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Geometric constructions: angle bisector

Sal constructs a line that bisects a given angle using compass and straightedge. Created by Sal Khan.

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  • primosaur ultimate style avatar for user Joshua Liddicott
    He says that we know the two triangles are congruent, but he doesn't prove it. How do we know?
    (2 votes)
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    • piceratops ultimate style avatar for user trek
      He did prove it. Sal showed that each pair of corresponding points is equidistant, thus demonstrating that the sides are all equal. By the SSS theorem, if all three sides of two triangles are the same length, the triangles are congruent.
      (23 votes)
  • blobby green style avatar for user George Coetzee
    I have been watching a couple of videos about trisecting arbitrary angles, using only an unmarked straightedge and compass. It is said to be impossible, because apparently Euclid said so.

    But the girl in this video https://www.youtube.com/watch?v=XZPY86eo3XU is doing exactly that. She uses an unmarked straightedge and compass to trisect an angle. Nothing else.

    Now on the wikipedia page it says the following, and I quote: "Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.

    The problem as stated is impossible to solve for arbitrary angles, as proved by Pierre Wantzel in 1837. However, although there is no way to trisect an angle in general with just a compass and a straightedge, some special angles can be trisected. For example, it is relatively straightforward to trisect a right angle (that is, to construct an angle of measure 30 degrees).

    It is possible to trisect an arbitrary angle by using tools other than straightedge and compass.
    For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools" end quote.

    Note, a marked straightedge, like a ruler or something. She uses a neusis construction with an unmarked straightedge and compass.

    Has she nailed it? If not, why not? She fullfills the requirement exactly. She trisects arbitrary angles using only a unmarked straightedge and compass using a neusis construction. How can this possibly not be a total win. XD
    (7 votes)
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    • winston default style avatar for user infinitely_infinite
      No, she has not. The problem require constructing it with only the straightedge and compass. Any other devices or constructions used are not allowed.

      The neusis requires marking of the straight edge. We can't know that for these problems. It's meant to be a way to make lines. In general, if the method used needs something else other than
      : a way to make a straight line
      : a way to make a circle of any size

      it is not allowed.
      (3 votes)
  • duskpin seed style avatar for user Manny
    Hey Sal, I loved your video but can you put some practice questions? It would really help, the video was awesome and I understood it!
    (5 votes)
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  • blobby green style avatar for user akena.adams
    angle 75
    How can we draw angle 75 with a compass
    (4 votes)
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    • duskpin ultimate style avatar for user Visishta
      So you could use a 90 degree angle and a 15 degree angle, so the 3rd angle would be 75 degrees. To get the 90, use a right triangle, and to get the 15, use an equilateral triangle, bisect the 60 degree in half, and then the 30 degree in half to get the 15 degree angle. But it's a pretty complex shape involving both a right triangle and an equilateral triangle combined. (Same answer as above)
      (2 votes)
  • duskpin tree style avatar for user sidhya
    Sometimes, the three circle point does not work for an angle because the points are not able to intersect the same circle. How do you find the angle bisector in this case?
    (3 votes)
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  • duskpin ultimate style avatar for user Mayhsa
    For a triangle, what is an angle bisector used to find?
    (2 votes)
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  • aqualine ultimate style avatar for user Z
    Is the angle bisector also the side bisector? And also is the angle bisector always perpendicular to the side opposite the angle? Pls help.
    (1 vote)
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    • female robot grace style avatar for user IOn
      Question 1:
      No, unless you have equilateral or isosceles triangle.
      actually name of side bisector is median it equally divides side into two parts.

      Question 2:
      Also No, unless you have equilateral or isosceles triangle.

      *in isosceles triangle it is only true if you construct
      bisector from not equal angles. so if isosceles triangle has angles: 30, 30 and 120. bisector will be median and also a perpendicular if you construct it from 120 degree angle.
      (4 votes)
  • purple pi teal style avatar for user Cael
    he forgot to check XD
    (2 votes)
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  • winston default style avatar for user hussain313
    How do you construct a bisector on a right and obtuse triangle
    (2 votes)
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  • blobby green style avatar for user craymond
    Where did the exercises go that used to be linked to this video?
    (2 votes)
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Video transcript

We're asked to construct an angle bisector for the given angle. So this is the angle they're talking about. And they want us to make a line that goes right in between that angle, that divides that angle into two angles that have equal measure, that have half the measure of the first angle. So let's first find two points that are equidistant from this point right over here on each of these rays. So to do that, let's draw one circle here. And I can make this of any radius. Wherever this intersects with the rays, that's where I'm going to put a point. So let's say, here and here. Notice both of these points, since they're both on this circle, are going to be equidistant from this point, which is the center of the circle. Now, what I want to do is construct a line that is equidistant from both of these points. And we've done that already when we looked at perpendicular bisectors for lines in this construction module. So let's do that. So let's add their compass. And so what I want to do, this circle is centered at this point. And it has a radius equal to the distance between this point and that point. And then I do that again. So this circle is centered at this point and has a radius equal to the distance between that point and that point. And then the two places where they intersect are equidistant to both of these points. And so we can now draw our angle bisector, just like that. And you might say, well, how do we really know that this angle is equal to this angle? Well, there's a couple ways we can tell. We know this distance right over here is equal to this distance right over there. We know that this distance over here is equal to this distance over here. And both of these triangles share this line. So essentially, if you look at this point, this point, and this point, that forms a triangle. And if you look at this point, this point, and this point, that forms a triangle. We know those two triangles are congruent, so this angle must be equal to this angle. These are the corresponding angles. So they're going to be congruent. This is an angle bisector.