Two triangles are congruent when one can be placed exactly on top of the other — same shape, same size. This chapter teaches you the shortcut rules (SSS, SAS, ASA, AAS, RHS) that prove congruence without measuring every part!
Congruent figures
Figures with exactly the same shape and the same size.
CPCT
Corresponding Parts of Congruent Triangles are equal.
Criteria
SSS, SAS, ASA, AAS and RHS prove congruence.
Inequalities
Bigger angle is opposite the longer side — and vice versa.
1. What does congruent mean?
Two geometrical figures are said to be congruent if they have exactly the same shape and the same size. If you cut out one figure and place it over the other, they cover each other completely. We write congruence with the symbol ≅. So △ABC ≅ △PQR is read as “triangle ABC is congruent to triangle PQR”. Two line segments are congruent if they have equal length; two circles are congruent if they have equal radii; two angles are congruent if they have equal measure.
2. Correspondence matters
When we say △ABC ≅ △PQR, the order of letters is not random. It tells us exactly which parts match: A ↔ P, B ↔ Q, C ↔ R. This means side AB = PQ, BC = QR, CA = RP, and angle A = angle P, angle B = angle Q, angle C = angle R. Writing △ABC ≅ △QRP would be a different (usually wrong) claim. Always match vertices in the correct order before listing equal parts.
3. CPCT — the most useful tool
Once two triangles are proved congruent, CPCT (Corresponding Parts of Congruent Triangles) lets you instantly say that every remaining pair of corresponding sides and angles is equal. In most exam proofs you first prove the triangles congruent using a criterion, and then quote CPCT to get the side or angle you actually want.
4. SAS criterion (Side-Angle-Side)
If two sides and the included angle (the angle between those two sides) of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent. This is taken as a basic axiom in the NCERT book. The word included is crucial — the angle must lie between the two equal sides, not anywhere else.
5. ASA and AAS criteria
ASA (Angle-Side-Angle): if two angles and the included side of one triangle equal two angles and the included side of another, the triangles are congruent. AAS (Angle-Angle-Side): if two angles and a non-included side match, the triangles are still congruent — because once two angles are equal, the third is automatically equal (angle sum = 180°), so any one side fixes the size.
6. SSS criterion (Side-Side-Side)
If all three sides of one triangle are equal to the three sides of another triangle, the two triangles are congruent. A triangle is rigid — once its three side lengths are fixed, its shape cannot change. This is why triangular frames are used in bridges and towers.
7. RHS criterion (Right angle-Hypotenuse-Side)
In two right-angled triangles, if the hypotenuse and one side of one triangle equal the hypotenuse and one side of the other, the triangles are congruent. RHS works only when both triangles have a right angle.
8. Properties of an isosceles triangle
An isosceles triangle has two equal sides. Two key theorems: (i) Angles opposite the equal sides are equal. (ii) The converse — sides opposite equal angles are equal. These are proved using the congruence criteria and CPCT, and they appear constantly in problems.
9. Inequalities in a triangle
Three important results: (i) If two sides are unequal, the angle opposite the longer side is larger. (ii) Conversely, the side opposite the larger angle is longer. (iii) The triangle inequality: the sum of the lengths of any two sides of a triangle is always greater than the third side. This is why you cannot build a triangle from sides 2 cm, 3 cm and 6 cm — the two shorter sides cannot reach across.
10. Why AAA and SSA do NOT work
Equal angles only (AAA) give similar triangles, not congruent ones — they may have the same shape but differ in size, like a small photo and its enlargement. SSA (two sides and a non-included angle) is also unreliable: with the same two sides and that angle you can sometimes draw two genuinely different triangles, so it is not accepted as a valid congruence rule. Always stick to the five proven criteria.
11. How to write a congruence proof
A neat proof follows a fixed pattern. First name the two triangles you are comparing. Then list the three equal parts one below the other, writing the reason for each (given, common, vertically opposite, construction, and so on). Next state the criterion you have satisfied and conclude that the triangles are congruent. Finally, use CPCT to bring out the side or angle the question actually asks for. Keeping this order makes your answer clear and earns full marks in the examination.
