Problem 19 · 2016 Math Kangaroo
Stretch
Spatial & Visual Reasoning
cube-views
In the diagram we see a cube and four marked angles. How big is the sum of those angles?
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Answer: B — 330°
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Hint 1 of 3
The four marked angles are the corners of the bold closed path on the cube, not flat 90° angles.
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Hint 2 of 3
Put the cube on unit coordinates and read each corner angle from the two bold edges meeting there.
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Hint 3 of 3
Watch for the corner where two face diagonals meet an edge to form an equilateral-triangle 60° angle.
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Approach: find each corner angle of the bold space quadrilateral
- Take a unit cube; the bold quadrilateral is built from cube edges and face diagonals.
- Three of its corners are where an edge meets a face diagonal at a right angle, giving 90° each.
- The remaining corner is the tip of an equilateral triangle formed by three equal face diagonals, giving 60°.
- The sum is 90° + 90° + 90° + 60° = 330°.
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