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Triangles are figures with three angles. However, this is not the maximum number of angles a figure can have. In this lesson, geometric figures with more than three angles are presented and studied, as well as some relationships between their angles.

Challenge

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?

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Discussion

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.

Concept

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.

Discussion

A vertex of a polygon is the point of intersection of two of its sides. A vertex can also be referred to as a **corner**.

Any polygon has as many vertices as it has sides.

Discussion

The sides of a polygon are line segments that connect two consecutive vertices. Segments that connect two non-consecutive vertices have a different name.

Concept

A diagonal of a polygon is a line segment connecting two non-consecutive vertices of the polygon. *consecutive* vertices of a polygon is a side of the polygon, not a diagonal. This means that triangles have no diagonals. The number of diagonals in an $n-$sided polygon can be calculated using the following formula.

It should be noted that a line segment that connects two

$Number of Diagonals=2n(n−3) $

Discussion

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.

Concept

A convex polygon is a polygon in which no line that contains a side of the polygon contains points in the interior of the polygon. Alternatively, a polygon is convex if all the interior angles measure less than $180_{∘}.$ Note that all triangles are convex polygons.

There are some characteristics that can be listed about convex polygons.

- All the vertices of the polygon point outwards.
- All the diagonals lie inside the polygon.
- The segment joining any two interior points of the polygon is fully contained in the polygon.

When a polygon is not convex, it is said to be concave.

Discussion

A concave polygon is a polygon in which at least one line containing a side of the polygon also contains points in the interior of the polygon. Alternatively, a polygon is concave if one or more interior angles have a measure greater than $180_{∘}.$ Note that only polygons with four or more sides can be concave.

Some features can be listed about concave polygons.

- At least one vertex of the polygon points inwards.
- Some of the diagonals may lie partly or entirely outside the polygon.
- There are points inside the polygon for which the segment joining them is not fully contained in the polygon.

Pop Quiz

Discussion

The interior angles of a polygon are the angles formed by two adjacent sides on the inside of the polygon.

Although the measures of the interior angles of a polygon can vary, the Polygon Interior Angles Theorem establishes a relationship between their sum and the number of sides of the polygon.Rule

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.

Example

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.

a Izabella notices that home plate is in the shape of a pentagon.

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b Izabella also observes that the field's border is a pentagon that looks like a diamond.

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a Use the Polygon Interior Angles Theorem to determine the sum of the interior angle measures of a pentagon. Solve the equation for $m∠D.$

b Use the Polygon Interior Angles Theorem. Set and solve an equation for $x.$ Substitute the value of $x$ into the appropriate angle measures.

$(n−2)180_{∘} $

As Izabella pointed out, the home plate is a pentagon. This means that $n=5.$
$(n−2)180_{∘}$

Substitute

$n=5$

$(5−2)180_{∘}$

SubTerm

Subtract term

$(3)180_{∘}$

Multiply

Multiply

$540_{∘}$

$m∠A+m∠B+m∠C+m∠D+m∠E=540_{∘} $

Take a look at the given graph to find some information about the angles.
Angles $A$ and $B$ are denoted with square angle markers. This means that both are right angles, so they measure $90_{∘}$ each. Angles $C$ and $E$ have a measure of $135_{∘}$ each. Substitute all these values into the equation and solve it for $m∠D.$
$m∠A+m∠B+m∠C+m∠D+m∠E=540_{∘}$

SubstituteValues

Substitute values

$90_{∘}+90_{∘}+135_{∘}+m∠D+135_{∘}=540_{∘}$

AddTerms

Add terms

$m∠D+450_{∘}=540_{∘}$

SubEqn

$LHS−450_{∘}=RHS−450_{∘}$

$m∠D=90_{∘}$

b The baseball field is a polygon with five sides. This means that the sum of the measures of the interior angles is $540_{∘}.$ Notice that the angle at the home plate is a right angle, so it measures $90_{∘}.$ Write an equation using all the given data. The degree symbol will be omitted for simplicity.

