1. Chapter at a Glance
Two ₹10 coins, two stamps from the same sheet, or the two blades of a pair of scissors — such objects have exactly the same shape and size. In geometry, figures that are exact copies of each other are called congruent ("geometric twins"). One can be placed on the other so that they cover each other completely. In this chapter you will learn what congruence means, how to match the parts of congruent figures, and the four short tests — SSS, SAS, ASA and RHS — that tell us when two triangles are congruent without checking every part.
- The meaning of congruence and the symbol ≅.
- Congruent line segments, angles and figures.
- Corresponding parts of congruent triangles.
- Congruence criteria: SSS, SAS, ASA (and AAS) and RHS.
- Why SSA and AAA are not valid tests.
- Using congruence to prove facts and solve problems.
2. What is Congruence?
Two figures are congruent if one can be moved (slid, turned or flipped) so that it fits exactly on the other. We write ≅ for "is congruent to".
- Line segments are congruent if they have the same length: AB ≅ CD when AB = CD.
- Angles are congruent if they have the same measure.
- Circles are congruent if they have the same radius.
- Squares are congruent if they have the same side.
Congruent figures have the same shape and the same size. Figures with the same shape but different sizes (like a photo and its enlargement) are similar, not congruent.
Are two rectangles 6 cm × 4 cm and 4 cm × 6 cm congruent?
Solution: Yes. Turning one through 90° makes it fit exactly on the other — it is the same rectangle in a different position.
3. Congruent Triangles and Corresponding Parts
Two triangles are congruent if all three sides and all three angles of one are equal to the matching sides and angles of the other. The matching is called a correspondence. If ΔABC ≅ ΔPQR, then A ↔ P, B ↔ Q, C ↔ R, and:
The order of the letters matters. Writing ΔABC ≅ ΔQPR would claim A matches Q, which may be wrong.
Once two triangles are known to be congruent, all their corresponding parts are equal. This is often written as CPCT — corresponding parts of congruent triangles.
4. Congruence Criteria
We don't need all six equalities. Certain three of them are enough to guarantee congruence, because those three already fix the triangle completely (as you saw in constructions).
| Criterion | What is equal | Why it works |
|---|---|---|
| SSS | All three sides | Three sides fix the triangle (arcs construction) |
| SAS | Two sides and the angle between them | The included angle fixes the third side |
| ASA | Two angles and the side between them | Rays from the ends of the side meet at one point |
| AAS | Two angles and a non-included side | The third angle is fixed by the angle sum, so it becomes ASA |
| RHS | Right angle, hypotenuse and one side | For right triangles only |
- AAA: equal angles only fix the shape, not the size (a small and a large equilateral triangle).
- SSA (two sides and a non-included angle): two different triangles may fit the same data, so it does not guarantee congruence. (RHS is the special case where it does work, because the angle is 90°.)
5. Using Congruence
Congruence lets us prove that lengths or angles are equal without measuring.
In ΔABC, AB = AC. Show that ∠B = ∠C.
Solution: Let D be the midpoint of BC and join AD. In ΔABD and ΔACD: AB = AC (given), BD = CD (D is the midpoint), AD = AD (common). So ΔABD ≅ ΔACD by SSS. Hence ∠B = ∠C (CPCT). This proves that the angles opposite equal sides are equal.
To find the width AB of a pond, a surveyor marks a point O, extends AO to C so that OC = OA, and BO to D so that OD = OB. Why is CD equal to AB?
Solution: In ΔAOB and ΔCOD: OA = OC, OB = OD, and ∠AOB = ∠COD (vertically opposite). So ΔAOB ≅ ΔCOD by SAS, and AB = CD (CPCT). Measuring CD on land gives the pond's width.
6. Solved Problems (NCERT Style)
In ΔDEF and ΔXYZ, DE = XY, ∠E = ∠Y and EF = YZ. Are they congruent? Which criterion?
Solution: ∠E lies between sides DE and EF, and ∠Y between XY and YZ. Two sides and the included angle match, so ΔDEF ≅ ΔXYZ by SAS.
ΔLMN ≅ ΔSTU. Write all the equal parts.
Solution: L ↔ S, M ↔ T, N ↔ U. Sides: LM = ST, MN = TU, NL = US. Angles: ∠L = ∠S, ∠M = ∠T, ∠N = ∠U.
Two right triangles each have hypotenuse 13 cm and one side 5 cm. Are they congruent?
Solution: Yes, by RHS (right angle, hypotenuse, side). Both third sides will be 12 cm.
In kite ABCD, AB = AD and CB = CD. Show that the diagonal AC bisects ∠A.
Solution: In ΔABC and ΔADC: AB = AD, CB = CD, AC common. So they are congruent by SSS, and ∠BAC = ∠DAC (CPCT). Hence AC bisects ∠A.
