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| 18 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Izabella is excited because her father got two tickets to a game played by their favorite baseball ⚾ team. On the way to the stadium, Izabella sees a stop sign and is curious about its shape. Her dad says that all the sides of the sign are the same length and all the interior angles have the same measure.
What is the measure of the labeled angle inside the stop sign?
Plane figures can be classified according to the number of angles they have. For example, triangles have three angles. Triangles and figures with more than three angles are also called polygons because they have several angles.
A polygon consists of three or more line segments, called sides, whose endpoints connect end-to-end to enclose an area. Some examples of polygons include triangles, squares, and rectangles. Polygons are denoted algebraically by writing the names of their vertices in consecutive order, either clockwise or counterclockwise.
Polygons come in different shapes and sizes, and there is no maximum limit to the number of line segments used to form a polygon. Polygons with more than four sides are commonly named using the Greek prefix for the number of sides, followed by -gon.
Think about the name in the applet before pressing ⬆️ and ⬇️ to increase or decrease the number of sides.
The sides of a polygon are line segments that connect two consecutive vertices. Segments that connect two non-consecutive vertices have a different name.
Depending on its shape, a polygon can be classified as convex or concave. Alternatively, this classification can be done by studying the measures of the interior angles.
The interior angles of a polygon are the angles formed by two adjacent sides on the inside of the polygon.
The sum of the measures of the interior angles of a polygon with n vertices is given by the following formula.
(n−2)180∘
The formula can be applied to any polygon.
Izabella is having a great time by the time she makes it inside the stadium. It is her first time in a baseball stadium and she is curious about everything.
Substitute values
Add terms
LHS−450∘=RHS−450∘
Add and subtract terms
LHS−60=RHS−60
LHS/6=RHS/6
Expression | x=80 | Measure |
---|---|---|
x+5 | 80+5 | 85∘ |
2x−20 | 2(80)−20 | 140∘ |
A regular polygon is a polygon whose sides and angles are all congruent.
Regular polygons with three sides are generally referred to as either equilateral or equiangular triangles. Regular polygons with four sides are called squares. Regular polygons with more than four sides do not typically have special names.
The greater the number of sides of a regular polygon, the more it resembles a circle. When a polygon is not regular, it is said to be an irregular polygon.
When a polygon is regular, all its sides and angles are congruent. Use the Polygon Interior Angles Theorem. Divide the sum of the measures of the interior angles by the total number of interior angles.
Start by recalling that in a regular polygon, all the angles have the same measure. Let x be the measure of each interior angle of a regular hexagon.
n=6
Subtract term
Multiply
LHS/6=RHS/6
Determine the missing angle measure. Here, the polygons where all the sides are the same length are regular polygons. Round the answer to the nearest integer if needed.
Interior Angle | Corresponding Exterior Angles | Sum of Measures |
---|---|---|
∠11 | ∠1 and ∠2 | m∠1+m∠11=180∘ m∠2+m∠11=180∘ |
∠12 | ∠3 and ∠4 | m∠3+m∠12=180∘ m∠4+m∠12=180∘ |
∠13 | ∠5 and ∠6 | m∠5+m∠13=180∘ m∠6+m∠13=180∘ |
∠14 | ∠7 and ∠8 | m∠7+m∠14=180∘ m∠8+m∠14=180∘ |
∠15 | ∠9 and ∠10 | m∠9+m∠15=180∘ m∠10+m∠15=180∘ |
In contrast, when polygon is concave, some exterior angles might lie inside the polygon. In this case, the measures of those angles are considered to be negative.
Although the sum of the measures of the interior angles of a polygon varies according to the number of sides the polygon has, the sum of the measures of the exterior angles is always constant.
The sum of the measures of the exterior angles of a polygon, one angle at each vertex, is 360∘.
Based on the diagram, the relation below holds true.
m∠1+ m∠2+ m∠3+ m∠4+ m∠5+ m∠6=360∘
When Izabella and her father got home, they went to the kitchen to prepare some pancakes and eat them with the honey they bought. What a way to end the day!
Add and subtract terms
LHS−10=RHS−10
LHS/35=RHS/35
Expression | x=10 | Measure |
---|---|---|
5x | 5(10) | 50∘ |
7x+5 | 7(10)+5 | 75∘ |
5x+2 | 5(10)+2 | 52∘ |
5x+3 | 5(10)+3 | 53∘ |
6x+1 | 6(10)+1 | 61∘ |
7x−1 | 7(10)−1 | 69∘ |
Determine the missing angle measure. Remember, every exterior angle forms a linear pair with its interior angle.
When Izabella was walking to the stadium, she saw a stop sign and wondered about the measure of one of its interior angles. According to her dad, all the sides of the stop sign are the same length and all the interior angles have the same measure.
Let's start by recalling the formula to calculate the sum of the interior angle measures of a polygon with n sides. This formula is given by the Polygon Interior Angles Theorem. Sum of Measures = (n-2)180^(∘) On the other hand, all the interior angles of a regular polygon have the same measure. Let x be the measure of the interior angles. Now, let's write a second expression representing the sum of the measures of the interior angles of a regular polygon with n sides. Sum of Measures &= x^(∘)+x^(∘)+⋯+x^(∘)_(n-times) &⇓ Sum of Measures &= n* x We can equate the right-hand sides of the two equations we wrote. (n-2)180^(∘) = n* x Next, we will substitute each of the given angle measures for x and solve the equation for n. Remember that n must be a positive integer greater than or equal to 3, as it represents the number of sides of the polygon. Let's begin by replacing x with 165^(∘).
We got that n=24, which is an integer. This means that a regular polygon with 24 sides has interior angle measures of 165^(∘). Therefore, 165^(∘) is a possible interior angle measure of a regular polygon. Let's repeat the same process with the other two measures. The results are summarized in the following table.
Angle | (n-2)180 = n* x | Simplify |
---|---|---|
164^(∘) | (n-2)180 = 164n | n=22.5 |
157.5^(∘) | (n-2)180 = 157.5n | n=16 |
For x=164^(∘), we found that n=22.5, which is not an integer. A polygon with 22.5 sides makes no sense. This means that 164^(∘) is not a possible interior angle measure of a regular polygon. In contrast, we got that n=16 when x=157.5^(∘), so, 157.5^(∘) is also a possible interior angle measure.
Of the given measures, only 165^(∘) and 157.5^(∘) are possible interior angle measures of a regular polygon.
Let's begin by considering a polygon with n sides. According to the Polygon Interior Angles Theorem, the sum of the measures of the interior angles is given by the following expression. ( n-2)180^(∘) Now, if we increase the number of sides by one, we will have a polygon with n+1 sides. We can use the Polygon Interior Angles Theorem again to write an expression that represents the sum of the measures of the interior angles of this new polygon. ( n+1-2)180^(∘) We want to determine by how much the sum of the measures of the interior angles increased. We can find it by subtracting the first expression from the second.
As we can see, the sum of the measures of the interior angles increased by 180^(∘). To help illustrate this concept, below are listed the sums of the measures of the interior angles of some polygons. We can see that the sum increases by 180^(∘) from one row to the next.
Polygon | Number of Sides | Sum of Interior Angle Measures |
---|---|---|
Triangle | 3 | ( 3-2)180^(∘) = 180^(∘) |
Quadrilateral | 4 | ( 4-2)180^(∘) = 360^(∘) |
Pentagon | 5 | ( 5-2)180^(∘) = 540^(∘) |
Hexagon | 6 | ( 6-2)180^(∘) = 720^(∘) |
Heptagon | 7 | ( 7-2)180^(∘) = 900^(∘) |