Congruence and isosceles and equilateral triangles
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Congruent legs and base angles of Isosceles Triangles
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Equilateral Triangle Sides and Angles Congruent
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Equilateral and Isosceles Example Problems
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Triangle types
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Triangle angles 1
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Another Isosceles Example Problem
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Example involving an isosceles triangle and parallel lines
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Figuring out all the angles for congruent triangles example
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Basic Triangle Proofs Module Example
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Basic Triangle Proofs Module Example 2
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Basic triangle proofs
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Fill-in-the-blank triangle proofs example 1
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Fill-in-the-blank triangle proofs example 2
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Fill-in-the-blank triangle proofs
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Wrong statements in proofs example 1
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Wrong statements in triangle proofs
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Problem involving angle derived from square and circle
Problem involving angle derived from square and circle Challenging problem to find an angle
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- So we're told that quadrilateral ABCD is a square
- which tells us that the 4 sides have equal length,
- and that they are all the interior angles are 90 degrees
- We also know that FG,FG is a perpendicular bisector of BC
- so we've already shown that it's perpendicular,
- that this a 90-degree angle, but it also bisects BC
- So this length is equal to that length right over there
- Then they tell us that arc AC, it's a curve on top,
- arc AC is part of circle B
- So this is a circle centered at B
- So this is the center of the circle,
- this is part of part of that circle
- It's really kind of the bottom left corner of that circle
- and then given that information, they want us
- to find what the measure of angle BED is
- So what is BED?
- So it's BED
- So we need to figure out the measure
- of this angle right over here
- And I encourage you to pause it and try it out
- And you might imagine, well I you could pause it
- and try it without any hints
- and now I would give you a hint if you if you try it the first time,
- and you weren't able to do it,
- and you should pause it again after this hint, is,
- try to draw some triangles that maybe split up this angle
- into a couple of different angles
- and that might be a little bit easier
- You might be able to use some of what we know about triangles
- Now that said, I will try to solve it
- and you should pause at any point
- where you think that you know exactly how to do this
- and try to do this yourself
- So the trick is to realize that this is a circle
- and so any any line that goes between B
- and any point on this arc
- is going to be equal to the radius of the circle
- So AB is equal to the radius of the circle
- BE is equal to the radius of the circle
- And we can keep drawing other things
- that are equal to the radius of the circle
- BC is equal to the radius of the circle
- So let's think about it a little bit
- If we were to draw, and a lot of trickier geometry problems
- really all revolve around drawing the right lines
- or or visualizing the right triangles, and I'll do one right here
- that might open up a lot for you in terms of thinking
- about how to do this problem
- So let me draw segment EC
- and draw that as straight as possible
- I can draw a better job of that
- So the segment EC
- Now something becomes interesting
- because what is what is the relationship
- between EBG and triangle ECG?
- Well they both definitely share, they both share
- this side right over here
- They both share side EG
- And then BG is equal to GC
- And they both have 90 degree angles
- They have 90 degree angle here
- and they have 90 degree angle there
- So you see by side angle side
- Side angle side, these 2 triangles are going to be congruent
- So we know that triangle EBG is congruent to triangle ECG
- ECG, I should emphasize the C not the E
- ECG by side angle side congruency,
- by side angel side congruency
- And that also tells that all of the corresponding angles
- and sides are going to be the same
- So, that tells us, that tells us right there
- that EC that EC is equal to EB,
- EC is equal to EB
- So we know that EB, EB is equal to EC,
- and what is what also is equal to that length?
- Once again, this is the radius of the circle
- BE, is one radius of the circle going from the center to the arc
- But so is BC
- It is also a radius of the circle going from the center to the arc
- So this is also equal to, this is also equal to BC
- So I could draw this other other 3 things right here
- I'm I'm referring to the entire thing
- not just one of the segments
- All of BC
- So what is, what kind of a triangle is this right over here?
- Triangle BEC
- Triangle BEC is equilateral
- Equilateral
- And we know that because all 3 sides are equal
- So that tells us that all of its angles are equal
- So that tells us that the measure of angle BEC, BEC,
- we're not done yet, but it gets us close, is 60 degrees
- So the measure of angle BEC right over there is 60 degrees
- So that gives us part of the problem
- BEC is part of the angle BED
- If we can just figure out the measure of angle CED now,
- if we can figure out this angle right over here,
- we just add that to 60 degrees and we're done
- We figured out the entire the entire BED
- Let's think about how we can do this right over here
- So there's a couple of interesting things
- that we already do know
- We know that this right over here
- is equal to the radius of the circle
- And we also know that this length down here,
- this is a square
- We know that this length down here
- is the same as this length up here
- That these are the exact same length,
- and this is equal to the radius of the circle
- We already put these 3 slashes here
- BC is the same as that length, is the same as that length,
- so all 4 sides are gonna be the same as that length
- because this is a square
- So let me write, let me write this down
- Because because it's a square, I'll just write it this way
- Because it's a square, we know that CD, we know that cd
- is equal to BC which is equal to,
- and we already established this as equal to CEC
- which is equal to EB, which is equal to EC
- What's important here is to realize that this and this
- are the same length
- And the reason why that is interesting
- is it let's us know that this is an isosceles triangle
- It's an isosceles triangle
- So whatever isosceles triangle, if you have your 2 legs of it,
- the 2 base angles are going to be congruent
- So whatever angle this green angle is,
- this angle is going to be as well
- So somehow, somehow,
- we can figure out this angle right over here
- We can subtract that from 180
- and then divide by 2 to figure out these 2
- because we know that they are the same
- So how can we figure out this angle?
- Well we know all the angles of these things off up,
- we can figure out all the angles of the larger ones up here
- We know that this is an equilateral triangle
- So this over here has to be 60 degrees as well
- That's 60 degrees, and that is also 60 degrees
- In fact, I can write it over here,
- which is equal to the angle of BCE
- The measure of angle BCE, BCE
- So this is 60 degrees, we know we are dealing with a square
- So this whole angle over here is a right angle
- What is the measure of angle ECD?
- What is this angle right over here?
- I'm using a new color
- This angle over here is gonna have to be 30 degrees
- So that is going to be 30 degrees
- And now, we are ready to solve
- If you know, now we are ready to solve for these 2 base angles
- If we call this x, if you call this x,
- and we know they have to be the same,
- we have x + x + 30 degrees,
- x + x + 30 degrees is going to be equal to 180 degrees
- That is the sum of all the interior angles of a triangle
- So you get 2x,
- 2x + 30, + 30 is equal to 180 degrees
- Now you can subtract 30 from both sides,
- and we are left with 2x is equal to 150
- Divide both sides by 2, you get x = 75
- So we figured that x = 75, and now we are at the home stretch
- We have to figure out angle BED
- Well, that's going to be angle C,
- so x is equal to the measure of angle CED,
- so BED is CED + CED + BEC
- So the 60 degrees + 75 degrees
- So it's going to be,
- ready for the drum roll?
- This is going to be equal to 75 degrees + 60 degrees
- which is equal to 135 degrees
- And we are done
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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