Congruence of Triangles

Class 7 Maths · Chapter 9: Geometric Twins · NCERT Ganita Prakash section P2 1.2

Learn

Meher and Kabir must copy a big triangular frame. They measure its sides: 40, 60 and 80 cm. Do they need the angles too? Building a 4, 6, 8 cm triangle with a compass, the arcs cross at two points, above and below the base, but the two triangles are mirror images: congruent! So equal sides are enough: the SSS condition.

SSS: three sides equal → congruent

Writing congruence: △ABC ≅ △XYZ means A ↔ X, B ↔ Y, C ↔ Z (first with first, second with second, third with third). Then AB = XY, BC = YZ, AC = XZ and ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z. △ACB ≅ △XZY says the same thing, but △ACB ≅ △XYZ is wrong.

△ABC ≅ △XYZ: A↔X, B↔Y, C↔Z

What else works? Equal angles alone do not (same shape, different sizes). Two sides and the angle between them (SAS) work; two angles and the side between them (ASA) work. Two angles and a side not between them (AAS) work too: the angle sum gives the third angle, and then ASA applies.

SSS · SAS · ASA · AAS · RHS

SSA fails: with PQ = 6 cm, ∠P = 30° and QR = 4 cm, the 4 cm arc from Q cuts the 30° line at two points, giving two different triangles. But for right triangles it works: with the right angle, the hypotenuse (the side opposite the right angle) and one more side equal, the triangles are congruent: RHS.

Using congruence: if O is the midpoint of both AD and BC, then AO = OD, BO = OC, and the vertically opposite angles at O are equal. By SAS, △AOB ≅ △DOC, so AB = DC. In a rectangle ABCD, the diagonal BD gives △ABD ≅ △CDB (SSS, with BD shared).

Explore: enough or not?

1. Sort the conditions.

Remember

• SSS, SAS, ASA, AAS and RHS guarantee congruence; AAA and SSA do not.
• In △ABC ≅ △XYZ, letters in the same position correspond.

Quick check

△SUN ≅ △BOX. Which side equals UN?

1. OX
2. BO
3. BX
4. SU

Think about it

△HEN ≅ △BIG. Write two other correct ways of saying this.

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.