Using only elementary geometry, determine angle x. Provide a step-by-step proof. You may only use elementary geometry, such as the fact that the angles of a triangle add up to 180 degrees and the basic congruent triangle rules (side-angle-side, etc.). You may not use more advanced trigonomery, such as the law of sines, the law of cosines, etc. There is a review of elementary geometry below. This is the hardest problem I have ever seen that is, in a sense, easy. It really can be done using only elementary geometry. This is not a trick question. Here is a very small hint. Here is a small hint.

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Using only elementary geometry, determine angle x. Provide a step-by-step proof. This is a variation of the problem above. This is also a very hard problem that is, in a sense, easy. Here is a very small hint. Here is a small hint.

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Sorry, but I'm not giving the answer nor the proof here. You will just have to work on it until you either solve it or are driven insane. If you email me at k.enevoldsen@wlonk.com, I may give you a bigger hint (if I feel like it). If you think you have solved it, you can ask me if your answer is correct, but please also tell me how you got the answer. The proof may be written informally, but you need to tell me all the steps, or at least the key steps, in your solution. It is helpful if you also send me a diagram. Try to persuade me that you are not just guessing. I have additional small, medium, and large hints, but you must first show your efforts to convince me that you have struggled valiantly.

Please don't search the the web for the answer -- that's cheating. You will only deprive yourself of many hours of delicious frustration.

I did not invent these problems. After I first read problem 1, I worked on it for many hours over several days before I eventually figured it out. A couple of years later I came back to the problem, but I had forgotten my proof. It took me many hours to figure it out again! Problem 2 also took me many hours to solve.

How hard are these problems? Any teenage student and some younger students can understand the proof, but very very few are able to discover the proof on their own. Of the hundreds of people that have emailed me, I'd estimate only one or two percent (mostly math professionals and college students) have solved it without significant hints. (The hints given above are not significant hints.) Most people who think they have found the solution are wrong.

These problems have been published in many places. Problem 2 first appeared here: Langley, "A Problem", Mathematical Gazette, 1922. Dr. Gary Gruber says his high school teacher showed him problem 1 in about 1955. Tom Rike says problem 1 first appeared in print here: Harry Schor, The New York State Mathematics Teachers' Journal, 1974. It also appeared here: "Problem 134", Eureka (now Crux Mathematicorum), 1976. Dr. Gruber popularized problem 1 in several papers (such as "The Genius Test") which appeared in newspapers throughout the 1990s (Universal Press Syndicate and Los Angeles Times Syndicate). That's where I discovered it.

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Here is everything you need to know to solve the above problems.

Lines and Angles: When two lines intersect, opposite angles are equal and the sum of adjacent angles is 180 degrees. When two parallel lines are intersected by a third line, the corresponding angles of the two intersections are equal.

Triangles: The sum of the interior angles of a triangle is 180 degrees. An isosceles triangle has two equal sides and the two angles opposite those sides are equal. An equilateral triangle has all sides equal and all angles equal. A right triangle has one angle equal to 90 degrees. Two triangles are called similar if they have the same angles (same shape). Two triangles are called congruent if they have the same angles and the same sides (same shape and size).

• Side-Angle-Side (SAS): Two triangles are congruent if a pair of corresponding sides and the included angle are equal.
• Side-Side-Side (SSS): Two triangles are congruent if their corresponding sides are equal.
• Angle-Side-Angle (ASA): Two triangles are congruent if a pair of corresponding angles and the included side are equal.
• Angle-Angle (AA): Two triangles are similar if a pair of corresponding angles are equal.

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Proofs may be written informally using plain English. Just be sure to include all the steps in your reasoning, or at least all the key steps. Providing a diagram is very helpful but not required. You can draw a diagram on the computer or you can draw it on paper and then scan it or photograph it with a digital camera. Name each point you use with a letter. If you don't provide a diagram, you will need to describe the named points with words (ex., say "the intersection of AE and DB is G "). Identify lines with two letters (ex., say "line AB" or simply "AB"). Identify triangles with three letters (ex., say "triangle ABC" or "tri ABC" or simply "ABC"). Identify angles with three letters, vertex in the middle (ex., say "angle ABC" or "ang ABC" or "<ABC" or simply "ABC"). It is helpful to number your steps to make it easy to refer back to earlier steps.