Exploring Polygons: Types, Angles, and Interior-Exterior Theorems

Expression	$x = 80$	Measure
$x + 5$	$80 + 5$	$8 5^{\circ}$
$2 x - 20$	$2 (80) - 20$	$14 0^{\circ}$

Expression

x = 80

Measure

x + 5

80 + 5

8 5^{\circ}

2 x - 20

2 (80) - 20

14 0^{\circ}

Interior Angle	Corresponding Exterior Angles	Sum of Measures
$∠ 11$	$∠ 1$ and $∠ 2$	$m ∠ 1 + m ∠ 11 = 18 0^{\circ}$ $m ∠ 2 + m ∠ 11 = 18 0^{\circ}$
$∠ 12$	$∠ 3$ and $∠ 4$	$m ∠ 3 + m ∠ 12 = 18 0^{\circ}$ $m ∠ 4 + m ∠ 12 = 18 0^{\circ}$
$∠ 13$	$∠ 5$ and $∠ 6$	$m ∠ 5 + m ∠ 13 = 18 0^{\circ}$ $m ∠ 6 + m ∠ 13 = 18 0^{\circ}$
$∠ 14$	$∠ 7$ and $∠ 8$	$m ∠ 7 + m ∠ 14 = 18 0^{\circ}$ $m ∠ 8 + m ∠ 14 = 18 0^{\circ}$
$∠ 15$	$∠ 9$ and $∠ 10$	$m ∠ 9 + m ∠ 15 = 18 0^{\circ}$ $m ∠ 10 + m ∠ 15 = 18 0^{\circ}$

Interior Angle

Corresponding Exterior Angles

Sum of Measures

∠ 11

∠ 1

and

∠ 2

m ∠ 1 + m ∠ 11 = 18 0^{\circ}

m ∠ 2 + m ∠ 11 = 18 0^{\circ}

∠ 12

∠ 3

and

∠ 4

m ∠ 3 + m ∠ 12 = 18 0^{\circ}

m ∠ 4 + m ∠ 12 = 18 0^{\circ}

∠ 13

∠ 5

and

∠ 6

m ∠ 5 + m ∠ 13 = 18 0^{\circ}

m ∠ 6 + m ∠ 13 = 18 0^{\circ}

∠ 14

∠ 7

and

∠ 8

m ∠ 7 + m ∠ 14 = 18 0^{\circ}

m ∠ 8 + m ∠ 14 = 18 0^{\circ}

∠ 15

∠ 9

and

∠ 10

m ∠ 9 + m ∠ 15 = 18 0^{\circ}

m ∠ 10 + m ∠ 15 = 18 0^{\circ}

Expression	$x = 10$	Measure
$5 x$	$5 (10)$	$5 0^{\circ}$
$7 x + 5$	$7 (10) + 5$	$7 5^{\circ}$
$5 x + 2$	$5 (10) + 2$	$5 2^{\circ}$
$5 x + 3$	$5 (10) + 3$	$5 3^{\circ}$
$6 x + 1$	$6 (10) + 1$	$6 1^{\circ}$
$7 x - 1$	$7 (10) - 1$	$6 9^{\circ}$

Expression

x = 10

Measure

5 x

5 (10)

5 0^{\circ}

7 x + 5

7 (10) + 5

7 5^{\circ}

5 x + 2

5 (10) + 2

5 2^{\circ}

5 x + 3

5 (10) + 3

5 3^{\circ}

6 x + 1

6 (10) + 1

6 1^{\circ}

7 x - 1

7 (10) - 1

6 9^{\circ}

Let's start by recalling what a convex and a concave polygon are.

Convex Polygon	Concave Polygon
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^(∘).	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^(∘).

In our case, we are not given angle measures. However, we can use some additional characteristics of convex and concave polygons.

Characteristics
Convex Polygon	Concave Polygon
All the vertices of the polygon point outwards.	At least one vertex of the polygon points inwards.
All diagonals lie inside the polygon.	Some of the diagonals may lie partly or entirely outside the polygon.
The segment joining any two interior points of the polygon is fully contained in the polygon.	There are points inside the polygon for which the segment joining them is not fully contained in the polygon.

For the given polygon, we can see that one of its vertices points inwards. Also, notice that at least one diagonal is not fully contained inside the polygon.

As a result, we can say that the given polygon is concave. We can also use a protractor and find out that one angle measure is greater than 180^(∘). This confirms our conclusion.

Let's list some characteristics that we can see in the given polygon.

All vertices point outwards.
All diagonals lie inside the polygon.
All interior angle measures seem to be less than 180^(∘). We can use a protractor to verify this.

If we compare these characteristics to the table we wrote in Part A, we can conclude that the given polygon is convex.

Let's begin by recalling what an exterior angle of a polygon is.

Exterior Angles of a Polygon |- An angle formed between one side of the polygon and the extension of an adjacent side.

We can see four extended sides in the given diagram. Additionally, we can see that angles 2 and 5 are formed by the extended sides. This does not match the definition, so these two angles are not exterior angles.

In contrast, angles 1, 3, 4, and 6 are all formed by one side of the polygon and the extension of an adjacent side. This means that these four angles are exterior angles of the polygon. Exterior Angles ∠ 1, ∠ 3, ∠ 4, ∠ 6

Notice that two of the marked angles lie inside the polygon. This does not necessarily contradict the definition of an exterior angle, though. We have to look for angles formed by one side of the polygon and an extension of an adjacent side. Angles 1 and 5 do not meet this condition because they are formed by two extended sides.

Angles 2, 3, and 4 are formed by one side of the polygon and the extension of an adjacent side. Although ∠ 3 lies inside the polygon, it meets the definition. As a consequence, these three angles are exterior angles. Exterior Angles ∠ 2, ∠ 3, ∠ 4

We can see that two of the labeled angles are inside the polygon. As we noted before, this does not mean that they cannot be exterior angles. Let's look for angles formed by one side of the polygon and an extension of an adjacent side. We can see that angles 1 and 2 are formed by one side and the extension of the same side.

This means that ∠ 1 and ∠ 2 are not exterior angles. On the contrary, angles 3 and 4 satisfy the definition, so they are exterior angles. Exterior Angles ∠ 3, ∠ 4

Polygons and Angles

Catch-Up and Review

Stop!

Figures With Three or More Angles

Polygon

Vertex of a Polygon

Connecting Non-consecutive Vertices

Diagonal of a Polygon

Classifying Polygons by Shape

Convex Polygon

Concave Polygon

Convex or Concave?

Interior Angles of a Polygon

Polygon Interior Angles Theorem

Play Ball!

Hint

Solution

Regular Polygons

Honeycombs

Hint

Solution

Finding the Missing Angle Measure

Exterior Angles of a Polygon

Sum of Measures of Exterior Angles

Polygon Exterior Angles Theorem

Designing a Logo

Hint

Solution

Finding an Exterior Angle Measure

Final Stop

Polygons and Angles

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