Mathematics • Year 8 • Unit 3 • Lesson 19
Congruent Figures
Build fluency with the four congruence tests (SSS, SAS, AAS, RHS) and with writing congruence statements where vertex order matches corresponding parts. One worked example, one guided fill-in, then eight independent problems.
1. I do — fully worked example
Read every line. Each step has a short reason so you can see why, not just what.
Problem. Triangle ABC has AB = 5 cm, BC = 8 cm, AC = 10 cm. Triangle DEF has DE = 5 cm, EF = 8 cm, DF = 10 cm. Show the triangles are congruent and write the congruence statement.
Step 1 — Decide which test to use.
We are given all THREE pairs of sides → SSS.
Reason: SSS = all three pairs of corresponding sides equal. No angle info needed.
Step 2 — Match corresponding pairs.
AB = DE = 5 cm ✓
BC = EF = 8 cm ✓
AC = DF = 10 cm ✓
Reason: list the matches explicitly so the vertex order is locked in: A↔D, B↔E, C↔F.
Step 3 — State the test and write the conclusion.
All three pairs of sides equal, so by SSS: △ABC ≅ △DEF.
Reason: the vertex order in the statement matches the pairing from Step 2.
Step 4 — Free bonus: corresponding angles are now equal too.
∠A = ∠D, ∠B = ∠E, ∠C = ∠F.
Reason: once two triangles are proven congruent, every matching part is automatically equal.
Answer: △ABC ≅ △DEF (SSS).
2. We do — fill in the missing steps
Same shape as Section 1, with the working faded. Fill in each blank. 4 marks
Problem. Two right-angled triangles each have a right angle, a hypotenuse of 13 cm, and one leg of 5 cm. Prove the triangles are congruent.
Step 1 — Decide which test: we are given a right angle, the hypotenuse, and one other side. The matching test is ______ (3 letters).
Step 2 — List the equal parts:
Right angle: 90° = ______° ✓
Hypotenuse: 13 cm = ______ cm ✓
One other side: ______ cm = ______ cm ✓
Step 3 — Conclude:
By the ______ test, the two triangles are ______.
Step 4 — Why is the missing leg the same too? Using a² + b² = c² gives leg² = ______ − ______ = ______, so the other leg = ______ cm in both triangles.
3. You do — independent practice
Show working. The first four are foundation (just name the test). The middle two are standard (test + congruence statement). The last two are extension (use congruence to find unknowns).
Foundation — name the test
3.1 Triangle 1 has sides 7 cm, 9 cm, 11 cm. Triangle 2 has sides 7 cm, 9 cm, 11 cm. Which test proves congruence? 1 mark
3.2 Two triangles each have sides 6 cm and 9 cm with an included angle of 50° between them. Which test applies? 1 mark
3.3 △PQR and △XYZ have ∠P = ∠X, ∠Q = ∠Y, and PQ = XY. Which test applies? 1 mark
3.4 Two right-angled triangles each have a hypotenuse of 17 cm and one leg of 8 cm. Which test applies? 1 mark
Standard — name the test AND write the congruence statement
3.5 In △KLM and △XYZ: KL = XY = 6 cm, ∠L = ∠Y = 70°, LM = YZ = 9 cm. State the test and write the full congruence statement (in matching vertex order). 2 marks
3.6 △ABC has ∠A = 40°, ∠C = 75°, AC = 12 cm. △DEF has ∠D = 40°, ∠F = 75°, DF = 12 cm. State the test and write the congruence statement. 2 marks
Extension — use congruence to find unknowns
3.7 △ABC ≅ △DEF. Given BC = 7 cm and ∠A = 55°, find EF and ∠D. Explain which pairs correspond. 2 marks
3.8 Two students each draw a triangle with two sides 5 cm and 7 cm and a non-included angle of 40°. Their triangles look different. Explain (using the lesson) why this does NOT contradict the congruence tests, and name the "fake test" they have wrongly tried to use. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (RHS fill-in)
Step 1: test = RHS.
Step 2: Right angle 90° = 90° ✓; Hypotenuse 13 cm = 13 cm ✓; One other side 5 cm = 5 cm ✓.
Step 3: By the RHS test, the two triangles are congruent.
Step 4: leg² = 13² − 5² = 169 − 25 = 144, so the other leg = 12 cm in both triangles. (5-12-13 triple.)
3.1 — sides 7, 9, 11
All three pairs of sides equal, so the test is SSS.
3.2 — sides 6 and 9, included angle 50°
Two sides plus the angle BETWEEN them (the included angle) → SAS.
3.3 — two angles + corresponding side
∠P = ∠X, ∠Q = ∠Y, and PQ = XY (a corresponding side) → AAS.
3.4 — right angle + hypotenuse 17 + leg 8
Right-angled triangles with equal hypotenuse and one equal other side → RHS. (8-15-17 triple, so the third side is 15 cm in both.)
3.5 — KLM and XYZ (SAS)
KL = XY, ∠L = ∠Y (the included angle, between KL and LM), LM = YZ → test is SAS.
Vertex pairing: K↔X, L↔Y, M↔Z, so △KLM ≅ △XYZ.
3.6 — ABC and DEF (AAS)
∠A = ∠D, ∠C = ∠F, and side AC = DF (corresponding side) → test is AAS.
Vertex pairing: A↔D, B↔E, C↔F, so △ABC ≅ △DEF.
3.7 — △ABC ≅ △DEF
Vertex pairing from the statement: A↔D, B↔E, C↔F.
BC corresponds to EF, so EF = 7 cm.
∠A corresponds to ∠D, so ∠D = 55°.
Reason: once triangles are congruent, every matching side and angle is automatically equal.
3.8 — two students, "different" triangles
The students used two sides (5 and 7) plus a non-included angle (40°). That is SSA, which is NOT a valid congruence test — two genuinely different triangles can be built from the same SSA data (the "ambiguous case"). The angle must be the INCLUDED angle (between the two given sides) for SAS to apply. So the lesson is not contradicted: SSA was never a real test.