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As the sum of the angles in a triangle is 180 ° and all four angles in a rectangle are 90 °, then sum of the two marked angles in the triangle 180 ° - 90 ° = 90 °.
Each interior angle of a regular hexagon is 120 ° and the sum of the angles in a quadrilateral is 360 °; hence the sum of the two marked angles in the quadrilateral is 360 ° - 90 ° - (360 ° - 120 °) = 30 °.
Hence the sum of the four marked angles is 90 ° + 30 ° = 120 °.
Inscribed Hexagon
Age 11 to 14
ShortChallenge Level 
As the sum of the angles in a triangle is 180 ° and all four angles in a rectangle are 90 °, then sum of the two marked angles in the triangle 180 ° - 90 ° = 90 °.
Each interior angle of a regular hexagon is 120 ° and the sum of the angles in a quadrilateral is 360 °; hence the sum of the two marked angles in the quadrilateral is 360 ° - 90 ° - (360 ° - 120 °) = 30 °.
Hence the sum of the four marked angles is 90 ° + 30 ° = 120 °.
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.
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