MathsICSE Class 9 · Triangles: Congruency and Inequalities

Congruence Conditions, Writing a Proof and Isosceles Triangles (ICSE Class 9)

Updated 17 October 2026 · Fact-checked

Two triangles are congruent if SSS, SAS, ASA, AAS or RHS holds; AAA and SSA are not enough. In an isosceles triangle the angles opposite equal sides are equal, and conversely. Write each proof as: given, statement with reason, match the letters in order, conclusion by the rule, then CPCT (corresponding parts of congruent triangles).

Bisector AD splits the triangle into two SAS-congruent triangles, so CPCT gives the equal base angles B and C.

Illustration drawn for QuestionBuddy.

Understand Congruence Conditions, Writing a Proof and Isosceles Triangles

Congruence conditions. $△ A B C ≅ △ P QR$ if:

SSS: all three sides equal;
SAS: two sides and the included angle equal;
ASA: two angles and the included side equal;
AAS: two angles and a non-included side equal;
RHS: right angle, hypotenuse and one side equal (right-angled triangles only).

AAA gives similar (same shape), not congruent triangles. SSA (two sides and a non-included angle) is not a test in general.

Writing a proof. (1) Draw and mark equal parts. (2) In $△ A B C$ and $△ P QR$ , list pairs: $A B = P Q$ (reason), $∠ B = ∠ Q$ (reason), $B C = QR$ (reason). (3) Write the correspondence in matching order. (4) State " $△ A B C ≅ △ P QR$ by SAS". (5) Use CPCT to deduce further equalities.

Isosceles triangle theorem (prove). If $A B = A C$ then $∠ B = ∠ C$ . *Proof.* Let $A D$ bisect $∠ A$ , meeting $B C$ at $D$ . In $△ A B D$ and $△ A C D$ : $A B = A C$ , $∠ B A D = ∠ C A D$ , $A D$ common, so SAS gives $△ A B D ≅ △ A C D$ , hence $∠ B = ∠ C$ (CPCT). Converse: if $∠ B = ∠ C$ then $A B = A C$ (prove by AAS using the bisector of $∠ A$ ).

Results. In an isosceles triangle the bisector of the vertex angle is the perpendicular bisector of the base, and an exterior angle at the apex equals twice a base angle. An equilateral triangle has three $6 0^{\circ}$ angles.

Altitudes. In isosceles $△ A B C$ with $A B = A C$ , the altitudes $B D ⊥ A C$ and $C E ⊥ A B$ are equal (AAS on $△ B D C$ and $△ C E B$ ). Numerically, if $A B = A C = 10$ and $B C = 12$ , the altitude from $A$ is $8$ , area $= 48$ , so $B D = C E = \frac{2 \times 48}{10} = 9.6$ .

Level. Class IX is a school-exam class. One-step angle questions are routine; the altitude proof and the surveyor-style application are at the hardest school-exam level.

Congruence Conditions, Writing a Proof and Isosceles Triangles: formulas to remember

SAS: $A B = P Q, ∠ B = ∠ Q, B C = QR$; Included angle.
ASA / AAS: $two angles and a side$; Any side for AAS.
RHS: $∠ B = ∠ Q = 9 0^{\circ}, A C = P R, A B = P Q$; Right triangles.
Isosceles theorem: $A B = A C \Leftrightarrow ∠ B = ∠ C$; And its converse.
Apex exterior angle: $exterior angle at A = 2 \times base angle$; Isosceles triangle.

How to solve Congruence Conditions, Writing a Proof and Isosceles Triangles questions

Mark, list matching pairs with reasons, state the rule, then use CPCT.

1Draw the figure and put tick marks on equal sides and arcs on equal angles.
2Choose the two triangles and write them in corresponding order.
3List three pairs of equal parts, each with a reason (given, common, vertically opposite, radii, and so on).
4Name the rule (SSS, SAS, ASA, AAS, RHS).
5Write the congruence and use CPCT for the required equality.
6For numerical parts, use angle sums or Pythagoras in the congruent triangles.

Congruence Conditions, Writing a Proof and Isosceles Triangles in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Measuring across a river

Suppose a surveyor wants the distance $A B$ across a river and stands at $C$ with $A C = 30$ m and $B C = 40$ m, $∠ A C B = 9 0^{\circ}$ . She extends $A C$ to $D$ with $C D = C A$ and $B C$ to $E$ with $C E = C B$ on the dry side.

Show that $D E = A B$ and find it.

