Geometry is one of the oldest branches of mathematics. It arose from very practical needs: measuring land, constructing with precision, observing the sky, and understanding the shapes around us. The Egyptians, Babylonians, and later Greek mathematicians gradually transformed these practical techniques into genuine reasoning based on properties and proofs. Even today, angles and geometric figures are indispensable. Architects use them to design buildings, engineers to calculate structures, astronomers to study the positions of celestial bodies, and navigation systems to determine directions.
A small problem involving angles can thus engage the same reasoning mechanisms as those used in much more complex applications.
A Little Story
Imagine a student facing this figure for the first time. They see 40°, 48°, 48°, two angles y, and an angle x. Their first instinct is to try to use all the numbers. Then they notice a much simpler detail: the two angles y are next to each other on the same straight line and are identical. At that moment, a significant part of the problem becomes clearer.
This is a good habit in mathematics: before calculating, observe the figure and search for the most useful information.
Quote of the Day
“Where there is matter, there is geometry.” — Johannes Kepler
This quote reminds us that geometry is not only present in school exercises. Shapes, angles, and proportions appear everywhere in the physical world, and learning to understand them enhances our ability to analyze the space around us.
What This Challenge Helps You Work On:
This challenge allows you to work on:
- Observing a geometric figure;
- The properties of angles on a line;
- The sum of the angles in a triangle;
- Identifying a right angle;
- Selecting genuinely useful information;
- Mental calculation;
- Logical reasoning;
- Concentration and speed.
CHALLENGE OF THE DAY
Carefully observe the figure.
We notably see angles of 40° and 48°, as well as two identical angles labeled y.
Your goal is to determine the value of angle x located at the top of the right triangle.
What is the value of x?
Note: not all visible information in the figure is necessarily needed to find the answer.
The Countdown Begins
30…
25…
20…
15…
10…
5…
4…
3…
2…
1…
Time’s up!
What Your Performance Can Reveal
If you found the result quickly and correctly, you have excellent logic and analytical skills.
If you found the result after some reflection, you know how to structure your reasoning and verify your steps.
If you found an incorrect result or couldn’t find one, no worries; this type of exercise is perfect for improving methods and precision.
Detailed Solution We Found (Not Necessarily the Only One, Sometimes Even the Right One)
Let’s start by observing the point located on the horizontal line, to the right of the figure. The two angles above this line are both labeled y. They together form a straight angle.
Thus, we can write the equation:
y + y = 180°
Therefore:
2y = 180°
And thus:
y = 90°
We now know that the interior angle at the base of the right triangle measures 90°.
Next, let’s look at the other angle at the base of this same triangle.
It is directly indicated on the figure:
48°
Therefore, we have a triangle with three angles:
48°, 90°, and x.
The sum of the angles in a triangle is always equal to 180°.
We can write the equation:
48° + 90° + x = 180°
Thus:
138° + x = 180°
We just need to subtract 138°:
x = 180° − 138°
x = 42°
Final Answer
x = 42°
Conclusion
This challenge intentionally contains several pieces of information that could distract attention. To determine x, it wasn’t necessary to use the angle of 40°, nor even the other 48° angle on the left. The key was to notice that the two angles y are equal and together form 180°. Each measures 90°. Then, it was enough to use the sum of the angles in the triangle to obtain 42°. This challenge thus illustrates an essential skill in mathematics: the ability to distinguish important information from that which is unnecessary for problem-solving.
Therefore, the answer to today’s challenge is indeed: 42°.
And before you get lost in the depths of the web, explore other games and tests by clicking here.
