And I'll just assume-- we already saw the case for four sides, five sides, or six sides. And we already know a plus b plus c is 180 degrees. 6-1 practice angles of polygons answer key with work at home. That is, all angles are equal. They'll touch it somewhere in the middle, so cut off the excess. If the number of variables is more than the number of equations and you are asked to find the exact value of the variables in a question(not a ratio or any other relation between the variables), don't waste your time over it and report the question to your professor. 6 1 word problem practice angles of polygons answers. So that would be one triangle there.
And it looks like I can get another triangle out of each of the remaining sides. The way you should do it is to draw as many diagonals as you can from a single vertex, not just draw all diagonals on the figure. So the remaining sides I get a triangle each. 6-1 practice angles of polygons answer key with work table. Not just things that have right angles, and parallel lines, and all the rest. What does he mean when he talks about getting triangles from sides? So the remaining sides are going to be s minus 4. Out of these two sides, I can draw another triangle right over there.
Let's say I have an s-sided polygon, and I want to figure out how many non-overlapping triangles will perfectly cover that polygon. So plus 180 degrees, which is equal to 360 degrees. So let's figure out the number of triangles as a function of the number of sides. I can get another triangle out of these two sides of the actual hexagon. Hexagon has 6, so we take 540+180=720. Did I count-- am I just not seeing something? The whole angle for the quadrilateral. One, two, and then three, four. And so if we want the measure of the sum of all of the interior angles, all of the interior angles are going to be b plus z-- that's two of the interior angles of this polygon-- plus this angle, which is just going to be a plus x. a plus x is that whole angle. I can get another triangle out of that right over there. 6-1 practice angles of polygons answer key with work today. One, two sides of the actual hexagon. So those two sides right over there.
Now let's generalize it. So let's try the case where we have a four-sided polygon-- a quadrilateral. And we know each of those will have 180 degrees if we take the sum of their angles. What are some examples of this? And it seems like, maybe, every incremental side you have after that, you can get another triangle out of it.
So in this case, you have one, two, three triangles. Once again, we can draw our triangles inside of this pentagon. As we know that the sum of the measure of the angles of a triangle is 180 degrees, we can divide any polygon into triangles to find the sum of the measure of the angles of the polygon. So if we know that a pentagon adds up to 540 degrees, we can figure out how many degrees any sided polygon adds up to. So let me make sure. I get one triangle out of these two sides. Created by Sal Khan. Plus this whole angle, which is going to be c plus y. So I have one, two, three, four, five, six, seven, eight, nine, 10.
But clearly, the side lengths are different. And so there you have it. That would be another triangle. We had to use up four of the five sides-- right here-- in this pentagon. Extend the sides you separated it from until they touch the bottom side again. And to generalize it, let's realize that just to get our first two triangles, we have to use up four sides. So if you take the sum of all of the interior angles of all of these triangles, you're actually just finding the sum of all of the interior angles of the polygon. There might be other sides here. Whys is it called a polygon? Same thing for an octagon, we take the 900 from before and add another 180, (or another triangle), getting us 1, 080 degrees. Let's experiment with a hexagon. 300 plus 240 is equal to 540 degrees. You could imagine putting a big black piece of construction paper. So we can assume that s is greater than 4 sides.
Imagine a regular pentagon, all sides and angles equal. In a square all angles equal 90 degrees, so a = 90. Orient it so that the bottom side is horizontal. How many can I fit inside of it? So it looks like a little bit of a sideways house there. And I am going to make it irregular just to show that whatever we do here it probably applies to any quadrilateral with four sides.
And then if we call this over here x, this over here y, and that z, those are the measures of those angles. Well there is a formula for that: n(no. You can say, OK, the number of interior angles are going to be 102 minus 2. This sheet covers interior angle sum, reflection and rotational symmetry, angle bisectors, diagonals, and identifying parallelograms on the coordinate plane. So out of these two sides I can draw one triangle, just like that. With two diagonals, 4 45-45-90 triangles are formed. I can draw one triangle over-- and I'm not even going to talk about what happens on the rest of the sides of the polygon.
So I could have all sorts of craziness right over here. So I got two triangles out of four of the sides. 180-58-56=66, so angle z = 66 degrees.