangement — the angle is NOT between the two given sides. SSA is not a valid congruence test in general: two different triangles can be drawn with those measurements (the "ambiguous case" / swinging-arm problem) — the 6 cm side could swing to two positions. To make a valid test, you would need either (i) the angle to be the INCLUDED angle (making it SAS), OR (ii) a right-angled triangle plus a hypotenuse (making it RHS).
3.8 — AAA is not enough
No. AAA (three angles equal) proves the triangles are SIMILAR (same shape) but not necessarily CONGRUENT — one could be a scaled-up copy of the other. To prove congruence you also need at least one pair of corresponding SIDES equal. (Then you'd use AAS.)