Exterior Angle Theorem and Why do arcs have the same angle measure as central angles?

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    Module 2 Week 2 Day 6 Your Turn Explanation Part 2

    How does Prof. Loh know to subtract the 40 by the 15 degrees in the video automatically? Timestamp is around 1:55.

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    How does Prof. Loh know to subtract the 40 by the 15 degrees in the video automatically? Timestamp is around 1:55.

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    Also what is Prof. Loh talking about after around timestamp 5:00? I do not understand a lot of it.


  • Hi Xiao Ma,

    Great questions. For the first one, one great property of triangles is that if you extend one side, the exterior angle which is made (in the problem, this is the 40 degrees), this is equal to the sum of the triangles other two angles. Armed with this knowledge, we can see that $$40 = x + 15 \implies x = 25$$
    You may ask "how do we know this property holds?" Consider that the sum of 40 degrees, and the interior angle right next to it, must be 180. Moreover, the sum of all the angles in the triangle must be 180. Let x be this interior angle. This gives us the following:
    $$15+25+x = 180$$ $$40+x = 180 $$ $$15+25+x = 40 + x$$ $$ 15+25= 40$$
    This gives us the property we want!

    For your second question, this is a little bit more advanced, but try and back up the video, perhaps slow it down, to better understand what Po is explaining. The important idea here is that secant lines and their angles of intersection have nice properties that we can derive using geometry. One thing which may be confusing is labeling an angle and an arc with a letter; what it means to measure an arc is to find the angle it makes when two radii are drawn to its endpoints. Draw from the center to one end, then from the center to the other, and measure that angle. Hopefully, that may clarify the picture at 5:00 a little bit.

    Hope this helps, let us know if you have any other questions.

    Happy Learning,

    The Daily Challenge Team

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    Sorry but I still don't get the first question and also how do we know that arc degree measures are the same as central degree angles?

  • MOD

    Hi again! Let's see if I can explain the first question in a different way, using a picture!

    91254b73-c38d-4058-9dc1-eccd2f4473b7-image.png

    Hopefully this is a lot clearer!

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    Why do arcs have the same angle measure as central angles?

  • MOD

    @TSS-Graviser There isn't much of a big "reason", it's simply how we define the "measure" of an "arc".

    If you think about it, arcs aren't really "angles", right? But, we find that it is very useful to just "say" that an arc has an "angle measure". But why, and what would it even represent?

    An arc is just part of a circle. So, we can think of the "angle measure" of an arc as how much the arc goes around the circle. For example, the full circle is 360 degrees - if you stood on a playground and drew a full circle around you with a stick, you'd spin a full "360 degrees". If you only went a quarter of the way around, you only draw "90 degrees" of an arc.

    As you can imagine, how much you go "around" is exactly the central angle you make!

    72888ed9-7bcc-40ef-aef1-531f11fd12b0-image.png

    So why is this useful? Off the top of my head, a good reason is that using "arcs" instead of "central angles" can be a good shortcut, and can be easier to think about. For example, inscribed angles along the same arc have the same measure!

    db49cd76-307c-4cdd-b311-750e921f0e38-image.png

    And we can say that the measure of each of those inscribed angles is half the measure of the intercepted arc, and we don't have to draw in the central angle.