12 multiple-choice questions, progressively harder.
Four interior angles of a convex hexagon have the same measure xxx. The other two are 130∘130^\circ130∘ and 150∘150^\circ150∘. Find xxx.
Solution
Correct answer: D
The interior sum is seven hundred twenty degrees.
The interior angle sum of a convex regular polygon is five times the sum of one exterior turn per vertex. How many sides does it have?
Correct answer: B
The exterior sum is one full turn.
Thus the fan contains ten triangles.
A convex polygon is replaced by another convex polygon with four more sides. How much larger is its interior angle sum?
Correct answer: C
The fan gains four triangles, independent of the original side count.
Which conclusion follows from requiring each interior angle of a convex regular polygon to be exactly 147∘147^\circ147∘?
Correct answer: A
The proposed exterior turn is thirty-three degrees.
The count is not an integer. Being smaller than a straight angle is necessary for a convex interior angle, but is not sufficient for a regular polygon specification.
Polygon A is convex and regular, with exterior turn 24∘24^\circ24∘.
Polygon B is convex and regular, with interior angle 162∘162^\circ162∘.
How many more sides does B have than A?
For A, divide the full turn by twenty-four.
For B, the exterior turn is eighteen degrees.
Compare the counts.
Could exactly five congruent convex regular polygon corners meet at every vertex of a regular tessellation?
Five equal corners would require seventy-two degrees each. Its exterior turn would be one hundred eight degrees.
This is not a polygon side count. Five general angles can total a full turn; the obstruction is the regular polygon requirement.
In a fan triangulation of a convex n-sided polygon from vertex A, which count explains the number of triangles?
The two sides touching A do not serve as triangle sides opposite A. Each remaining boundary side does, exactly once, so the fan has n minus two triangles. The number of diagonals is one smaller than that triangle count.
Three congruent regular heptagon corners meet at one point. Each corner measures nine hundred sevenths of a degree. What is the local angle result?
Add the three corners and compare with a full turn in sevenths.
The total exceeds a full turn, forcing overlap.
Two congruent regular octagon corners and one square corner, all with matching side lengths, total a full turn. What further evidence would demonstrate that this local arrangement extends to a plane tiling?
The angle sum settles only a neighborhood of one point. A repeating construction must also match every shared edge and fill each adjacent region. Recalculating the local sum gives no evidence about that continuation.
Failure of regular pentagons to pass the full-turn corner test proves which statement?
The test used the one hundred eight degree angles of congruent regular pentagons and an integer number meeting whole side to whole side. It rules out that particular tiling family. Changing the polygon or mixing shapes changes the angle test.
Seven exterior turns of a convex octagon are each 42∘42^\circ42∘. Find the interior angle at the remaining vertex.
The seven known turns total two hundred ninety-four degrees.
The remaining interior angle is its supplement.
Every square in a square-grid tessellation is divided along the same diagonal. Which conclusion follows?
Each cut partitions an existing square without moving or removing any piece, so coverage and absence of overlap are preserved. The triangles have a right angle and cannot be equilateral. Thus this tiling does not add another type of regular polygon to the three regular tessellations.
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