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Difficulty: HardAngles, Parallel Lines, and Polygons

In a convex polygon of nn sides, three of its interior angles are right angles, and each of the remaining interior angles is equal to 150∘150^\circ. What is the total number of diagonals in this polygon?

  1. 9Answer
  2. B
    15
  3. C
    6
  4. D
    18

Answer

9 diagonals
The total sum of interior angles of an nn-sided polygon is (n−2)×180∘(n - 2) \times 180^\circ. Setting this equal to 3(90∘)+(n−3)(150∘)3(90^\circ) + (n - 3)(150^\circ) simplifies to 180n−360=150n−180180n - 360 = 150n - 180, which solves to n=6n = 6. Substituting n=6n = 6 into the diagonal formula D=n(n−3)2D = \frac{n(n - 3)}{2} yields 6×32=9\frac{6 \times 3}{2} = 9 diagonals.

Step-by-Step Solution

1
Formulate the interior angle sum equation for the nn-sided convex polygon.
The total interior angle sum is (n−2)×180∘(n - 2) \times 180^\circ. Summing the individual angles gives 3×90∘+(n−3)×150∘3 \times 90^\circ + (n - 3) \times 150^\circ. Thus: (n−2)×180∘=270∘+(n−3)×150∘(n - 2) \times 180^\circ = 270^\circ + (n - 3) \times 150^\circ.
The sum of all interior angles of any nn-sided polygon is given by (n−2)×180∘(n - 2) \times 180^\circ.
2
Solve the equation for the number of sides nn.
180n−360=270+150n−450  ⟹  180n−360=150n−180  ⟹  30n=180  ⟹  n=6180n - 360 = 270 + 150n - 450 \implies 180n - 360 = 150n - 180 \implies 30n = 180 \implies n = 6.
Expanding terms and isolating nn reveals that the polygon has 6 sides (a hexagon).
3
Calculate the total number of diagonals using the formula D=n(n−3)2D = \frac{n(n - 3)}{2}.
D=6(6−3)2=6×32=9D = \frac{6(6 - 3)}{2} = \frac{6 \times 3}{2} = 9.
Each of the nn vertices can connect to n−3n - 3 non-adjacent vertices, and dividing by 2 avoids double-counting each diagonal line segment.

Key Concept

Polygon interior angle sum theorem and diagonal calculation
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Topics

Angles, Parallel Lines, and Polygons

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