Chapter 7 — Triangles: Congruence Theorems

Section 7.1

Rigidity of Triangles

Triangles are the most "rigid" polygon — specifying three sides or two sides and an angle uniquely determines the triangle (unlike quadrilaterals which can flex). This is why triangles are used in bridges and trusses.

🔺 Why Triangles are Rigid

If you fix the three sides of a triangle, there is only ONE possible shape. You cannot deform it without changing a side length. This is NOT true for a square or rectangle — they can be pushed into parallelograms.

🔷 Quadrilaterals are NOT Rigid

A square with hinged joints can be pushed into a rhombus. A rectangle becomes a parallelogram. Quadrilaterals can change shape while keeping all side lengths the same. This is why bridges use triangular trusses, not square ones.

📌 What is Congruence?

Two figures are congruent (≅) if they have exactly the same shape and size — one can be placed exactly over the other (allowing flips, rotations, and translations). For triangles: △ABC ≅ △DEF means corresponding sides are equal AND corresponding angles are equal.

CPCT: Corresponding Parts of Congruent Triangles are equal (= CPCTC in some texts). After proving triangles congruent, you can use CPCT to conclude any corresponding part is equal.

Section 7.2 — The MOST Important Section

Congruence Criteria (Rules)

CBSE tests all five criteria. You must know: (1) the name, (2) exactly what information is needed, (3) when to use each. Memorise the mnemonics.

SAS

Side-Angle-Side

Two sides and the included angle (angle between the two sides) of one triangle equal the corresponding parts of another triangle.

The angle MUST be between the two sides

SSS

Side-Side-Side

All three sides of one triangle equal the corresponding three sides of another triangle. No angle information needed.

All 3 sides equal → congruent

ASA

Angle-Side-Angle

Two angles and the included side (side between the two angles) of one triangle equal the corresponding parts of another.

Side MUST be between the two angles

RHS

Right Angle-Hypotenuse-Side

In right-angled triangles: hypotenuse and one side of one triangle equal the hypotenuse and corresponding side of another right-angled triangle.

Only for right-angled triangles

AAS

Angle-Angle-Side

Two angles and one non-included side of one triangle equal corresponding parts of another. (The side is NOT between the two angles.)

AAS ≅ ASA + angle-sum theorem

SAS, SSS, and ASA — the three most commonly used congruence criteria in CBSE proofs

Criteria	What you need	Key condition	Tip / Mnemonic
SAS	2 sides + included ∠	Angle BETWEEN the sides	"Sandwich" — angle is sandwiched between sides
SSS	3 sides	All 3 sides equal	"Three strikes" — 3 sides proves congruence
ASA	2 angles + included side	Side BETWEEN the angles	"Meat in a sandwich" — side between angles
RHS	Right angle + hyp + 1 side	Must be right-angled triangle	"Right Hypotenuse Side" — only for right triangles
AAS	2 angles + non-included side	Side NOT between the angles	Same as ASA since 3rd angle = 180°−other two

⚠️

SSA and AAA do NOT work! Two sides and a non-included angle (SSA / ASS) is NOT a valid criterion — it can give two different triangles. Three equal angles (AAA) only ensures similarity, not congruence.

Section 7.3

Isosceles Triangle Theorem & Converse

One of the most tested theorems in CBSE. Both the theorem and its converse must be proved.

📐 Theorem 7.2 — Isosceles Triangle Theorem

If two sides of a triangle are equal, then the angles opposite to them are also equal.
i.e., In △ABC, if AB = AC, then ∠ABC = ∠ACB (base angles are equal)

Proof

Prove that base angles of an isosceles triangle are equal (AB = AC → ∠B = ∠C). [3 Marks]

1

Given: In △ABC, AB = AC. To prove: ∠ABC = ∠ACB

2

Construction: Draw AD, the bisector of ∠BAC, meeting BC at D.

