Hey, this is Presh Talwalkar, and you're watching Mind Your Decisions.
Here's a really fun geometry problem:
Start out with a triangle where one angle is equal to 20 degrees,
and another angle is equal to 80 degrees.
Now draw a line segment such that this length right here is equal to this side line.
The challenge is to solve for the measure of this angle using only elementary geometry.
You're not allowed to use trigonometry or other methods.
I thank Barry, and I also thank Akshay Dhivare from India for emailing me this problem. Can you figure it out?
Give this problem a try and when you're ready, keep watching the video for a solution.
So I admit this was a very challenging problem, and I was not able to figure out the trick on my own.
But I was able to find the solution online and, I'd like to present it because it uses a clever trick.
So, let's get started by considering the big triangle.
We'll solve for the third angle.
This will be equal to 180 - 20 - 80 = 80 degrees.
Now this triangle has two angles that are equal to 80 degrees,
so these sides opposite them will be equal. This will be an isosceles triangle
We'll put these sides in green to show that they have equal length
Now let's zoom out a little bit and here's the clever trick
Along the top side of the triangle construct an equilateral triangle as illustrated.
So, all of these side lengths have equal lengths to the green side.
Since it's an equilateral triangle, all of the angles will be equal to 60 degrees.
So this is 60 degrees
This other angle will be 60 degrees in total. So this will be 40 degrees and here we have another 60 degree angle
Since the very top angle is 80 degrees, we know that this will be 80 - 60 = 20 degrees
Now let's connect these two points.
Furthermore, we'll focus on this triangle
Now two of the side lengths are equal to each other because they both have the same length as a greenside
So this is an isosceles triangle with a central angle of 40 degrees,
which means its other two equal angles are (180 - 40)/2
= 70 degrees. So here we have the 70 degree angle and the other angle is also equal to 70 degrees.
So if we remove 60 degrees from it, this will be equal to 10 degrees.
Now let's go back to this diagram, and we want to solve for this angle over here.
How can we do that? Well, here's the other insight.
Look at these two triangles.
Now they will be congruent to each other by SAS,
because both of these triangles have a blue side then a 20 degree angle and then a green side.
This means their obtuse angles will be equal to each other,
so this angle here will be equal to 80 + 70 degrees, which is 150 degrees.
Finally, we can solve for the angle that we want to.
This will be equal to 180 - 150 which is 30 degrees. And that's it!
Did you figure it out?
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