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Isosceles & equilateral triangles problems

Three example problems involving isosceles and equilateral triangles (partly taken from Art of Problem Solving, by Richard Rusczyk). Created by Sal Khan.

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  • leafers ultimate style avatar for user fisherlam→ΣβФ
    What are conjugate angles?
    (23 votes)
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  • mr pink red style avatar for user Eoghan Perkins
    I know complementary angles add up to 90 degrees and supplementary angles add up to 180 degrees but is there a word for angles that add up to 360 degrees?
    (3 votes)
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  • winston default style avatar for user Semetrius Nixon
    If the expressions are the equal legs of an isosceles, or equilateral triangle, then we should go

    • Create an equation with the equivalent expressions, by setting them equal to each other,

    then solve for x using algebraic methods, (by keeping the equation balanced by performing the same math operations on both sides), to…

    • Isolate x to one side of the equation am I right?
    (5 votes)
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  • starky tree style avatar for user cooperellis
    huh? I still do not understand this😅
    (4 votes)
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    • duskpin tree style avatar for user Duskpin
      What part? Finding the angles in general? Finding supplementary? Well, I don't know which one you need help with, but here is some tips:
      "Straight" for Supplementary (Because it is straight, 180 degrees)
      "Corner" for Complementary (Because it makes a corner, 90 degrees)
      All of a triangle's angles add up to 180 degrees, but not squares, though! Square's angles add up to 360 degrees, if that make since? Half of a square is a triangle, so that is correct. :3

      If you need more help, feel free to ask for help!
      -Duskpin, the avatar
      (5 votes)
  • blobby green style avatar for user 3026156
    Through time stamps and Sal states that angle ABE would be 105 degrees. wouldn't it be 210 degrees as angle "e" is 45 degrees and angle b is 105 degrees and angle "a" is 60 degrees? 60+105+45=210 right? so it should be 210 degrees or am i mistaken?
    (4 votes)
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  • aqualine ultimate style avatar for user Arsheya Jaishiva
    How do you remember the difference between supplementary and complimentary angles?
    (5 votes)
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  • hopper cool style avatar for user Jazz R
    Why does a triangle add up to 180 ?
    (4 votes)
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  • blobby blue style avatar for user TheOneAndOnlyME
    At Khan said that "the base angles are going to be congruent". But why though?
    (4 votes)
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  • aqualine ultimate style avatar for user Sarrah Cooper
    How can you remember the difference between equilateral and isosceles triangles?
    (4 votes)
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    • cacteye yellow style avatar for user Emma🖤
      Hmmm....well, equilateral is equal all around that easy.
      Isosceles is a bit harder; you could remember that there is one isolated side that’s not like the other two. Since isolated sounds like isosceles a little.
      That’s all I can think of right now.
      Merry Christmas and Jesus loves you❤️
      (3 votes)
  • spunky sam blue style avatar for user 💎Chυcκ Lørrε💎
    How to prove a 30, 60, 90 degrees angle triangle's hypotenuse is always two times the side that opposite the 30° angle?
    (2 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      Nice question!

      Draw a 30-60-90 triangle and its reflection about the leg opposite the 60° angle. These two 30-60-90 triangles together form a larger triangle. This larger triangle has three 60° angles and is therefore equilateral!

      The hypotenuse of either one of the 30-60-90 triangles is one of the sides of the equilateral triangle. The sides opposite the 30° angles of the two 30-60-90 triangles are equal in length, and the two of them together form another side of the equilateral triangle.

      It then follows that the length of the side opposite the 30° angle of a 30-60-90 triangle is half the length of the hypotenuse!