- SAS: two sides + included angle equal → congruent.
- ASA: two angles + included side equal → congruent.
- AAS: two angles + any one side equal → congruent.
- SSS: all three sides equal → congruent.
- RHS: right angle + hypotenuse + one side → congruent.
- Angle sum of a triangle = 180°.
- Triangle inequality: sum of any two sides > third side.
- Longer side is opposite the larger angle.
In quadrilateral ACBD, AC = AD and AB bisects angle A. Show that △ABC ≅ △ABD, and hence that BC = BD.
- In △ABC and △ABD: AC = AD (given).
- angle CAB = angle DAB (AB bisects angle A, given).
- AB = AB (common side to both triangles).
- Two sides and the included angle are equal, so by SAS, △ABC ≅ △ABD.
- Therefore BC = BD by CPCT.
ABC is an isosceles triangle with AB = AC. The bisector of angle B and the bisector of angle C meet BC-line at points giving BO = CO style setup. Simpler: in △ABC, AB = AC. Show that angle B = angle C.
- Draw AD, the bisector of angle A, meeting BC at D. So angle BAD = angle CAD.
- In △ABD and △ACD: AB = AC (given).
- angle BAD = angle CAD (by construction).
- AD = AD (common side).
- By SAS, △ABD ≅ △ACD.
- Hence angle ABD = angle ACD by CPCT, that is angle B = angle C.
Remember the valid rules with the phrase “Some Stupid Apes Always Run” → SSS, SAS, ASA, AAS, RHS. And remember the two fakers: AAA and SSA are NOT congruence rules!
The biggest mistake is using SAS with a non-included angle, or writing the congruence statement with vertices in the wrong order. Always check that the equal angle lies between the two equal sides for SAS, and match corresponding vertices carefully before quoting CPCT.
Q1. AD and BC are equal perpendiculars to a line segment AB. Show that CD bisects AB (they meet at O).
Answer: In △BOC and △AOD: angle BOC = angle AOD (vertically opposite angles); angle CBO = angle DAO = 90° (given perpendiculars); BC = AD (given). By AAS, △BOC ≅ △AOD. Therefore BO = AO by CPCT, which means O is the midpoint of AB, so CD bisects AB.
Q2. Line l is the bisector of an angle A, and B is any point on l. BP and BQ are perpendiculars from B to the arms of the angle. Show that BP = BQ.
Answer: In △APB and △AQB: angle PAB = angle QAB (l bisects angle A); angle APB = angle AQB = 90° (perpendiculars); AB = AB (common). By AAS, △APB ≅ △AQB. Hence BP = BQ by CPCT — every point on an angle bisector is equidistant from the two arms.
Q3. In △ABC, AB = AC and the perpendicular AD is drawn from A to BC. Show that BD = DC.
Answer: In right triangles △ABD and △ACD: angle ADB = angle ADC = 90° (AD ⊥ BC); hypotenuse AB = AC (given); AD = AD (common). By RHS, △ABD ≅ △ACD. Therefore BD = DC by CPCT, so AD bisects BC.
Q4. Can a triangle have sides 4 cm, 5 cm and 10 cm? Justify using the triangle inequality.
Answer: Check the sum of the two shorter sides against the longest side: 4 + 5 = 9 cm, which is less than 10 cm. The triangle inequality requires the sum of any two sides to be greater than the third side. Since 9 < 10, the two shorter sides cannot meet, so no such triangle exists.
- ✅ Congruent = same shape AND same size; symbol ≅.
- ✅ Valid criteria: SSS, SAS, ASA, AAS, RHS. AAA and SSA are not valid.
- ✅ Use CPCT after proving congruence to get extra equal parts.
- ✅ Isosceles: equal sides → equal opposite angles, and converse.
- ✅ Triangle inequality: sum of any two sides > the third side.
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