Geometry (Classes VI–VIII) • Topic 5 of 6

Congruence & Symmetry

Two figures are congruent if they have exactly the same shape AND size — one can be slid, turned or flipped onto the other so they coincide. For triangles, CTET tests the congruence criteria by name: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS, and RHS (right angle–hypotenuse–side) for right triangles. The trap is the non-criteria: AAA proves only similarity (same shape, possibly different size), and SSA/ASS is not a valid congruence rule. Symmetry is the other half: line (reflection) symmetry — a figure folds onto itself across a line of symmetry (a square has 4, a rectangle 2, an equilateral triangle 3, a circle infinitely many) — and rotational symmetry, measured by the order (how many times the figure maps onto itself in one full turn; a square has order 4). The misconception is conflating 'same shape' with 'congruent' (it must also be the same size) and assuming every figure with line symmetry also has rotational symmetry of order > 1. CTET tests naming the right criterion for a given marked diagram, counting lines of symmetry, and the order of rotational symmetry.

✅ Solved examples

1. In two triangles, two sides and the angle between those two sides of one equal the corresponding parts of the other. By which criterion are the triangles congruent?
SAS (Side–Angle–Side) — two sides and the INCLUDED angle are equal. The angle must be the one between the two given sides.
2. Why is AAA not a valid test for the congruence of two triangles?
Equal angles guarantee the same shape but not the same size — AAA gives similar triangles, which can be scaled up or down. Congruence needs at least one pair of equal sides (as in SSS, SAS, ASA, AAS or RHS).
3. How many lines of symmetry does a rectangle (that is not a square) have?
Two — the lines through the midpoints of opposite sides. (Its diagonals are NOT lines of symmetry, unlike in a square, which has four lines.)
4. What is the order of rotational symmetry of a square?
Order 4 — in one full 360° turn the square maps exactly onto itself four times (every 90°).

✏️ Practice — try these, take hints as needed

1. For two right-angled triangles, the criterion that uses the right angle, the hypotenuse and one side is called:
It applies only to right triangles.
Right angle – Hypotenuse – Side.
RHS
2. Two figures with the same shape but different sizes are said to be:
Not congruent — they differ in size.
AAA gives this.
Similar
3. The number of lines of symmetry of an equilateral triangle is:
One through each vertex.
Same as its number of sides.
3
4. The order of rotational symmetry of a circle is:
It looks identical after any rotation.
Unlimited.
Infinite

📝 Topic test — 8 questions

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