3

In △ABD and △ACD:
AB = AC (given)
∠BAD = ∠CAD (AD bisects ∠A)
AD = AD (common)

4

∴ △ABD ≅ △ACD SAS

5

∴ ∠ABD = ∠ACD CPCT
i.e., ∠ABC = ∠ACB □

∠ABC = ∠ACB (proved by SAS congruence + CPCT)

📐 Theorem 7.3 — Converse of Isosceles Triangle Theorem

If two angles of a triangle are equal, then the sides opposite to them are also equal.
i.e., In △ABC, if ∠ABC = ∠ACB, then AB = AC

💡

Remember: Equal sides → equal opposite angles (Isosceles Theorem). Equal angles → equal opposite sides (Converse). Both proved by congruence. CPCT links the proof to the conclusion.

Section 7.4 — CBSE Exam Pattern Proofs

Standard Proof Techniques

Most CBSE 3–5 mark questions follow the same pattern: identify triangles → write Given, To Prove, Construction → match 3 parts → state criterion → use CPCT.

📋 The Standard Proof Template

1

Given: State all given information clearly.

2

To Prove: State what needs to be proved.

3

Construction: If needed, draw any auxiliary line (e.g., bisector, median, altitude).

4

Proof: In △XYZ and △PQR: write 3 equal pairs with reasons. Then state the congruence criterion.

5

Conclusion: State what follows by CPCT. QED/□

Example 1 · 3 Marks

In △ABC, D is the midpoint of BC. AD ⊥ BC. Prove that AB = AC (triangle is isosceles).

G

Given: D is midpoint of BC (BD=DC), AD⊥BC (∠ADB=∠ADC=90°)

TP

To Prove: AB = AC

P

In △ADB and △ADC:
AD = AD (common)
BD = DC (D is midpoint of BC)
∠ADB = ∠ADC = 90° (AD⊥BC)

C

∴ △ADB ≅ △ADC SAS

✓

∴ AB = AC CPCT □

AB = AC — proved. The triangle is isosceles.

Example 2 · 3 Marks

Prove that diagonals of a rectangle are equal. (Using congruence)

G

Given: ABCD is a rectangle. Diagonals AC and BD.

TP

To Prove: AC = BD

P

In △ABC and △DCB:
AB = DC (opposite sides of rectangle)
∠ABC = ∠DCB = 90° (angles of rectangle)
BC = BC (common)

C

∴ △ABC ≅ △DCB SAS

✓

∴ AC = DB CPCT □

Diagonals of a rectangle are equal — proved using SAS congruence.

Example 3 · 5 Marks

△ABC and △DBC are two isosceles triangles on the same base BC with vertices A and D on opposite sides. Prove that AD bisects ∠A and ∠D.

G

Given: △ABC: AB=AC. △DBC: DB=DC. A and D on opposite sides of BC.

TP

To Prove: AD bisects ∠BAC and ∠BDC

P1

In △ABD and △ACD:
AB = AC (△ABC isosceles)
DB = DC (△DBC isosceles)
AD = AD (common)

C1

∴ △ABD ≅ △ACD SSS
∴ ∠BAD = ∠CAD CPCT → AD bisects ∠A ✓

C2

Also ∠BDA = ∠CDA CPCT → AD bisects ∠D ✓ □

AD bisects both ∠BAC and ∠BDC — proved using SSS congruence + CPCT.

More Examples

Additional Worked Problems

Angle calculation and converse applications.

Example 4 · 2M

In △ABC, AB = AC and ∠B = 65°. Find ∠A.

1

AB = AC → ∠B = ∠C = 65° (isosceles triangle theorem)

2

∠A + ∠B + ∠C = 180° → ∠A + 65° + 65° = 180° → ∠A = 50°

∠A = 50°

Example 5 · 2M

In △PQR, ∠P = 80°, ∠Q = 50°. Find the longest side.

1

∠R = 180° − 80° − 50° = 50°

2

Largest angle = ∠P = 80°. The side opposite to ∠P is QR

3

Note: ∠Q = ∠R = 50° → PQ = PR (isosceles). QR is the base and longest side.

QR is the longest side (opposite to largest angle ∠P = 80°)

NCERT Exercise

Exercise 7.1 & 7.2 — Solved

Standard CBSE exercise questions with complete proof format.

