12 multiple-choice questions, progressively harder.
Five interior angles of a convex hexagon measure 100∘100^\circ100∘, 110∘110^\circ110∘, 120∘120^\circ120∘, 130∘130^\circ130∘, and 140∘140^\circ140∘. Find the sixth angle.
Solution
Correct answer: B
The hexagon interior sum is four triangle sums.
The five known angles total six hundred degrees.
A convex polygon has interior angle sum 1980∘1980^\circ1980∘. How many sides does it have?
Correct answer: D
Divide the total by a triangle sum to count the triangles.
Find one interior angle of a convex regular eighteen-sided polygon.
Correct answer: A
The exterior turn is
Take its supplement.
A convex regular polygon has an exterior turn of 8∘8^\circ8∘. Find its side count.
Correct answer: C
The side count must account for one complete turn.
The result is a whole number at least three.
Four exterior turns of a convex pentagon are 40∘40^\circ40∘, 50∘50^\circ50∘, 65∘65^\circ65∘, and 85∘85^\circ85∘. Find the fifth turn.
The four known turns total two hundred forty degrees.
At each vertex of a convex regular polygon, the interior angle and exterior turn have ratio seven to two. How many sides does it have?
The linear pair has nine equal ratio parts. Each part is twenty degrees, so the exterior turn uses two parts.
Can a convex regular polygon have each interior angle exactly 155∘155^\circ155∘?
Its exterior turn would be twenty-five degrees.
Rounding changes the angle, so no such polygon exists. The claimed sixty-degree restriction is false; a regular octagon has a different interior angle.
Four congruent regular pentagon corners of 108∘108^\circ108∘ are placed successively about one vertex. By how much does their angle total exceed a full turn?
Total the four corners, then compare with a full turn.
Four equilateral triangle corners and one square corner meet at one point. What does their angle total show?
Four triangle corners contribute two hundred forty degrees. Adding the right angle gives
At one point, two regular hexagon corners and two equilateral triangle corners are placed in order, with equal tile side lengths. What does the angle calculation establish?
Hexagon corners contribute one hundred twenty degrees each; triangle corners contribute sixty degrees each.
This settles the angles at that point. Matching and extending the placement across the plane still needs a construction.
A convex polygon has interior angle sum nine hundred degrees. How many diagonals are drawn in a fan from one vertex?
The total gives five triangles. The polygon therefore has seven sides. From one vertex, exclude that vertex and its two neighbors to count diagonals.
For a convex polygon, which reasoning proves that exactly one exterior turn per vertex totals a full turn?
Each interior angle and its selected turn make a straight angle. With all measures in degrees,
No equality between the separate turns is required.
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