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5 solutions
Moderator note:
I was surprised that this solution only needed to calculate one intermediate step, in order to get to the answer.
It shows how having the "right perspective" to the problem can greatly simplify the work that you have to do.
That's very direct!
I was playing around with this diagram with different angles, and thought that this required multiple steps. Good to see that you have a 2-step solution. Mine required 4 lol.
the solution is right but the draw is wrong: 110° angle in A seems to be larger than 120° angle in E and 30° angle in D seems to be larger than 40° angle in B which is the solution !
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Geometry diagrams that we provide are rarely drawn to scale, unless otherwise stated.
In part, this reduced the "solution by accurately measuring the angle on the screen".
I should have take the time to do the 180 degrees. I assumed the drawing was accurate and to scale, so I simply estimated it visually, did not do the work.
best solution
Nice Explanation!
Very clever
Right, it's all about finding the supplementary angles. Neat work!
As a bonus, can we find ALL the interior angles of this star-shaped polygon?
b=e, since it is vertically opposite, c is not needed
There are two kinds of solutions. The elegant creations which demonstrate the superior insight of the author. And the pedestrian ones which show how to ploddingly get to the right answer without a need for an intuitive leap. This is the second kind.
We are given the red angles. So what do we know immediately? We know the blue ones, because they are supplementary. (Even if we happen not to remember the word for it, we know that they have to make up 1 8 0 ∘ with the red ones next to them.)
In a triangle A B G , we now know two angles. So the remaining one ∠ A G B = 1 8 0 ∘ − 7 0 ∘ − 3 0 ∘ = 8 0 ∘ .
That’s good, because that means F G E = 8 0 ∘ as well.
Now we have two known angles in the triangle F E G . What’s more, the remaining one is the one we want: x = 1 8 0 ∘ − 8 0 ∘ − 6 0 ∘ = 4 0 ∘ .
Great job on color coding the order in which we discovered the angles!
The original problem (picture) is a mess. The green angle is larger than the yellow. Obviously. Therefore, yellow is less than thirty. No need to go beyond that. Unless we assume that the author had no idea how to measure and draw angles. Pretty weak.
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Typically the original picture is not meant to be to scale. It informs us about the locations of the angles but it is not to be relied on for hints about the answer. Maybe it could have specifically stated that it is not to scale..
by the rules of euclidian geometry
This is the one I like. Look around at the triangles you've got. I was thrown off for a moment, cuz yo as drawn is clearly less than 30 degrees. But the algebra is impeccable, implying the drawing is imperfect, evidently intentionally.
Does anyone know the "proof" that any triangle is isosceles? It also involves an misleading drawing.
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Oh, I didn't know the "proof" so I looked it up, and it almost tricked me. I took me a while to figure out where the fault in the reasoning was. Pólya's quote is quite relevant here: "Geometry is the science of correct reasoning on incorrect figures."
sorry for my bad drawing :(
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No problem. I could understand which triangles you are talking about by comparing it with the star diagram. As long as the angles are labelled correctly, it does not matter much what is the value of the angle drawn in the diagram. Proving results by measurement are not considered as valid proofs.
Could you explain how you got the values of X, Y, Z, and A?
Take into consideration the △ D E C . According to the angle sum property, ∠ D C E should be 1 8 0 ∘ − ( 3 0 + 1 2 0 ) ∘ = 3 0 ∘ .
Now see the △ A B C . According to the angle sum property, ∠ C A B = 1 8 0 ∘ − ( 1 1 0 + 3 0 ) ∘ = 4 0 ∘ .