Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Angles in Three Squares

Why do this problem?

Geometry sometimes offers surprising results that can be proved very elegantly. This is a great example of how finding the right representation (in this case a diagram) allows a problem to be solved more easily.

Possible approach

Show students the image:

Invite them to draw it and measure angles a, b and c to verify that a+b=c, and give them some time to think about how they might prove it. There are many methods that use advanced techniques such as trigonometry or vectors, but there is also an elegant geometric method. Once they have had time to engage with the difficulty of the problem, show them the following image:

"Can you find angles in the diagram that are the same size as angles a and b?"

"What can you say about the angle at B?"

"What can you say about the lengths AB and BC?"

"What can you deduce about the angles at A and C?"

Possible extension

Students may be interested to read 8 Methods for Three by One which outlines a number of different methods for this proof using more advanced mathematics.

Possible support

Angles in Polygons Short Problems worksheets offer some more straightforward examples of geometrical reasoning problems.

## You may also like

### At a Glance

### No Right Angle Here

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

Or search by topic

Age 14 to 16

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Why do this problem?

Possible approach

Invite them to draw it and measure angles a, b and c to verify that a+b=c, and give them some time to think about how they might prove it. There are many methods that use advanced techniques such as trigonometry or vectors, but there is also an elegant geometric method. Once they have had time to engage with the difficulty of the problem, show them the following image:

"Can you find angles in the diagram that are the same size as angles a and b?"

"What can you say about the angle at B?"

"What can you say about the lengths AB and BC?"

"What can you deduce about the angles at A and C?"

Possible extension

Possible support

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.