NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 2: Geometry - Problems and Solutions

Question 46

Idea: An angle is made by two rays starting from the same point (the vertex).

We will list angles that are clearly less than \(180^{\circ}\). These are called minor angles.

1. Read rays at vertex \(B\):

   Rays from \(B\): \(BA\), \(BC\), \(BD\), \(BE\), \(BF\).

   Form angles by taking pairs of these rays and choosing the smaller turn.

2. Angles at \(B\) using ray \(BA\):

   \(\angle ABC\), \(\angle ABD\), \(\angle ABE\), \(\angle ABF\).

   Each is the turn from \(BA\) to \(BC/BD/BE/BF\) that is \(<\;180^{\circ}\).

3. More angles at \(B\) starting from \(BC\):

   \(\angle CBD\), \(\angle CBE\), \(\angle CBF\).

   Again, take the smaller turn each time \(<\;180^{\circ}\).

4. Continue at \(B\) with next rays:

   \(\angle DBE\), \(\angle DBF\), \(\angle EBF\).

   All are clearly less than a straight angle \(180^{\circ}\).

5. Read rays at vertex \(A\):

   Rays from \(A\): \(AB\), \(AC\), \(AD\), \(AE\), \(AF\).

   Pick pairs that form minor angles in the figure:

   \(\angle BAC\), \(\angle BAD\), \(\angle BAE\), \(\angle CAF\), \(\angle DAF\).

6. Check the count:

   Total listed = \(4 + 3 + 3 + 5 = 15\) angles.

7. Final check:

   None of the chosen pairs make a straight line, so each angle is \(<\;180^{\circ}\).

   If any pair looked straight, we would skip it because a straight angle equals \(180^{\circ}\).

Note: Other correct sets of 15 minor angles are also possible if you pick different valid pairs.