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Difficulty: HardQuadrilaterals and Polygons

Match each geometric description to the correct polygon or quadrilateral it uniquely describes.

  • A regular polygon in which the sum of the interior angles is exactly five times the sum of its exterior angles.Dodecagon
  • A convex polygon in which the total number of diagonals is exactly 2.52.5 times the number of its vertices.Octagon
  • A quadrilateral whose diagonals are equal in length and bisect each other, but are not mutually perpendicular.Rectangle
  • A regular polygon in which the measure of each interior angle is exactly 100∘100^\circ greater than its adjacent exterior angle.Nonagon

Answer

The geometric descriptions correspond perfectly to the Dodecagon, Octagon, Rectangle, and Nonagon based on their algebraic angle properties and specific diagonal traits.
Each provided description translates to a specific algebraic or geometric property that uniquely identifies a single polygon. Setting up and solving the respective equations for nn (number of sides) or applying logical diagnostic properties of quadrilaterals produces the correct matching pairs.

Step-by-Step Solution

1
Analyze the polygon where the interior angle sum is five times the exterior angle sum.
180(n−2)=5(360)⇒180(n−2)=1800⇒n−2=10⇒n=12180(n-2) = 5(360) \Rightarrow 180(n-2) = 1800 \Rightarrow n-2 = 10 \Rightarrow n=12.
Applying the universal constant for exterior angle sums (360∘360^\circ) allows solving algebraically for the 1212-sided dodecagon.
2
Solve for the polygon where the number of diagonals equals 2.52.5 times the number of vertices (nn).
n(n−3)2=2.5n⇒n−3=5⇒n=8\frac{n(n-3)}{2} = 2.5n \Rightarrow n-3 = 5 \Rightarrow n=8.
Equating the formula for the total number of diagonals to 2.5n2.5n allows us to simplify and identify the 88-sided octagon.
3
Deduce the quadrilateral with congruent, bisecting, non-perpendicular diagonals.
The shape is uniquely identified as a rectangle (excluding a square).
Bisecting diagonals prove it is a parallelogram, congruent diagonals prove it is a rectangle, and non-perpendicularity proves it is not a square.
4
Calculate the sides of the polygon where the interior angle is 100∘100^\circ larger than the exterior angle.
I+E=180∘I + E = 180^\circ and I−E=100∘I - E = 100^\circ. Solving the system gives E=40∘E = 40^\circ. Since n=360En = \frac{360}{E}, n=36040=9n = \frac{360}{40} = 9.
Recognizing that interior and exterior angles form a linear pair lets us set up a system of equations. Finding the exterior angle easily leads to the 99-sided nonagon.

Key Concept

Advanced Polygon Analysis and Quadrilateral Properties
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Topics

Quadrilaterals and Polygons

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