Angles with triangles and polygons
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Proof - Sum of Measures of Angles in a Triangle are 180
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Triangle Angle Example 1
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Triangle Angle Example 2
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Triangle Angle Example 3
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Challenging Triangle Angle Problem
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Proof - Corresponding Angle Equivalence Implies Parallel Lines
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Finding more angles
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Angles 1
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Angles 2
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Sum of Interior Angles of a Polygon
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Angles of a polygon
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Sum of the exterior angles of convex polygon
Sum of Interior Angles of a Polygon Showing a generalized way to find the sum of the interior angles of any polygon
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- We already know that the sum of the interior angles of a triangle
- add up to 180 degrees
- If the measure of this angle is A,
- the measure of this angle over here is B, and
- the measure of this angle is C, we know that A + B + C = 180 degrees
- What happens when we have polygons with more than 3 sides?
- a quadrilateral
- probably applies to any quadrilateral with four sides,
- not just things that have right angles and parallel lines
- and all the rest
- Actually, that looks a little too close to being parallel
- so let me draw it like this
- The way you can think about it, with the 4-sided quadrilateral is,
- we already know about this:
- the measures of the interior angles of a triangle add up to 180
- so maybe we can divide this into 2 triangles
- From this point right over here, if we draw a line like this,
- Then if measure of this angle is A, measure of this is B,
- measure of that is C
- We know that A + B + C = 180 degrees
- Then if we call this over here, X, this over here, Y and that, Z
- Those are the measures of those angles
- We know that X + Y + Z = 180 degrees
- So, if we want the measure of the sum of all of the interior angles,
- all of the interior angles are going to be:
- plus this angle, which is going to be A + X
- A + X is that whole angle for the quadrilateral
- Plus this whole angle, which is going to be C + Y
- And where you know A + B + C is 180 degrees
- And we know that Z + X + Y = 180 degrees
- So plus 180 degrees which is equal to 360 degrees
- I think you see the general idea here
- We just have to figure out how many triangles
- we can divide something into
- Then we just multiply it by 180 degrees,
- since each of those triangles will have 180 degrees
- can we fit into that thing
- Let me draw an irregular pentagon
- 1, 2, 3, 4, 5
- Looks more like a bit of a side-ways house there
- Once again, we can draw our triangles inside of this pentagon
- That would be one triangle there
- That would be another triangle
- that perfectly cover this pentagon
- This is one triangle, the other triangle and the other one
- We know each of those have 180 degrees,
- if we take the sum of their angles
- We also know the sum of all those interior angles are equal
- to the sum of the interior angles of the polygon as a whole
- To see that, clearly this interior angle is
- one of the angles of the polygon
- This is, as well
- When you take the sum of this one and this one,
- We take the sum of that one and that one, you get that entire one
- Then when you take the sum of that one, plus that one,
- plus that one, you get that entire interior angle
- So if you take the sum of all the interior angles of
- all of the interior angles of the polygon
- In this case you have 1, 2, 3 triangles
- 3 times 180 degrees is equal to what?
- 300 +240 = 540 degrees
- To generalize it,
- we have up to use up four sides
- We have to use up all the four sides of this quadrilateral
- We had to use up four of the five sides right here in this pentagon
- 1, 2, and then 3, 4
- So four sides give you two triangles
- It seems like maybe every incremental side you have after that,
- you can get another triangle out of it
- 1, 2, 3, 4, 5, 6 sides
- and I can get one triangle out of these 2 sides
- 1, 2 sides of the actual hexagon
- I can get another triangle out of these 2 sides of the actual hexagon
- And it looks like I can get another triangle
- out of each of the remaining sides
- So one out of that one, and then,
- one out of that one right over there
- S-sided polygon
- or 6 sides
- So we can assume that S is greater than 4 sides
- I want to figure out how many non-overlapping triangles
- that perfectly cover that polygon
- How many can I fit inside of it
- Then I just have to multiply the number of triangles
- times 180 degrees
- to figure out what are the sum of the interior angles of that polygon
- as a function of the number of sides
- Once again, four of the sides are going to be used
- to make two triangles
- and we have two sides right over there
- I can have, I can draw one triangle
- what happens to the rest of the sides of the polygon
- You can imagine putting a big black piece of construction paper
- There might be other sides here
- So out of these 2 sides I can draw one triangle just like that
- Out of these two sides, I can draw another triangle right over there
- So 4 sides used for two triangles
- Then, no matter how many sides I have left over,
- if I have all sorts of craziness here
- Let me draw a little bit neater than that
- So I can have all sorts of craziness right over here
- It looks like every other incremental side
- I can get another triangle out of it
- one triangle out of that side, one triangle out of that side
- and then one triangle out of this side
- For example, this figure that I have drawn is a very irregular
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, is that right?
- 1, 2, 3, 4, 5, 6, 7 ,8 ,9 10 It is a decagon
- In this decagon, four of the sides were used for two triangles
- Then the other 6 sides I was able to get a triangle each
- I have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
- Did I count, am I just not seeing something?
- Oh I see, I have to draw another line right over here
- These are two different sides
- I can get another triangle out of that right over there
- There you have it
- I have these two triangles out of 4 sides
- Then out of the other 6 remaining sides I get a triangle each
- Plus 6 triangles, I got a total of 8 triangles
- So we can generally think about it
- Let me write this down
- Our number of triangles is going to be equal to 2,
- The remaining sides, I get a triangle each
- The remaining sides are going to be S minus 4
- The number of triangles are going to be 2 plus S minus 4
- So, if I have an S-sided polygon, I can get S minus two triangles
- that perfectly cover that polygon
- Which tells us that an S-sided polygon if it has S minus 2 triangles,
- that the interior angles in it are going to be
- S minus 2 times 180 degrees, which is a pretty cool result
- So someone told you that they had a 102-sided polygon
- So, S is equal to 102 sides
- You can say, okay the number of interior angles are going to be
- Which is equal to 180, with two more zeros behind it
- of a 102-sided polygon
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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