Mathematics • Year 7 • Unit 3 • Lesson 14
Congruent Figures — Mixed Challenge
All four tests together (SSS, SAS, AAS, RHS), the classic "AAA does NOT prove congruence" mistake, and an open-ended challenge to write two of your own valid congruence problems.
1. Mixed problems — pick the test and write the statement
For each pair, name the test (SSS / SAS / AAS / RHS) and write the congruence statement matching vertices correctly. 2 marks each
1.1 △ABC and △DEF: AB = DE = 6, BC = EF = 8, AC = DF = 10.
1.2 △PQR and △STU: PQ = ST = 5, ∠Q = ∠T = 70°, QR = TU = 9.
1.3 Right-angled △KLM (∠L = 90°) and right-angled △NOP (∠O = 90°): hypotenuses KM = NP = 17, KL = NO = 8.
1.4 △ABC and △DEF: ∠A = ∠D = 65°, ∠B = ∠E = 50°, BC = EF = 11.
1.5 △XYZ and △RST: XY = RS = 7, YZ = ST = 7, XZ = RT = 9. Identify the test AND state what kind of triangle each is (isosceles / scalene / equilateral).
1.6 △ABC and △DEF: AB = DE = 4, ∠A = ∠D = 90°, AC = DF = 3. (Note: the right angle IS at A and D — between sides AB and AC.)
2. Find the mistake
Exactly one step contains a mistake. Spot it, explain, then redo. 3 marks
Student's question: In △ABC and △DEF, ∠A = ∠D = 60°, ∠B = ∠E = 80°, ∠C = ∠F = 40°. Prove △ABC ≡ △DEF.
Step 1: List the equal parts.
Step 2: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F (three pairs of angles equal).
Step 3: Three angles equal → use the AAA test.
Step 4: ∴ △ABC ≡ △DEF (AAA).
(a) Which step contains the mistake?
(b) Explain in one or two sentences why that step is wrong.
(c) Write the corrected conclusion. (Hint: with only angles given, can you prove congruence at all? What MORE information would you need?)
Stuck? AAA is NOT one of the four tests. AAA proves SIMILAR, not congruent. To prove congruence you need at least one matching side.3. Open-ended challenge — author your own problems
You will be the question-setter. 4 marks
3.1 Invent TWO of your own pairs of congruent triangles — one that uses the SAS test and one that uses the RHS test. For each pair:
(i) State the two triangles (vertex labels of your choice).
(ii) Write down EXACTLY the three pieces of given information (with side lengths in cm and angles in degrees).
(iii) State which test applies and write a full congruence statement (with the test name in brackets).
(iv) Briefly explain WHY the given information matches that test (and not another).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Three sides match (SSS)
All three pairs of sides equal → SSS. ∴ △ABC ≡ △DEF (SSS).
1.2 — Two sides + included angle (SAS)
∠Q is between PQ and QR (included). Two sides + included angle → SAS. ∴ △PQR ≡ △STU (SAS).
1.3 — Right angle, hypotenuse, side (RHS)
Right angles equal (90° at L and O), hypotenuses equal (17), one other side equal (8) → RHS. ∴ △KLM ≡ △NOP (RHS).
1.4 — Two angles + matching side (AAS)
Two pairs of angles equal + one matching side (BC = EF) → AAS. ∴ △ABC ≡ △DEF (AAS).
1.5 — SSS, isosceles
All three sides match (7, 7, 9) → SSS. ∴ △XYZ ≡ △RST (SSS). Each triangle has two equal sides (7, 7) → isosceles.
1.6 — SAS at the right angle
AB = DE (4), ∠A = ∠D (90°, INCLUDED between AB and AC), AC = DF (3) → SAS. ∴ △ABC ≡ △DEF (SAS). (Both are right-angled triangles; you could also argue RHS once the hypotenuse BC = EF = 5 is established by Pythagoras, but SAS is the direct justification.)
2 — Find the mistake
(a) The mistake is on Step 3.
(b) AAA is NOT one of the four valid congruence tests (SSS, SAS, AAS, RHS). Three pairs of equal angles only prove the triangles are SIMILAR (same shape, possibly different size). You can have a small triangle and a giant triangle with the same three angles.
(c) Corrected conclusion: With only angles equal, the triangles are similar but NOT necessarily congruent. To prove congruence you would need at least ONE pair of corresponding SIDES equal as well (then you'd use AAS).
3 — Author your own (sample solutions)
SAS pair (sample): △ABC ≡ △PQR with AB = PQ = 6 cm, ∠B = ∠Q = 55°, BC = QR = 9 cm. ∠B is between AB and BC, so it's the included angle. Two sides + included angle → SAS. ∴ △ABC ≡ △PQR (SAS).
Why SAS not SSS/AAS/RHS: we only have two sides (not three, so not SSS), only one angle (not two, so not AAS), and no stated right angle (so not RHS).
RHS pair (sample): △DEF ≡ △XYZ with ∠E = ∠Y = 90°, hypotenuses DF = XZ = 25 cm, DE = XY = 7 cm. Right angle + hypotenuse + one other side → RHS. ∴ △DEF ≡ △XYZ (RHS).
Why RHS: both triangles are right-angled, the longest side (hypotenuse) is given equal, and one short side is given equal. The Pythagorean theorem forces the third side to be √(25² − 7²) = 24 cm in both, but RHS lets us skip that calculation.
Marking: 2 for a valid SAS pair (sides + included angle clearly identified); 2 for a valid RHS pair (right angle + hypotenuse + one side clearly identified). Many correct designs exist — accept any that genuinely fit the test.