To find the value of $x$ in each question, we use the
Polygon Exterior Angle Sum Theorem. This theorem states that for any convex polygon, the sum of the measures of its exterior angles (one at each vertex) is always
$360^\circ$.
We will set up an equation for each shape by adding all the given exterior angles and setting the sum equal to 360. Then, we solve for $x$.
1. Quadrilateral (Top Left)
The exterior angles are: $3x$, $2x$, $160$, and $90$ (indicated by the square symbol).
Equation:
$$3x + 2x + 160 + 90 = 360$$
Steps:
1. Combine like terms: $5x + 250 = 360$
2. Subtract 250 from both sides: $5x = 110$
3. Divide by 5: $x = 22$
2. Quadrilateral (Top Middle)
The exterior angles are: $90$ (square symbol), $2x$, $3x$, and another $90$ (square symbol).
Equation:
$$90 + 2x + 3x + 90 = 360$$
Steps:
1. Combine like terms: $5x + 180 = 360$
2. Subtract 180 from both sides: $5x = 180$
3. Divide by 5: $x = 36$
3. Hexagon (Top Right)
The exterior angles are: $(100 - x)$, $(5x - 50)$, $(90 - \frac{x}{2})$, $(50 - x)$, $(\frac{x}{2} - 15)$, and $65$.
Equation:
$$(100 - x) + (5x - 50) + (90 - \frac{x}{2}) + (50 - x) + (\frac{x}{2} - 15) + 65 = 360$$
Steps:
1. Group the $x$ terms: $-x + 5x - \frac{x}{2} - x + \frac{x}{2} = 3x$ (The $-\frac{x}{2}$ and $+\frac{x}{2}$ cancel out).
2. Group the constant numbers: $100 - 50 + 90 + 50 - 15 + 65 = 240$.
3. Simplify the equation: $3x + 240 = 360$
4. Subtract 240 from both sides: $3x = 120$
5. Divide by 3: $x = 40$
4. Pentagon (Bottom Left)
The exterior angles are: $(x + 30)$, $90$ (square symbol), $(8x - 25)$, $150$, and $7x$. Note: The angle labeled $(31x)^\circ$ is an interior reflex angle or simply placed confusingly, but the exterior angle at that vertex is clearly marked as $7x$. We use the exterior angles.
Equation:
$$(x + 30) + 90 + (8x - 25) + 150 + 7x = 360$$
Steps:
1. Combine $x$ terms: $x + 8x + 7x = 16x$
2. Combine constants: $30 + 90 - 25 + 150 = 245$
3. Simplify the equation: $16x + 245 = 360$
4. Subtract 245 from both sides: $16x = 115$
5. Divide by 16: $x = 7.1875$
5. Octagon (Bottom Right)
The exterior angles are: $\frac{4}{5}x$, $(120 + \frac{1}{3}x)$, $(178 - \frac{x}{5})$, $\frac{4}{3}x$, $0.6x$, $\frac{2x}{5}$, $\frac{4x}{3}$, and $(2x - 70)$.
*Note: $0.6x$ is the same as $\frac{3}{5}x$ or $\frac{6}{10}x$.*
Equation:
$$\frac{4}{5}x + (120 + \frac{1}{3}x) + (178 - \frac{x}{5}) + \frac{4}{3}x + 0.6x + \frac{2x}{5} + \frac{4x}{3} + (2x - 70) = 360$$
Steps:
1. Group constants: $120 + 178 - 70 = 228$.
2. Group $x$ terms. Let's convert decimals to fractions to make it easier ($0.6x = \frac{3}{5}x$):
- Fifths: $\frac{4}{5}x - \frac{1}{5}x + \frac{3}{5}x + \frac{2}{5}x = \frac{8}{5}x$
- Thirds: $\frac{1}{3}x + \frac{4}{3}x + \frac{4}{3}x = \frac{9}{3}x = 3x$
- Whole $x$: $2x$
- Total $x$ sum: $\frac{8}{5}x + 3x + 2x = 1.6x + 5x = 6.6x$
3. Simplify the equation: $6.6x + 228 = 360$
4. Subtract 228 from both sides: $6.6x = 132$
5. Divide by 6.6: $x = 20$
Final Answer:
1. x = 22
2. x = 36
3. x = 40
4. x = 7.1875
5. x = 20
Parent Tip: Review the logic above to help your child master the concept of polygon angle worksheet.
