NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 9: Symmetry and Practical Geometry - Fill in the Blanks

Question 39

39. A (30^circ)–(60^circ)–(90^circ) set-square has _____ line/lines of symmetry.

Answer: no.

Beginner-Friendly Explanation (Step by Step)

1. What is a line of symmetry?
   If you can fold a shape along a line and both halves match exactly, that line is called a line of symmetry.

2. What triangle are we talking about?
   It’s a right triangle with angles:

   (30^circ)

   (60^circ)

   (90^circ)

3. Check if any two angles are equal.
   For symmetry in a triangle, usually at least two angles (and the opposite sides) are equal (like in an isosceles triangle).
   Here we have three different angles:

   (30^circ eq 60^circ)

   (60^circ eq 90^circ)

   (30^circ eq 90^circ)

   So, no two angles are equal.

4. Look at the side lengths of a (30^circ)–(60^circ)–(90^circ) triangle.
   Their ratio is:

   (1 : sqrt{3} : 2)

   All three sides are different.

5. Try possible folds and see if halves match.

   Fold through the right angle (the (90^circ) vertex): one corner is (30^circ) and the other is (60^circ).

   (30^circ eq 60^circ) → the corners won’t match after folding.

   Fold through any other vertex: again you’ll try to match different angles/sides, so the halves won’t coincide.

6. Conclusion:
   Because all angles and sides are unequal, there is no line along which the triangle can be folded to make two identical halves.

   Therefore, number of lines of symmetry = (0).

Common mistake: Thinking every right triangle has a symmetry line. Only an isosceles right triangle (angles (45^circ), (45^circ), (90^circ)) has one line of symmetry. A (30^circ)–(60^circ)–(90^circ) triangle does not.