6.1: Angles in Polygons
Learning Objectives
- Extend the concept of interior and exterior angles from triangles to convex polygons.
- Find the sums of interior angles in convex polygons.
- Identify the special properties of interior angles in convex quadrilaterals.
Review Queue
- Find the measure of \begin{align*}x\end{align*}
x and \begin{align*}y\end{align*}y . -
- Find \begin{align*}w^\circ, x^\circ, y^\circ\end{align*}
w∘,x∘,y∘ , and \begin{align*}z^\circ\end{align*}z∘ . - What is \begin{align*}w^\circ + y^\circ + z^\circ\end{align*}
w∘+y∘+z∘ ? - What two angles add up to \begin{align*}y^\circ\end{align*}?
- What are \begin{align*}72^\circ, 59^\circ\end{align*}, and \begin{align*}x^\circ\end{align*} called? What are \begin{align*}w^\circ, y^\circ\end{align*}, and \begin{align*}z^\circ\end{align*} called?
- Find \begin{align*}w^\circ, x^\circ, y^\circ\end{align*}
Know What? To the right is a picture of Devil’s Post pile, near Mammoth Lakes, California. These posts are cooled lava (called columnar basalt) and as the lava pools and cools, it ideally would form regular hexagonal columns. However, variations in cooling caused some columns to either not be perfect or pentagonal.
First, define regular in your own words. Then, what is the sum of the angles in a regular hexagon? What would each angle be?
Interior Angles in Convex Polygons
Recall from Chapter 4, that interior angles are the angles inside a closed figure with straight sides. Even though this concept was introduced with triangles, it can be extended to any polygon. As you can see in the images below, a polygon has the same number of interior angles as it does sides.
From Chapter 1, we learned that a diagonal connects two non-adjacent vertices of a convex polygon. Also, recall that the sum of the angles in a triangle is \begin{align*}180^\circ\end{align*}. What about other polygons?
Investigation 6-1: Polygon Sum Formula
Tools Needed: paper, pencil, ruler, colored pencils (optional)
1. Draw a quadrilateral, pentagon, and hexagon.
2. Cut each polygon into triangles by drawing all the diagonals from one vertex. Count the number of triangles.
Make sure none of the triangles overlap.
3. Make a table with the information below.
Name of Polygon | Number of Sides | Number of \begin{align*}\triangle s\end{align*} from one vertex | (Column 3) \begin{align*}\times\end{align*} (\begin{align*}^\circ\end{align*} in a \begin{align*}\triangle\end{align*}) | Total Number of Degrees |
---|---|---|---|---|
Quadrilateral | 4 | 2 | \begin{align*}2 \times 180^\circ\end{align*} | \begin{align*}360^\circ\end{align*} |
Pentagon | 5 | 3 | \begin{align*}3 \times 180^\circ\end{align*} | \begin{align*}540^\circ\end{align*} |
Hexagon | 6 | 4 | \begin{align*}4 \times 180^\circ\end{align*} | \begin{align*}720^\circ\end{align*} |
4. Do you see a pattern? Notice that the total number of degrees goes up by \begin{align*}180^\circ\end{align*}. So, if the number sides is \begin{align*}n\end{align*}, then the number of triangles from one vertex is \begin{align*}n - 2\end{align*}. Therefore, the formula would be \begin{align*}(n - 2) \times 180^\circ\end{align*}.
Polygon Sum Formula: For any \begin{align*}n-\end{align*}gon, the sum of the interior angles is \begin{align*}(n - 2) \times 180^\circ\end{align*}.
Example 1: Find the sum of the interior angles of an octagon.
Solution: Use the Polygon Sum Formula and set \begin{align*}n = 8\end{align*}.
\begin{align*}(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ\end{align*}
Example 2: The sum of the interior angles of a polygon is \begin{align*}1980^\circ\end{align*}. How many sides does this polygon have?
Solution: Use the Polygon Sum Formula and solve for \begin{align*}n\end{align*}.
\begin{align*}(n - 2) \times 180^\circ & = 1980^\circ\\ 180^\circ n - 360^\circ & = 1980^\circ\\ 180^\circ n & = 2340^\circ\\ n & = 13 \qquad \text{The polygon has} \ 13 \ \text{sides.}\end{align*}
Example 3: How many degrees does each angle in an equiangular nonagon have?
Solution: First we need to find the sum of the interior angles in a nonagon, set \begin{align*}n = 9\end{align*}.
\begin{align*}(9 - 2) \times 180^\circ = 7 \times 180^\circ = 1260^\circ\end{align*}
Second, because the nonagon is equiangular, every angle is equal. Dividing \begin{align*}1260^\circ\end{align*} by 9 we get each angle is \begin{align*}140^\circ\end{align*}.
Equiangular Polygon Formula: For any equiangular \begin{align*}n-\end{align*}gon, the measure of each angle is \begin{align*}\frac{(n-2)\times 180^\circ}{n}\end{align*}.
