1. The Acute Angle Problem
An Investigation

2. The acute angle problem.
In this problem, the angle between the two outside lines is always acute.
1. The diagram shows an acute angle AOB. The angle is made with the lines OA and OB.
2. If a third line is added to the diagram,
how many acute angles can now be
found? {the answer is not 2} O B A O B A C

3. 3. Adding a fourth line gives yet more acute angles.
How many can you find ?
It might help if you make a list.
4. With five lines, counting the total number of acute angles becomes a bit harder. See how many you can find. Make a list. O D C B A E D C B A O

4. In the table on the next slide, enter the results you have so far.
From the table, see if you can spot a pattern, which will allow you to fill in the rest of the table without having to draw any more diagrams or count any more angles.

5. Number
of lines
Number of
acute angles 2 3 4 5 6 7 8 9 10

6. Can you find a formula for the number of acute angles when there are n lines.
Use this formula to see how many acute angles you can form if there are
(i) 20 lines (ii) 40 lines

7. Entries in the table are 1, 3, 6, 10, 15, 21, 28, 36, 45 and 55
Solution
Entries in the table are
1, 3, 6, 10, 15, 21, 28, 36, 45 and 55
Using a difference table or otherwise pupils should get the formula
Using this formula they get 210 and 840 acute angles respectively.