If you are citizen of an European Union member nation, you may not use this service unless you are at least 16 years old.
You already know Dokkio is an AI-powered assistant to organize & manage your digital files & messages. Very soon, Dokkio will support Outlook as well as One Drive. Check it out today!
Watch this funny video for a review of angles. They call same-side interior angles by the name co-interior angles. I think I like that better; it is short.
1. are created by line segments that intersect (touch) each other only at their endpoints; these segments are called sides.
2. these sides are not collinear (do not lay on top of each other!)
Be sure you understand what aconvex polygon is. The definition is useful, butrecognizing their
appearance is more important. A polygon is either concave or convex.
There are only 2 concave ones in the diagram above, the pink ones. The other 3 polygons are convex.
Two sides intersect at their endpoints to create the point called vertex. The plural for vertex is vertices.
The polygon with 7 sides will have 7 vertices. Look at the concave polygon above that has 7 sides.
( It looks like a crown!) Be sure you can count the vertices; there are 7.
The vertices are the points where 2 sides intersect.
The NUMBER of SIDES that a Polygon can have ranges from 3 to however many you would like to draw or build!
Quadrilateral has 4 sides and 4 interior angles and 4 vertices.
Pentagon has 5 sides and 5 interior angles and 5 vertices.
Hexagon has 6 sides and 6 interior angles and 6 vertices.
Heptagon has 7 sides and 7 interior angles and 7 vertices.
Octagon has 8 sides and 8 interior angles and 8 vertices.
Nonagon has 9 sides and 9 interior angles and 9 vertices.
Decagon has 10 sides and 10 interior angles and 10 vertices.
What polygon has only 3 sides? TRIANGLE
NOTICE in the DIAGRAM ABOVE that:
The triangles in the top pentagon overlap each other's angles so we cannot use it to find the total degrees of the pentagon.
The triangles in the lower pentagon do not overlap. The angles of the 3 YELLOW triangles create the 5 angles of the pentagon.
Draw a convex octagon.
That is a STOP sign that you see at the intersection of two roads.
How many sides does it have? An octagon has 8 sides and 8 interior angles too.
Choose one vertex and draw all the diagonals that you can from it.
I' ll wait while you do this!!!!
Now count thediagonals. How many did you get? Did you get 8????
I hope NOT.
There are8 sides and 8 vertices in an octagon.
BUT you should be able to draw only 5 diagonals from ONE VERTEX.
That should have created 6 triangles. Count the triangles.
Yes, the number of triangles is always 2 less than the number of sides of the polygon.
We have 8 sides, but only 6 triangles created by the 5 diagonals.
In the pentagon we had 5 sides and 3 triangles.
So in a hexagon we will have 6 sides and how many triangles? look: 4
I sure hope you drew the octagon.
Did you draw all the diagonals that you couldfrom one vertex?
You should have 6 triangles.
WHY is this important?? LOOK back at the yellow pentagon above. We can SEE that all the angles of the triangles are really the interior angles of the polygon.
We know that a triangle has 180 degrees in its angles.
Thus we can now find the number of degrees in the interior angles of any polygon
if we know the number of TRIANGLES!!
The total degrees for the interior angles of 1 triangle is 180 degrees.
So guess whatthe total degrees of the interior angles of a polygon are ?
The TOTAL degrees (sum) of the INTERIOR ANGLES of a convex polygon:
180 times the number of triangles that we drew using only the one vertex.
The total degrees for the 4 interior angles of a 4-sided convex polygon is 180 times 2= 360 degrees.
The total degrees for the 5 interior angles of a 5-sided convex polygon is 180 times 3= 540 degrees.
The total degrees for the 7 interior angles of a 7-sided convex polygon is 180 times 5= 900 degrees.
REMEMBER the pattern!!
Or the formula:
Total Degrees of interior angles of polygon= 180(n-2) where n-2 means 2 less than the number of sides.
The number of triangles in the convex polygon (drawn from 1 vertex)
is always 2 less than the number of sides; we write n-2.
BUT what about the exterior angles (recall how we extend the sides) of TRIANGLES?
We practiced that in another Lesson.
The TOTAL degrees (sum) of the EXTERIOR ANGLES of any convex polygon is the SAME for all polygons.
It is always 360 degrees. We do not have to figure it out! The sum of the exterior angles is 360 degrees for any convex polygon.
Yes, the sum of the exterior angles for
a pentagon or
an octagon or
a hexagon or
a decagon or dodecagon (12 sides)
any convex polygon
is 360 degrees.
If we have REGULAR octagon ( regular means all sides are congruent and all angles are congruent too! 8 congruent sides, 8 congruent interior angles, 8 congruent exterior angles)
then one exterior angle is 360 divided by 8, or 45 degrees.
If we have REGULAR nonagon (9 congruent sides, 9 congruent interior angles, 9 congruent exterior angles)
then one exterior angle is 360 divided by 9, or 40 degrees.
congruent means equal measurement
Here is a video for this lesson
Can you win all 2500 points? TRY THIS GAME more than once;
Look at exterior angle then look at interior angle that is adjacent to it.
Do you see those at a vertex of the polygon?
The interior angle + its exterior angle = 180 degrees.
These 2 angles are called a linear pair.
If we know the exterior angle then we can now find the interior angle!
A REGULAR Nonagon has 40 degrees for one exterior angle.
Thus one interior angle is 180-40 = 140 degrees.
We can check that with the total of the interior angles.
RECALL the triangles drawn from one diagonal for a nonagon is 9-2 or 7.
The regular nonagon has:
9 congruent sides and
7 triangles (draw 6 diagonals from 1 vertex) and
9 congruent interior angles and
9 congruent exterior angles.
Total of the interior angles = 180 degrees times (9-2) = 180(7) = 1260 degrees.
That is 180 times the triangles!
So one (1) interior angle of a nonagon must be 1260 divided by 9.
Do the division and you will get 140 degrees.
YEA, that is what we had when we used the 360/9 = 40 for the exterior angle. So, Interior angle is 180-40 = 140.
We can find the interior angle two ways.
One way is to begin with the formula 180(n-2) and
the other way is to begin with the exterior angle.
You must know both ways to find the interior angle.
Can you do these in reverse? RECALL the interior angle and the exterior angle are a LINEAR PAIR. If we have 150 degrees for the interior angles then what is the measure of the exterior angle?_____ 180-150=30 degrees. So if the exterior angles are each 30 then how many sides or interior angles does this polygon have? Recall that all the exterior angles will add to 360. How many angles of 30 degrees each are needed to make the 360?
That is just dividing! 360/30 = 12. 12 sided polygon is called a dodecagon.
CAN you win $1 million at Quia? Play this game until you do.
Can you find the measure of the exterior angles and the interior angles?
Can you find the number of sides (or interior angles)?
Take a screenshot of your win for your Lesson Notes. Remember SAVE it to your desktop so you can drag it to the Lesson Notes. Or you can learn to UPLOAD a file.
