The Acute Angle Problem

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Presentation transcript:

The Acute Angle Problem An Investigation

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. 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

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.

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

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

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.