Regular Polygon: When a polygon is equilateral and equiangular.
It is important to note that in the Equiangular Polygon Formula, the word equiangular can be substituted with regular.
Example 4: Algebra Connection Find the measure of \begin{align*}x\end{align*}.
Solution: From our investigation, we found that a quadrilateral has \begin{align*}360^\circ\end{align*}. We can write an equation to solve for \begin{align*}x\end{align*}.
\begin{align*}89^\circ+(5x-8)^\circ+(3x+4)^\circ+51^\circ&=360^\circ\\ 8x&=224^\circ\\ x&=28^\circ\end{align*}
Exterior Angles in Convex Polygons
Recall that an exterior angle is an angle on the outside of a polygon and is formed by extending a side of the polygon (Chapter 4).
As you can see, there are two sets of exterior angles for any vertex on a polygon. It does not matter which set you use because one set is just the vertical angles of the other, making the measurement equal. In the picture to the left, the color-matched angles are vertical angles and congruent.
In Chapter 4, we introduced the Exterior Angle Sum Theorem, which stated that the exterior angles of a triangle add up to \begin{align*}360^\circ\end{align*}. Let’s extend this theorem to all polygons.
Investigation 6-2: Exterior Angle Tear-Up
Tools Needed: pencil, paper, colored pencils, scissors
- Draw a hexagon like the hexagons above. Color in the exterior angles as well.
- Cut out each exterior angle and label them 1-6.
- Fit the six angles together by putting their vertices together. What happens?
The angles all fit around a point, meaning that the exterior angles of a hexagon add up to \begin{align*}360^\circ\end{align*}, just like a triangle. We can say this is true for all polygons.
Exterior Angle Sum Theorem: The sum of the exterior angles of any polygon is \begin{align*}360^\circ\end{align*}.
Proof of the Exterior Angle Sum Theorem
Given: Any \begin{align*}n-\end{align*}gon with \begin{align*}n\end{align*} sides, \begin{align*}n\end{align*} interior angles and \begin{align*}n\end{align*} exterior angles.
Prove: \begin{align*}n\end{align*} exterior angles add up to \begin{align*}360^\circ\end{align*}
NOTE: The interior angles are \begin{align*}x_1, x_2, \ldots x_n\end{align*}.
The exterior angles are \begin{align*}y_1, y_2, \ldots y_n\end{align*}.
Statement | Reason |
---|---|
1. Any \begin{align*}n-\end{align*}gon with \begin{align*}n\end{align*} sides, \begin{align*}n\end{align*} interior angles and \begin{align*}n\end{align*} exterior angles. | Given |
2. \begin{align*}x_n^\circ\end{align*} and \begin{align*}y_n^\circ\end{align*} are a linear pair | Definition of a linear pair |
3. \begin{align*}x_n^\circ\end{align*} and \begin{align*}y_n^\circ\end{align*} are supplementary | Linear Pair Postulate |
4. \begin{align*}x_n^\circ+ y_n^\circ=180^\circ\end{align*} | Definition of supplementary angles |
5. \begin{align*}(x_1^\circ+x_2^\circ+\ldots+x_n^\circ)+(y_1^\circ+ y_2^\circ+\ldots+ y_n^\circ)=180^\circ n\end{align*} | Sum of all interior and exterior angles in an \begin{align*}n-\end{align*}gon |
6. \begin{align*}(n-2)180^\circ=(x_1^\circ+ x_2^\circ+\ldots+x_n^\circ)\end{align*} | Polygon Sum Formula |
7. \begin{align*}180^\circ n=(n-2)180^\circ+(y_1^\circ+ y_2^\circ+\ldots+ y_n^\circ)\end{align*} | Substitution PoE |
8. \begin{align*}180^\circ n=180^\circ n-360^\circ+(y_1^\circ+ y_2^\circ+\ldots+ y_n^\circ)\end{align*} | Distributive PoE |
9. \begin{align*}360^\circ=(y_1^\circ+ y_2^\circ+\ldots+ y_n^\circ)\end{align*} | Subtraction PoE |
Example 5: What is \begin{align*}y\end{align*}?
Solution: \begin{align*}y\end{align*} is an exterior angle, as well as all the other given angle measures. Exterior angles add up to \begin{align*}360^\circ\end{align*}, so set up an equation.
\begin{align*}70^\circ + 60^\circ + 65^\circ + 40^\circ + y & = 360^\circ\\ y & = 125^\circ\end{align*}
Example 6: What is the measure of each exterior angle of a regular heptagon?
Solution: Because the polygon is regular, each interior angle is equal. This also means that all the exterior angles are equal. The exterior angles add up to \begin{align*}360^\circ\end{align*}, so each angle is \begin{align*}\frac{360^\circ}{7} \approx 51.43^\circ\end{align*}.