Q1 · 2 Marks

In △ABC, ∠B = ∠C and D is the midpoint of BC. Show that AD ⊥ BC.

G

∠B=∠C → AB=AC (converse of isosceles), D=midpoint of BC so BD=DC

1

In △ABD and △ACD: AB=AC, BD=DC, AD=AD (common) → △ABD≅△ACD (SSS)

2

∴ ∠ADB=∠ADC (CPCT). Since ∠ADB+∠ADC=180° (linear pair) → each = 90°

✓

∴ AD ⊥ BC □

Q2 · 3 Marks

Line segment AB is bisected at C. Prove AC = CB using congruence. If AC = 5cm, find AB.

1

C bisects AB → AC = CB by definition of bisection. AB = AC + CB = 5 + 5 = 10 cm

AB = 10 cm

Q3 · 5 Marks

In △ABC, AB = AC. D is a point on BC. Prove that AD is the perpendicular bisector of BC iff BD = DC.

P1

If BD=DC (given D is midpoint), prove AD⊥BC:
In △ABD and △ACD: AB=AC, BD=DC, AD=AD → △ABD≅△ACD (SSS)
→ ∠ADB=∠ADC (CPCT) → each=90° → AD⊥BC ✓

P2

If AD⊥BC, prove BD=DC:
In △ABD and △ACD: AB=AC (given), ∠ADB=∠ADC=90° (AD⊥BC), AD=AD → △ABD≅△ACD (RHS)
→ BD=DC (CPCT) ✓ □

Both directions proved: AD⊥BC ⟺ BD=DC. AD is the perpendicular bisector of BC.

Smart Study

10 Study Tips for Triangles: Congruence

1

Name triangles in MATCHING order

△ABC ≅ △PQR means A↔P, B↔Q, C↔R. Always write vertices in the order of correspondence — wrong order = wrong CPCT.

2

SAS: angle MUST be included

Side-Angle-Side works only when the angle is between (included by) the two sides. SA-S with angle not between = SSA = INVALID.

3

CPCT is your conclusion tool

After stating congruence (e.g., △ABD≅△ACD by SAS), use CPCT to get corresponding angles/sides equal. Always write "by CPCT" explicitly.

4

RHS only for right triangles

Right Hypotenuse Side only applies when one angle is 90°. The right angle itself counts as the "R" part — you only need to state hyp and one side.

5

Construction is often necessary

Many proofs need you to draw a median, altitude, or angle bisector. State the construction clearly: "Draw AD, the bisector of ∠A, meeting BC at D."

6

Write proof in 5-step format

Given → To Prove → Construction → Proof (list 3 equal pairs with reasons) → Conclusion with CPCT. Missing any step costs marks.

7

Isosceles: sides equal ↔ base angles equal

AB=AC → ∠B=∠C (theorem). ∠B=∠C → AB=AC (converse). Both appear frequently. Know both directions.

8

Biggest side opposite biggest angle

In any triangle, the largest angle is opposite the longest side. Used to identify which side is largest given angles.

9

Sum of angles in a triangle = 180°

Use this frequently in congruence problems. If two angles are determined, the third = 180° − sum of the other two.

10

Exterior angle = sum of non-adjacent interior angles

Exterior angle theorem: exterior ∠ = sum of the two interior angles not adjacent to it. Very useful in angle calculation proofs.

Quick Reference

Chapter 7 — Summary Table

Criterion	Minimum info needed	Key constraint	Works for
SAS	2 sides + 1 angle	Angle between the two sides	All triangles
SSS	3 sides	All 3 sides equal	All triangles
ASA	2 angles + 1 side	Side between the two angles	All triangles
AAS	2 angles + 1 side	Side NOT between the angles	All triangles
RHS	Hypotenuse + 1 side	Right angle must exist	Right-angled triangles only
INVALID: SSA (or ASS) and AAA — do NOT prove congruence

CBSE Pattern Practice

50 Practice Questions

MCQ · 1 Mark · 2 Marks · 3 Marks · 5 Marks · Case-Based — all with complete solutions.

Triangles:Congruence Theorems