Maths Week Scotland 2025 - Challenge 5 - Mystery angles

Challenge 5 is all about working out an unknown angle.

Maths teacher Chris Smith and pupils from Grange Academy are here to explain.

The Maths Week Scotland Daily Challenges have been set by the Scottish Mathematical Council.

So here's the challenge:

This problem is all about angles.

There’s a mystery about the size of the angles in this quadrilateral.

Here are some clues:

The red angle is twice the size of the blue angle.

The yellow angle is three times the size of the blue angle.

The black angle is 52° greater than the blue angle.

What size is the black angle?

• If you are unsure, try thinking about what kind of shape this is.
• Think about how the different angles compare to each other.
• You might need to use an equation.

Worked out the answer? Here's how you can do it.

Step 1

The key to solving this puzzle is knowing that because the shape is a quadrilateral, all the angles in it add up to 360°.

We can work out the angles by starting with the blue one.

We can make the size of the blue angle 𝑥° then the size of the red angle is 2𝑥°, yellow is 3𝑥°, and the black angle is 𝑥 + 52°.

Step 2

We can put the information we have into an equation.

𝑥 + 2𝑥 + 3𝑥 + 𝑥 + 52° = 360°

Step 3

We can simplify this equation by gathering all the 𝑥 terms together:

7𝑥 + 52 = 360°

Step 4

If we subtract 52° from each side, we end up with

7𝑥 = 308°

Step 5

Then we can divide each side by 7 to find the value of 𝑥.

308° ÷ 7 = 44°

𝑥 = 44°

Step 6

The size of the blue angle is 44°, and so the black angle will be 52° greater:

44° + 52° = 96°

The black angle is 96°