      Have a blessed, wonderful day!
      (4 votes)

Video transcript

Let's do some example problems using our newly acquired knowledge of isosceles and equilateral triangles. So over here, I have kind of a triangle within a triangle. And we need to figure out this orange angle right over here and this blue angle right over here. And we know that side AB or segment AB is equal to segment BC, which is equal to segment CD. Or we could also call that DC. So first of all, we see that triangle ABC is isosceles. And because it's isosceles, the two base angles are going to be congruent. This is one leg. This is the other leg right over there. So the two base angles are going to be congruent. So we know that this angle right over here is also 31 degrees. Well, if we know two of the angles in a triangle, we can always figure out the third angle. They have to add up to 180 degrees. So we could say 31 degrees plus 31 degrees plus the measure of angle ABC is equal to 180 degrees. You can subtract 62. This right here is 62 degrees. You subtract 62 from both sides. You get the measure of angle ABC is equal to-- let's see. 180 minus 60 would be 120. You subtract another 2. You get 118 degrees. So this angle right over here is 118 degrees. Let me just write it like this. This is 118 degrees. Well, this angle right over here is supplementary to that 118 degrees. So that angle plus 118 is going to be equal to 180. We already know that that's 62 degrees. 62 plus 118 is 180. So this right over here is 62 degrees. Now, this angle is one of the base angles for triangle BCD. I didn't draw it that way, but this side and this side are congruent. BC has the same length as CD. Those are the two legs of an isosceles triangle. You can kind of imagine it was turned upside down. This is the vertex. This is one base angle. This is the other base angle. Well, the base angles are going to be congruent. So this is going to be 62 degrees, as well. And then finally, if you want to figure out this blue angle, the blue angle plus these two 62-degree angles are going to have to add up to 180 degrees. So you get 62 plus 62 plus the blue angle, which is the measure of angle BCD, is going to have to be equal to 180 degrees. These two characters-- let's see. 62 plus 62 is 124. You subtract 124 from both sides. You get the measure of angle BCD is equal to-- let's see. If you subtract 120, you get 60, and then you have to subtract another 4. So you get 56 degrees. So this is equal to 56 degrees. And we're done. Now, we could do either of these. Let's do this one right over here. So what is the measure of angle ABE? So they haven't even drawn segment BE here. So let me draw that for us. And so we have to figure out the measure of angle ABE. So we have a bunch of congruent segments here. And in particular, we see that triangle ABD, all of its sides are equal. So it's an equilateral triangle, which means all of the angles are equal. And if all of the angles are equal in a triangle, they all have to be 60 degrees. So all of these characters are going to be 60 degrees. Well, that's part of angle ABE, but we have to figure out this other part right over here. And to do that, we can see that we're actually dealing with an isosceles triangle kind of tipped over to the left. This is the vertex angle. This is one base angle. This is the other base angle. And the vertex angle right here is 90 degrees. And once again, we know it's isosceles because this side, segment BD, is equal to segment DE. And once again, these two angles plus this angle right over here are going to have to add up to 180 degrees. So you call that an x. You call that an x. You've got x plus x plus 90 is going to be 180 degrees. So you get 2x plus-- let me just write it out. Don't want to skip steps here. We have x plus x plus 90 is going to be equal to 180 degrees. x plus x is the same thing as 2x, plus 90 is equal to 180. And then we can subtract 90 from both sides. You get 2x is equal to 90. Or divide both sides by 2. You get x is equal to 45 degrees. And then we're done because angle ABE is going to be equal to the 60 degrees plus the 45 degrees. So it's going to be this whole angle, which is what we care about. Angle ABE is going to be 60 plus 45, which is 105 degrees. And now we have this last problem over here. This one looks a little bit simpler. I have an isosceles triangle. This leg is equal to that leg. This is the vertex angle. I have to figure out B. And the trick here is like, wait, how do I figure out one side of a triangle if I only know one other side? Don't I need to know two other sides? And we'll do it the exact same way we just did that second part of that problem. If this is an isosceles triangle, which we know it is, then this angle is going to be equal to that angle there. And so if we call this x, then this is x as well. And we get x plus x plus 36 degrees is equal to 180. The two x's, when you add them up, you get 2x. And then-- I won't skip steps here. 2x plus 36 is equal to 180. Subtract 36 from both sides, we get 2x-- that 2 looks a little bit funny. We get 2x is equal to-- 180 minus 30 is 150. And then you want to subtract another 6 from 150, gets us to 144. Did I do that right? 180 minus 30 is 150, yep, 144. Divide both sides by 2. You get x is equal to 72 degrees. So this is equal to 72 degrees. And we are done.