I saw this on "Andymath" but I haven't looked at the solution there. The problem is to find the marked angle.
We have two equilateral triangles, not necessarily of equal size, arranged as shown in the diagram, but no information about the orientation of the upper triangle or its relative size.
One reason geometry problems are so interesting is that often the solution requires some divergent thinking, "out of the box" as the saying goes. This can often involve adding auxiliary elements such as extra lines or circles to the given diagram, and seeing what to add can require considerable creativity.
But experience plays a role too, and one rule of thumb is that it is usually worth drawing in any standout perpendiculars and seeing if they do anything for us. So let's add line CD.
Now we notice that angles CGD and CBD are both interior angles of equilateral triangles, so they are equal. They both stand on CD and the converse of the angles in the same segment theorem tells us CD must be a chord of a circle passing through BG and B. Let's add that circle.
But now we see DB is a chord in that circle with angles DCB and DGB being angles in the same segment and therefore equal. DCB is obviously 30 degrees and so the angle DGB that we want is 30 degrees also. Solved!
A different approach is to say that we were not told the orientation of the upper triangle relative to the lower one and yet expected to solve the puzzle. So we conclude the orientation does not affect the answer and we can choose it to be anything we like. Very well, let's position the upper triangle directly above the lower, like this.
Now it's self-evident the angle sought is 30 degrees and from the argument just given we see it must have this value irrespective of the orientation of the upper triangle. Another example where the absence of what seems a crucial piece of information turns out to be the key that unlocks the puzzle!
Two Tangled Triangles
I saw this on "Andymath" but I haven't looked at the solution there. The problem is to find the marked angle.
We have two equilateral triangles, not necessarily of equal size, arranged as shown in the diagram, but no information about the orientation of the upper triangle or its relative size.
One reason geometry problems are so interesting is that often the solution requires some divergent thinking, "out of the box" as the saying goes. This can often involve adding auxiliary elements such as extra lines or circles to the given diagram, and seeing what to add can require considerable creativity.
But experience plays a role too, and one rule of thumb is that it is usually worth drawing in any standout perpendiculars and seeing if they do anything for us. So let's add line CD.
Now we notice that angles CGD and CBD are both interior angles of equilateral triangles, so they are equal. They both stand on CD and the converse of the angles in the same segment theorem tells us CD must be a chord of a circle passing through BG and B. Let's add that circle.
But now we see DB is a chord in that circle with angles DCB and DGB being angles in the same segment and therefore equal. DCB is obviously 30 degrees and so the angle DGB that we want is 30 degrees also. Solved!
A different approach is to say that we were not told the orientation of the upper triangle relative to the lower one and yet expected to solve the puzzle. So we conclude the orientation does not affect the answer and we can choose it to be anything we like. Very well, let's position the upper triangle directly above the lower, like this.
Now it's self-evident the angle sought is 30 degrees and from the argument just given we see it must have this value irrespective of the orientation of the upper triangle. Another example where the absence of what seems a crucial piece of information turns out to be the key that unlocks the puzzle!