The riddle
Why Did the Obtuse Angle Go to the Beach?
What makes an angle obtuse
An obtuse angle measures more than 90 degrees and less than 180. At exactly 90 degrees an angle is a right angle, and at 180 it is a straight line. Obtuse comes from a Latin word meaning blunt, which fits the wide, open look of the corner. A triangle can have only one obtuse angle, because the three angles of a triangle add up to 180 degrees, and two angles of more than 90 would already be over that.
The weather side
The beach part of the joke is about heat. On the Fahrenheit scale used for weather in the United States, 90 degrees is a hot day, about 32 degrees on the Celsius scale. A day over 90 sends people to the water, and an obtuse angle is over 90 too.
One word, two jobs
Degree measures both angles and temperature. Merriam-Webster traces the word to a Latin word for a step, because a degree is one step on a scale. A full turn around a circle is 360 degrees, and water freezes at 32 on the Fahrenheit scale. The units are unrelated, and the riddle works because they share a name. A hot afternoon of 90 degrees and a 90 degree corner have nothing else in common.
Using it in a math class
A quick follow-up is to ask the group to name an angle of exactly 90 degrees and then one that is acute. The corner of a sheet of paper is a right angle, and a door swung wide open shows an obtuse one. The joke is better remembered when the shapes are in hand.