World’s Hardest Simple Triangle Problem

Using only elementary geometry, determine angle x (angle AED). 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 trigonometry, 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. You may add construction lines if it makes the solution easier.

 

World’s Second Hardest Simple Triangle Problem

Using only elementary geometry, determine angle x (angle AED). 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 trigonometry, such as the law of sines, the law of cosines, etc. There is a review of elementary geometry below.

This is the second 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. You may add construction lines if it makes the solution easier.

 

Elementary Geometry

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.

Sphere Puzzle

Shown here is the section sketch of a sphere with a large hole drilled right through the center of it. As shown the hole is 10 inches long. What is the volume of the ring that is left of the sphere. The ring is shown hatched in the sketch.

Before you start complaining let me state that you have sufficient information to calculate the solution.