Grade 10 Congruency: Short Notes

10 ශ්‍රේණිය · 1 වාරය · 5 පරිච්ඡේදය · අංගසාම්‍යය

100%

of Grade 10 term tests include it (54 of 54)

724

marks in past papers (119 questions)

10 min

full lesson with quick checks

🎯 What you'll learn

Explain what congruent figures are and which parts are equal
Name congruent triangles with vertices in matching order
Use the four cases SAS, AAS, SSS and RHS
Write a short congruency proof with reasons
Use corresponding parts to find unknown sides and angles

The carpenter's three checks

A carpenter in Moratuwa is making 12 triangular roof trusses for a new house. They must all be exactly the same, or the roof will sag. He doesn't check every side and angle: he checks just three measurements. Which three are enough? That is what congruency tells us.

What 'congruent' means

Two figures are congruent if one fits exactly on top of the other: same shape and same size.Then all corresponding sides are equal, all corresponding angles are equal, and the areas are equal.We write △ABC ≡ △PQR (≡ means 'is congruent to').Same shape but different size is not congruent: that is 'similar'.

The order of letters matters

△ABC ≡ △PQR matches the letters in order:A ↔ P, B ↔ Q, C ↔ R.So the equal sides are AB = PQ, BC = QR, AC = PR, and the equal angles are ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R.When you name congruent triangles, write the vertices in matching order.

Four ways to be sure

If these marked parts of one triangle equal the same parts of another, the triangles are congruent: (1) SAS (2) AAS (3) SSS (4) RHS.

The four cases in words

• SAS: two sides and the angle between them (the included angle)• AAS (or ASA): two angles and one corresponding side• SSS: all three sides• RHS: both right-angled, with equal hypotenuses and one other pair of equal sidesThree matching facts are enough; you don't need all six.

⚠️ AAA and SSA don't work

Three equal angles only give the same shape: a small and a big triangle can share all angles. So AAA is not a case.In SAS the angle must be between the two sides. If the equal angle is not the included one, the triangles need not be congruent.

Writing a congruency proof

1. Name the two triangles.2. Write three pairs of equal parts, each with its reason: given, common side, vertically opposite angles, alternate angles, radii...3. State the case: '∴ △ABD ≡ △ACD (SSS)'.4. Use it: corresponding parts of congruent triangles are equal.

✏️ A kite-shaped frame

In quadrilateral PQRS, PQ = PS and RQ = RS. Diagonal PR is drawn. ∠PQR = 112°. Show that △PQR ≡ △PSR and find ∠PSR.

1In △PQR and △PSR: PQ = PS (given).
2RQ = RS (given), and PR is a common side.
3∴ △PQR ≡ △PSR (SSS).
4Q ↔ S, so ∠PSR = ∠PQR (corresponding angles) = 112°.

✅ ∠PSR = 112°

✏️ The line from the top of an isosceles triangle

AB = AC and D is the midpoint of BC. Prove △ABD ≡ △ACD and show that AD ⟂ BC.

1AB = AC (given); BD = DC (D is the midpoint); AD is common.
2∴ △ABD ≡ △ACD (SSS).
3So ∠ADB = ∠ADC (corresponding angles).
4They lie on straight line BC: ∠ADB + ∠ADC = 180°, so each is 90°.
5∴ AD ⟂ BC.

✅ △ABD ≡ △ACD (SSS), so AD ⟂ BC

⚠️ No reasons, no marks

Examiners report year after year that candidates write 'AB = AC' with no reason. Every equal pair needs one: (given), (common), (vertically opposite angles), (alternate angles, AB ∥ DC). Also write the case in brackets at the end, e.g. (AAS).

⚠️ Mixing up corresponding parts

A common slip is naming △ABO ≡ △CDO when O ↔ O, A ↔ C and B ↔ D are not in matching places, then reading off the wrong side. Before naming, match vertices with equal angles: the side opposite equal angles are corresponding sides.

📌 Remember

• Congruent = same shape and size; corresponding sides, angles and areas are equal.• Write △ABC ≡ △PQR with vertices in matching order.• Four cases: SAS (included angle), AAS/ASA, SSS, RHS. AAA is not one.• In a proof give a reason for every pair, then the case.• After proving congruence, use corresponding parts to find sides and angles.

🧾 Quick revision sheet

What does △ABC ≡ △PQR mean?	The triangles are congruent with A ↔ P, B ↔ Q, C ↔ R.
SAS	Two sides and the included angle equal.
AAS	Two angles and a corresponding side equal.
SSS	All three pairs of sides equal.
RHS	Right angle, equal hypotenuses and one more equal side.
Why is AAA not a congruency case?	Equal angles give the same shape but not always the same size.
Areas of congruent triangles?	Equal.
Three common reasons in proofs	Common side; vertically opposite angles; alternate angles (parallel lines).