Try it first, then show the solutionSolution

In $△ A B C$ and $△ D E C$ : $A C = D C$ , $B C = E C$ and $∠ A C B = ∠ D C E$ (vertically opposite).
By SAS they are congruent, so $D E = A B$ (CPCT).
$A B = 3 0^{2} + 4 0^{2} = 50$ m, so $D E = 50$ m.

Answer: DE = AB = 50 m.

Memory hook: Mirror triangle on the dry side.

A symmetric roof frame

Imagine an isosceles roof triangle with equal rafters $A B = A C$ and apex angle $4 0^{\circ}$ .

Find the angles at the base.

Try it first, then show the solutionSolution

$∠ B = ∠ C = \frac{180 - 40}{2} = 7 0^{\circ}$ .

Answer: 70° each.

Memory hook: Equal sides, equal base angles.

Tricks to remember Congruence Conditions, Writing a Proof and Isosceles Triangles

Short memory aids. Say each one aloud once, then use it on a question the same day.

AAA is not enough
Three equal angles only give the same shape.
Use it when: Avoiding wrong reasons.
Match the order
Write ΔABC ≅ ΔPQR so that A↔P, B↔Q, C↔R.
Use it when: CPCT correctness.
Sides across, angles in
In an isosceles triangle, angles opposite equal sides are equal.
Use it when: Base angles.

Congruence Conditions, Writing a Proof and Isosceles Triangles: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Congruence tests · 1 mark · MCQBoard level

Two triangles have two pairs of equal sides and one pair of equal angles that is not between those sides. This information alone:
(a) proves congruence by SAS
(b) proves congruence by ASA
(c) does not prove congruence
(d) proves congruence by SSS

Try it first, then show the solutionSolution

SAS needs the angle to be included between the two sides; a non-included angle (SSA) does not fix the triangle.

Answer: (c)

Example 2: Isosceles angles · 2 marks · Short answerBoard level

In $△ A B C$ , $A B = A C$ and $∠ A = 4 0^{\circ}$ . Find $∠ B$ and $∠ C$ .

Try it first, then show the solutionSolution

$∠ B = ∠ C$ (angles opposite equal sides).
$∠ B + ∠ C = 180 - 40 = 140$ , so each is $7 0^{\circ}$ .

Answer: $7 0^{\circ}$ each

Example 3: Altitude of an isosceles triangle · 4 marks · Short answerHarder board level

In $△ A B C$ , $A B = A C = 13$ cm and $B C = 10$ cm. $D$ is the mid-point of $B C$ . Prove that $A D ⊥ B C$ and find $A D$ and the area.

Try it first, then show the solutionSolution

In $△ A B D$ and $△ A C D$ : $A B = A C$ , $B D = C D$ , $A D$ common; SSS gives $△ A B D ≅ △ A C D$ .
So $∠ A D B = ∠ A D C$ ; being a linear pair, each is $9 0^{\circ}$ .
$A D = 1 3^{2} - 5^{2} = 12$ cm.
Area $= \frac{1}{2} \times 10 \times 12 = 60$ cm².

Answer: Proved; AD = 12 cm; area 60 cm²

Example 4: Isosceles with algebraic angles · 3 marks · Short answerHarder board level

In $△ A B C$ , $A B = A C$ , $∠ B = (3 x - 10)^{\circ}$ and $∠ C = (2 x + 15)^{\circ}$ . Find $x$ and all three angles.

Try it first, then show the solutionSolution

$∠ B = ∠ C$ : $3 x - 10 = 2 x + 15$ , $x = 25$ .
$∠ B = ∠ C = 6 5^{\circ}$ .
$∠ A = 180 - 130 = 5 0^{\circ}$ .

Answer: $x = 25$ ; $6 5^{\circ}, 6 5^{\circ}, 5 0^{\circ}$

Example 5: Equal altitudes · 5 marks · Long answerHOTS / competency

In $△ A B C$ , $A B = A C$ . $B D ⊥ A C$ and $C E ⊥ A B$ . Prove that $B D = C E$ . If $A B = A C = 10$ cm and $B C = 12$ cm, find $B D$ .

Try it first, then show the solutionSolution

In $△ B D C$ and $△ C E B$ : $∠ B D C = ∠ C E B = 9 0^{\circ}$ ; $∠ D C B = ∠ E B C$ (angles opposite equal sides $A B = A C$ ); $B C$ is common.
By AAS, $△ B D C ≅ △ C E B$ , so $B D = C E$ (CPCT).
The altitude from $A$ to $B C$ : $1 0^{2} - 6^{2} = 8$ , so area $= \frac{1}{2} \times 12 \times 8 = 48$ cm².
Also area $= \frac{1}{2} \times A C \times B D$ , so $B D = \frac{2 \times 48}{10} = 9.6$ cm.