$x+5+2x−20+2x−20+x+5+90=540 $

Next, solve the equation for $x.$
$x+5+2x−20+2x−20+x+5+90=540$

AddSubTerms

Add and subtract terms

$6x+60=540$

SubEqn

$LHS−60=RHS−60$

$6x=480$

DivEqn

$LHS/6=RHS/6$

$x=80$

Expression | $x=80$ | Measure |
---|---|---|

$x+5$ | $80+5$ | $85_{∘}$ |

$2x−20$ | $2(80)−20$ | $140_{∘}$ |

${90 ,85 ,85 ,140 ,140 } $

Discussion

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.

Example

The game ended with an incredible victory for Izabella's team. What a game! On the way home, she and her father stop at a bee farm to buy honey 🍯. There, Izabella learns that honeycombs are made up of many hexagons.
### Hint

### Solution

The measure of each interior angle of a regular hexagon is $120_{∘}.$

Assume each hexagon is regular. What is the measure of each interior angle?

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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.

The sum of the measures of the interior angles is $6x.$ The total sum of these measures can be calculated using the Polygon Interior Angles Theorem. In this case, $n=6.$$6x=(n−2)180_{∘}$

Substitute

$n=6$

$6x=(6−2)180_{∘}$

SubTerm

Subtract term

$6x=(4)180_{∘}$

Multiply

Multiply

$6x=720_{∘}$

DivEqn

$LHS/6=RHS/6$

$x=120_{∘}$

Pop Quiz

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.

Discussion

An exterior angle of a polygon is an angle formed between one side of the polygon and the extension of an adjacent side. An $n-$sided polygon has $2n$ exterior angles, two at each vertex.

In a convex polygon, each exterior angle lies outside the polygon and forms a linear pair with its corresponding interior angle. This means that each exterior angle is supplementary to its interior angle.

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.

Discussion

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.

Rule

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.

$m∠1+$ $m∠2+$ $m∠3+$ $m∠4+$ $m∠5+$ $m∠6=360_{∘}$

Example

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!

External credits: @upklyak

a While eating, Izabella noticed the cute name and cool logo on the honey pot.

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a Use the Polygon Exterior Angles Theorem.

b Use the Polygon Exterior Angles Theorem. Solve the resulting equation for $x.$ Substitute the value of $x$ into each of the expressions for the angle measures.

a Start by noticing that each of the labeled angles is an exterior angle corresponding to the pentagon in the middle of the logo.

$x+68_{∘}+68_{∘}+70_{∘}+70_{∘}=360_{∘} $

Solve the equation for $x.$
$x+68_{∘}+68_{∘}+70_{∘}+70_{∘}=360_{∘}$

AddTerms

Add terms

$x+276_{∘}=360_{∘}$

SubEqn

$LHS−344_{∘}=RHS−344_{∘}$

$x=84_{∘}$

b The angles labeled in Izabella's logo are exterior angles corresponding to the hexagon in the middle of the drawing.

$5x+7x+5+5x+2+5x+3+6x+1+7x−1=360$

AddSubTerms

Add and subtract terms

$35x+10=360$

SubEqn

$LHS−10=RHS−10$

$35x=350$

DivEqn

$LHS/35=RHS/35$

$x=10$

Expression | $x=10$ | Measure |
---|---|---|

$5x$ | $5(10)$ | $50_{∘}$ |

$7x+5$ | $7(10)+5$ | $75_{∘}$ |

$5x+2$ | $5(10)+2$ | $52_{∘}$ |

$5x+3$ | $5(10)+3$ | $53_{∘}$ |

$6x+1$ | $6(10)+1$ | $61_{∘}$ |

$7x−1$ | $7(10)−1$ | $69_{∘}$ |

Pop Quiz

Determine the missing angle measure. Remember, every exterior angle forms a linear pair with its interior angle.

Closure

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.

The stop sign has eight angles, so it is a regular octagon, based on Izabella's dad description. All the interior angles have a measure of $x_{∘}.$ The sum of all interior angles is $8x.$ Additionally, this sum can be calculated using the Polygon Interior Angles Theorem. The measure of each interior angle of the stop sign is $135_{∘}.$ Before moving on, keep in mind that polygons are everywhere in the real world. For example, they can be seen on kitchen tiles, traffic signals, honeycombs, windows, and many more objects.Loading content