7. Master Formula Table
| Criterion | Condition |
|---|---|
| SSS | 3 sides equal |
| SAS | 2 sides + included angle |
| ASA | 2 angles + included side |
| AAS | 2 angles + a matching non-included side |
| RHS | right angle + hypotenuse + one side |
| Not valid | AAA, SSA |
| CPCT | congruent ⇒ all matching parts equal |
8. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Congruence | Same shape and size | Identify congruent figures |
| Correspondence | Order of vertices matters | Write equal parts |
| Criteria | SSS, SAS, ASA/AAS, RHS | Choose the right one |
| Invalid tests | AAA, SSA | Explain why |
| Applications | Prove equal lengths and angles | Isosceles, kite, surveying |
9. Mind Map
- Same shape and size
- Symbol ≅
- Correspondence
- CPCT
- SSS, SAS
- ASA, AAS
- RHS
- Isosceles base angles
- Distances across rivers
10. Tips and Tricks
- Mark equal sides with ticks and equal angles with arcs before choosing a criterion.
- Look for a common side shared by both triangles and for vertically opposite angles — free equal parts.
- In SAS, check that the angle is between the two sides.
- Writing the congruence with vertices in the wrong order.
- Using SSA or AAA as a congruence test.
- Confusing congruent (same size) with similar (same shape only).
11. Important One-liners
- Congruent figures have the same shape and size.
- Line segments are congruent if their lengths are equal.
- Angles are congruent if their measures are equal.
- In ΔABC ≅ ΔPQR, A matches P, B matches Q and C matches R.
- SSS, SAS, ASA, AAS and RHS are valid congruence criteria.
- AAA and SSA do not guarantee congruence.
- RHS applies only to right triangles.
- Corresponding parts of congruent triangles are equal (CPCT).
- Angles opposite equal sides of a triangle are equal.
12. Exam-pattern Questions with Detailed Answers
Two circles are congruent if they have equal:
- centres
- radii
- chords
- positions
Show answer
A circle's size depends only on its radius.
Answer: (B) radii
If ΔABC ≅ ΔXYZ, which side equals BC?
Show answer
B ↔ Y and C ↔ Z, so BC = YZ.
Answer: YZ
Which criterion is used when all three sides of two triangles are equal?
Show answer
Three sides → SSS.
Answer: SSS
In ΔPQR and ΔDEF, ∠P = ∠D, ∠Q = ∠E and PQ = DE. Are they congruent? State the criterion.
Show answer
PQ is between ∠P and ∠Q; DE is between ∠D and ∠E. Two angles and the included side → ASA.
Answer: Yes, by ASA (ΔPQR ≅ ΔDEF)
Two triangles have angles 50°, 60°, 70° each. Must they be congruent?
- Yes, by AAA
- No, they may differ in size
- Yes, by ASA
- Only if right-angled
Show answer
Equal angles give the same shape but not necessarily the same size. AAA is not a congruence criterion.
Answer: (B)
Lines AB and CD bisect each other at O. Prove that AC = BD.
Show answer
In ΔAOC and ΔBOD: OA = OB and OC = OD (bisect each other); ∠AOC = ∠BOD (vertically opposite). So ΔAOC ≅ ΔBOD by SAS, and AC = BD (CPCT).
Answer: Proved by SAS and CPCT
In right triangles ABC and DEF (right-angled at B and E), AC = DF = 10 cm and BC = EF = 6 cm. Find AB and DE.
Show answer
By RHS the triangles are congruent, so AB = DE. Using the right-angle relation (6, 8, 10), AB = DE = 8 cm.
Answer: 8 cm each
ΔABC ≅ ΔPQR with ∠A = 40° and ∠Q = 75°. Find ∠R and ∠C.
Show answer
∠P = ∠A = 40°, ∠B = ∠Q = 75°. ∠C = ∠R = 180° − 115° = 65°.
Answer: 65° each
P is a point equidistant from the ends A and B of a segment AB, and M is the midpoint of AB. Show that PM is perpendicular to AB.
Show answer
ΔPAM ≅ ΔPBM by SSS (PA = PB, AM = BM, PM common). So ∠PMA = ∠PMB. They form a linear pair (sum 180°), so each is 90°.
Answer: ∠PMA = ∠PMB = 90°
Explain with an example why SSA does not guarantee congruence.
Show answer
Take AB = 5 cm, ∠A = 30° and BC = 3 cm. Draw ∠A on a base line, mark B on the slanting arm, and draw an arc of radius 3 cm from B. The arc can cut the base line at two different points C₁ and C₂, giving two different triangles with the same SSA data.
Answer: Two different triangles can share the same SSA data
In quadrilateral ABCD, AB = CD and AD = CB. Prove that ∠B = ∠D.
Show answer
Draw diagonal AC. In ΔABC and ΔCDA: AB = CD, BC = DA, AC common. So ΔABC ≅ ΔCDA by SSS, hence ∠B = ∠D (CPCT).
Answer: Proved by SSS
Two ladders of equal length lean against two walls from points on the ground at equal distance from the walls. Are the heights they reach equal? Why?
Show answer
Each ladder, wall and ground form a right triangle with equal hypotenuses (ladders) and equal sides (distances from the walls). By RHS the triangles are congruent, so the heights are equal (CPCT).
Answer: Yes, by RHS
13. Quick Revision Summary
- Congruent = same shape and same size; written ≅.
- Write congruences in matching order; corresponding parts are equal (CPCT).
- Valid tests: SSS, SAS (included angle), ASA/AAS, RHS (right triangles).
- Invalid tests: AAA, SSA.
- Uses: proving isosceles base angles equal, perpendicular bisectors, distances across obstacles.