Know What? Revisited A regular polygon has congruent sides and angles. A regular hexagon has \begin{align*}(6-2)180^\circ=4\cdot180^\circ=720^\circ\end{align*} total degrees. Each angle would be \begin{align*}720^\circ\end{align*} divided by 6 or \begin{align*}120^\circ\end{align*}.
Review Questions
- Fill in the table.
# of sides | # of \begin{align*}\triangle s\end{align*} from one vertex | \begin{align*}\triangle s \times 180^\circ\end{align*} (sum) | Each angle in a regular \begin{align*}n-\end{align*}gon | Sum of the exterior angles |
---|---|---|---|---|
3 | 1 | \begin{align*}180^\circ\end{align*} | \begin{align*}60^\circ\end{align*} | |
4 | 2 | \begin{align*}360^\circ\end{align*} | \begin{align*}90^\circ\end{align*} | |
5 | 3 | \begin{align*}540^\circ\end{align*} | \begin{align*}108^\circ\end{align*} | |
6 | 4 | \begin{align*}720^\circ\end{align*} | \begin{align*}120^\circ\end{align*} | |
7 | ||||
8 | ||||
9 | ||||
10 | ||||
11 | ||||
12 |
- What is the sum of the angles in a 15-gon?
- What is the sum of the angles in a 23-gon?
- The sum of the interior angles of a polygon is \begin{align*}4320^\circ\end{align*}. How many sides does the polygon have?
- The sum of the interior angles of a polygon is \begin{align*}3240^\circ\end{align*}. How many sides does the polygon have?
- What is the measure of each angle in a regular 16-gon?
- What is the measure of each angle in an equiangular 24-gon?
- What is the measure of each exterior angle of a dodecagon?
- What is the measure of each exterior angle of a 36-gon?
- What is the sum of the exterior angles of a 27-gon?
- If the measure of one interior angle of a regular polygon is \begin{align*}160^\circ\end{align*}, how many sides does it have?
- How many sides does a regular polygon have if the measure of one of its interior angles is \begin{align*}168^\circ\end{align*}?
- If the measure of one interior angle of a regular polygon is \begin{align*}158 \frac{14}{17}^\circ\end{align*}, how many sides does it have?
- How many sides does a regular polygon have if the measure of one exterior angle is \begin{align*}15^\circ\end{align*}?
- If the measure of one exterior angle of a regular polygon is \begin{align*}36^\circ\end{align*}, how many sides does it have?
- How many sides does a regular polygon have if the measure of one exterior angle is \begin{align*}32 \frac{8}{11}^\circ\end{align*}?
For questions 11-20, find the measure of the missing variable(s).
- The interior angles of a pentagon are \begin{align*}x^\circ, x^\circ, 2x^\circ, 2x^\circ\end{align*}, and \begin{align*}2x^\circ\end{align*}. What is the measure of the larger angles?
- The exterior angles of a quadrilateral are \begin{align*}x^\circ, 2x^\circ, 3x^\circ\end{align*}, and \begin{align*}4x^\circ\end{align*}. What is the measure of the smallest angle?
- The interior angles of a hexagon are \begin{align*}x^\circ, (x + 1)^\circ, (x + 2)^\circ, (x + 3)^\circ, (x + 4)^\circ\end{align*}, and \begin{align*}(x + 5)^\circ\end{align*}. What is \begin{align*}x\end{align*}?
- Challenge Each interior angle forms a linear pair with an exterior angle. In a regular polygon you can use two different formulas to find the measure of each exterior angle. One way is \begin{align*}\frac{360^\circ}{n}\end{align*} and the other is \begin{align*}180^\circ - \frac{(n-2)180^\circ}{n}\end{align*} (\begin{align*}180^\circ\end{align*} minus Equiangular Polygon Formula). Use algebra to show these two expressions are equivalent.
- Angle Puzzle Find the measures of the lettered angles below given that \begin{align*}m \ || \ n\end{align*}.
Review Queue Answers
- \begin{align*}72^\circ + (7x+3)^\circ + (3x+5)^\circ = 180^\circ\!\\ {\;}\qquad \qquad \qquad \quad \ 10x + 80^\circ = 180^\circ\!\\ {\;}\qquad \qquad \qquad \qquad \ \ \quad \ 10x = 100^\circ\!\\ {\;}\qquad \qquad \qquad \qquad \ \ \qquad \ x = 10^\circ\end{align*}
- \begin{align*}(5x+17)^\circ +(3x - 5)^\circ = 180^\circ\!\\ {\;}\qquad \qquad \quad \ \ 8x +12^\circ = 180^\circ\!\\ {\;}\qquad \qquad \qquad \qquad \ 8x = 168^\circ\!\\ {\;}\qquad \qquad \qquad \qquad \ \ \ x = 21^\circ\end{align*}
- \begin{align*}w = 108^\circ, x = 49^\circ, y = 131^\circ, z = 121^\circ\end{align*}
- \begin{align*}360^\circ\end{align*}
- \begin{align*}59^\circ + 72^\circ\end{align*}
- interior angles, exterior angles
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