The following geometry puzzle has often been described as:

The World's hardest, easy, geometry puzzle

The aim of the puzzle is to determine the angle x. That's it!

At first glance, this appears to be a fairly trivial problem.

You should by able to find a solution using geometry; there is no need to resort to trigonometry (and no cheating by drawing to scale and using a protractor to measure angles! We want an exact answer).

There are various versions of this puzzle floating around with different angles (some also with slightly different triangle setups). I'll not get into a debate about if some versions are 'harder' or 'easier' than others. This is the one I've selected for this article; It's my blog. (If you disagree, start your own blog!)

It is possible to solve this problem using the simple principles taught in High School.

(The diagram is not necessarily to scale).

Answer

Give it a go. Did you work out a solution? Once you have your answer, click 'Reveal Answer' to compare with mine.

x = 30°

Solution

Here is one of the (many) ways that this problem can be solved: Click button to reveal.

Step-by-step Solution …

We known a few things about triangles. Here are a couple of facts we can use:

All the angles of a triangle add up to 180°.

If a triangle has two equal sides, we call this an Isoceles Triangle.
An Isoceles triangle also has two equal angles (the angles opposite the equal sides).

If a triangle has three equal sides, we call this an Equilateral Triangle.
An Equilateral triangle has three equal angles. These angles are all 60°.

Let's put these facts to use.

First, let's label the points so we know how to describe them.

The points are labelled: A, B, C, D, E.

The angle at A is 60°+20° = 80°

The angle at B is 50°+30° = 80°

This triangle is, therefore, Isoceles and length AE = BE

Because we know the angles at A and B, we can calculate the angle E to be 20°
(We will see later that we never actually use this fact!)

However, I'm going to draw a new line on the diagram.

We're going to create a new point, P, on the triangle such that AP = AB.

This will make another isoceles triangle.

Because this new triangle is isoceles and AP = AB, and we know angle B = 80°, we can now determine that angle β is 80° also.

Knowing these two angles allows us to calculate that α = 20°

We now have all the information needed to break down all the angles at point A.

For the triangle ACB, we know two of the angles.

This allows us to calculate γ = 50°

As this is the same angle (50°) as the angle at B, this makes triangle ACB isoceles.

Therefore, length AC is equal to length AB. (From before, this is also the length AP)

Now we are making progress …

Because AC = AP and angle CAP = 60°, we know the triangle CAP has to be an equilateral triangle. Length CP = AP = CA, and all internal angles are 60°

Half-way there

We're about half-way there. Let's pause to recap and label all the angles we known so far (and edges that are the same length):

We can also complete the angles around point P as we know this is a straight line, and the angles there add up to 180° (40°+60°+80°).

Shown left.

From the triangle ABD, the angle δ = 40°. Shown below.

Triangle ADP is isoceles because it has two angles at 40°

This means that the length AP = DP.

(and also DP = AC = AP = AB = CP).

And we are there …

Triangle CDP is an isoceles because CP = DP

We know the angle at P is 40°, so the remaining 140° has to be split equally between the other two corners (70° each).

At D, (x+δ) = 70°, and we know δ = 40°

Result: x = 30°

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