Online Course Discussion Forum
Math Challenge IC Handout 6
I'm struggling with the proofs for 6.23 and 6.24. I know it has 'something' to do with isosceles triangles but for 6.23, I don't know how to use the radius to help me since APB is 'outside' of O, the center. For 6.24, I can't find the isosceles triangles. I also need help with 6.26 and 6.29. Drawing doesn't seem to help with the situation. Please and thank you!!
*thanks for the help on the last handout!
For problem 23, try writing $\angle APB$ as the difference of two angles. It might help to figure out what angles are helpful if you draw the diameter that goes through $P$.
For problem 24 try using some inscribed angles, you will need to draw some extra lines in your diagram. (after problem 23 you have now proved that the angle measure of any inscribed angle is half of the angle measure of the arc that it opens to).
For 26, if you have a quadrilateral $ABCD$ inscribed in a circle, what can you say about $\angle A + \angle C$?
For 29, remember that a tangent line intersects the circle perpendicularly, that is, if you draw the radius from the point where the line meets the circle, the line and that radius are perpendicular. Also, what do we know about the slopes of two perpendicular lines?
Thank you for your help!!
I'm really sorry, but I still don't get problem 23, 24, and 26
for 23, isn't the whole point to not have a diameter as the line can't pass through the center of the circle because the question said so? How do you write it as the difference of two angles?
for 24... how? which extra lines?
for 26, Okay I see what you mean by angle a plus angle c is 180 degrees, but how do you proceed from there? I tried drawing little radius lines, but they were not helping. I tried to use the Pythagorean theorem, but I don't think that is even usable.
I have no idea what to do, would you mind being a little bit more specific as to how, because I don't think I'm seeing the main idea :(
Indeed in problem 23 the angle you want to work with does not go through the diameter, however, that doesn't mean you can't draw some other lines to help you figure it out. Drawing the diameter will let you write the angle you want in terms of two angles you already know how to work with, that is, you can write $\angle APB$ as the difference of two angles that go through the diameter.
On 24 one extra line that should help is $BC$. Try writing the angle you want in terms of this two inscribed angles, and use the fact that the angle measure of these two inscribed angles is half of the angle measure of the arcs they open to.
Try using the fact that $\angle A + \angle C = 180^\circ$, together with the fact that the quadrilateral is a trapezoid. What do you know about the angles of a trapezoid? (Look at the angles formed between parallel lines!)
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