Exterior angles of angular polygon are in the ratio of 2:3. a) Find the size of the exterior angle. b) What is the interior angle sum of the hexagon?

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5) If the exterior angles of a regular polygon are in the ratio of 2:3, find the size of the exterior angle. The phrasing of this question is unusual, as all exterior angles of a regular polygon are equal. We will assume it means the ratio of the exterior angle to the interior angle is 2:3.

a) Find the size of the exterior angle.

Step 1: Let the exterior angle be $E$ and the interior angle be $I$. We assume the ratio of the exterior angle to the interior angle is $2:3$. $\frac{E}{I} = \frac{2}{3} \implies E = \frac{2}{3}I$

Step 2: Use the property that the interior angle and exterior angle of a polygon sum to $180^\circ$. $I + E = 180^\circ$

Step 3: Substitute $E = \frac{2}{3}I$ into the sum equation and solve for $I$. $I + \frac{2}{3}I = 180^\circ$ $\frac{5}{3}I = 180^\circ$ $I = 180^\circ \times \frac{3}{5} = 36^\circ \times 3 = 108^\circ$

Step 4: Calculate the exterior angle $E$. $E = 180^\circ - I = 180^\circ - 108^\circ = 72^\circ$ Alternatively, $E = \frac{2}{3}I = \frac{2}{3}(108^\circ) = 72^\circ$. The exterior angle is $\boxed{72^\circ}$. (This corresponds to a regular pentagon, as $360^\circ / 72^\circ = 5$ sides).

b) What is the interior angle sum of a heptagon?

Step 1: Identify the number of sides for a heptagon. A heptagon has $n=7$ sides.

Step 2: Use the formula for the sum of interior angles of a polygon. The sum of interior angles of an $n$-sided polygon is given by: $Sum = (n-2) \times 180^\circ$

Step 3: Substitute $n=7$ into the formula and calculate the sum. $Sum = (7-2) \times 180^\circ = 5 \times 180^\circ = 900^\circ$ The interior angle sum of a heptagon is $\boxed{900^\circ}$.

7) Find the value of x in the figures below.

a) The figure is a quadrilateral. The sum of the interior angles of a quadrilateral is $360^\circ$. The angles are $(x-5)^\circ$, $(x-1)^\circ$, $(x+11)^\circ$, and $(x+19)^\circ$.

Step 1: Set up the equation by summing all interior angles. $(x-5) + (x-1) + (x+11) + (x+19) = 360$

Step 2: Combine like terms. $4x + (-5 - 1 + 11 + 19) = 360$ $4x + 24 = 360$

Step 3: Solve for $x$. $4x = 360 - 24$ $4x = 336$ $x = \frac{336}{4}$ $x = 84$ The value of $x$ is $\boxed{84}$.

b) The figure is a 5-pointed star (pentagram). The sum of the angles at the points of a pentagram is $180^\circ$. All five angles are given as $x$.

Step 1: Set up the equation by summing the angles at the points. $x + x + x + x + x = 180$ $5x = 180$

Step 2: Solve for $x$. $x = \frac{180}{5}$ $x = 36$ The value of $x$ is $\boxed{36}$.

c) The figure is a rectangle. In a rectangle, opposite sides are equal in length. The left side is $2x$ and the right side is $x+1$.

Step 1: Set the expressions for the opposite sides equal to each other. $2x = x+1$

Step 2: Solve for $x$. $2x - x = 1$ $x = 1$ The value of $x$ is $\boxed{1}$.

d) The figure shows two angles on a straight line. Angles on a straight line sum to $180^\circ$. The angles are $x$ and $2x$.

Step 1: Set up the equation by summing the angles. $x + 2x = 180$

Step 2: Combine like terms. $3x = 180$

Step 3: Solve for $x$. $x = \frac{180}{3}$ $x = 60$ The value of $x$ is $\boxed{60}$.

e) The figure is a hexagon. The sum of the interior angles of an $n$-sided polygon is $(n-2) \times 180^\circ$. For a hexagon, $n=6$. The angles are $(x+9)^\circ$, $(x-1)^\circ$, $(x+1)^\circ$, $(x+3)^\circ$, $(x+5)^\circ$, and $(x+7)^\circ$.

Step 1: Calculate the sum of interior angles for a hexagon. $Sum = (6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ$

Step 2: Set up the equation by summing all given angles. $(x+9) + (x-1) + (x+1) + (x+3) + (x+5) + (x+7) = 720$

Step 3: Combine like terms. $6x + (9 - 1 + 1 + 3 + 5 + 7) = 720$ $6x + 24 = 720$

Step 4: Solve for $x$. $6x = 720 - 24$ $6x = 696$ $x = \frac{696}{6}$ $x = 116$ The value of $x$ is $\boxed{116}$.

f) The figure shows three angles around a point. Angles around a point sum to $360^\circ$. The angles are $x$, $110^\circ$, and $150^\circ$.

Step 1: Set up the equation by summing the angles. $x + 110 + 150 = 360$

Step 2: Combine the constant terms. $x + 260 = 360$

Step 3: Solve for $x$. $x = 360 - 260$ $x = 100$ The value of $x$ is $\boxed{100}$.

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