Answer: Proved; BD = 9.6 cm

Where students lose marks in Congruence Conditions, Writing a Proof and Isosceles Triangles

The slips examiners see most often, and what to write instead.

Mistake: Using SSA or AAA as a rule
These are not valid conditions.
Wrong working
ΔABC ≅ ΔPQR by AAA.
Right working
Use SSS, SAS, ASA, AAS or RHS.
Do this: Check the list.
Usually costs: Whole proof.
Mistake: Wrong order in the congruence statement
CPCT depends on matching letters.
Wrong working
$△ A B C ≅ △ QP R$
Right working
$△ A B C ≅ △ P QR$ when $A \leftrightarrow P$ , $B \leftrightarrow Q$ .
Do this: Match vertices.
Usually costs: Reason marks.
Mistake: No reason for each equality
Each equal pair needs a reason.
Wrong working
$A B = A C$ .
Right working
$A B = A C$ (given).
Do this: Write the reason in brackets.
Usually costs: Half the marks.

Exam tips

Class IX is examined in school papers; the isosceles theorem is a standard 4 mark proof.
Always state "by CPCT" when you deduce an equality from congruent triangles.
Use RHS only for right-angled triangles.
Mark equal parts with matching ticks and arcs.

Practice questions on Congruence Conditions, Writing a Proof and Isosceles Triangles

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
In $△ A B C$ and $△ P QR$ , $A B = P Q$ , $∠ B = ∠ Q$ and $B C = QR$ . The triangles are congruent by:
(a) SSS
(b) SAS
(c) ASA
(d) RHS
Hint
The angle is between the two sides.
Answer
(b) SAS
Q2 · 3 marks · Short answerHarder board level
In an isosceles triangle the exterior angle at the apex is $10 0^{\circ}$ . Find the base angles and the apex angle.
Hint
Exterior angle = sum of base angles.
Answer
Base angles sum to $10 0^{\circ}$ , so $5 0^{\circ}$ each; the apex angle is $180 - 100 = 8 0^{\circ}$ .
Q3 · 4 marks · Short answerHarder board level
Right triangles $A B C$ (right angle at $B$ ) and $P QR$ (right angle at $Q$ ) have $A C = P R = 13$ cm and $A B = P Q = 5$ cm. Prove them congruent and find $B C$ and the area.
Hint
RHS.
Answer
Right angle, hypotenuse $A C = P R$ and $A B = P Q$ : congruent by RHS. $B C = 169 - 25 = 12$ cm, area $= \frac{1}{2} \times 5 \times 12 = 30$ cm².
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): Two triangles with three pairs of equal angles need not be congruent.
Reason (R): AAA is not a congruence condition.
(a) Both (A) and (R) are true and (R) explains (A).
(b) Both are true but (R) does not explain (A).
(c) (A) is true, (R) is false.
(d) (A) is false, (R) is true.
Hint
Think of a big and a small equilateral triangle.
Answer
(a)
Q5 · 4 marks · Case-basedHOTS / competency
Case study: A surveyor at $C$ wants the width $A B$ of a river. She measures $A C = 30$ m and $B C = 40$ m with $∠ A C B = 9 0^{\circ}$ , then extends $A C$ to $D$ with $C D = C A$ and $B C$ to $E$ with $C E = C B$ .
(a) Name the congruence rule that makes $△ A B C ≅ △ D E C$ .
(b) What does CPCT give?
(c) Find $D E$ .
Hint
Vertically opposite angles at C.
Answer
(a) SAS (b) DE = AB (c) 50 m

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Name the five congruence conditions.Show answer
SSS, SAS, ASA, AAS and RHS.
2.Why is AAA not a congruence rule?Show answer
It only guarantees the same shape, not the same size.
3.State the isosceles triangle theorem.Show answer
Angles opposite equal sides are equal.
4.What does CPCT stand for?Show answer
Corresponding parts of congruent triangles.

Congruence Conditions, Writing a Proof and Isosceles Triangles: frequently asked questions

Do I need to prove the isosceles theorem?

Yes; the proof and its converse are part of the syllabus.

Is AAS in the syllabus?

It follows from ASA since the third angle is determined.

What is the difference between congruent and similar?

Congruent triangles have the same size and shape; similar triangles have the same shape.