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Do you need a two-column proof or just an outline of the proof?
just outline should be fine
Ok. Here's the idea for this proof. A full circle is an arc of measure 360 deg.
Let's look at two arcs on the figure.
so would we have to figure they both have to equal 180 right?
No. We don't know how much either arc measures. We do know that the sum of their measures is 360. Also, if we make an arc 180, and the other one has to also be 180, then this proof would only work when the arcs are 180 deg. We want a proof that works for all inscribed quadrilaterals, no matter what any of their angle measures are.
Because of the above, we call the measure of one arc x. Since the sum of the measures of the arc is the entire circle, the arcs add up to 360 degrees.
so if one arc is x, the other arc must be 360 - x. Ok so far?
so we would do 360 minus x for both angles?
We are not dealing with angles yet. So far, we are only dealing with arc measures. The measure of arc EHG is x. The measure of arc EFG is 360 - x. This is all we have so far. Do you follow this so far?
oh yeah I get it now.
the arc is two times the measure of the angle correct?
Exactly. An inscribed angle is half the measure of its arc. (A central angle is the same measure as the arc, but we're not dealing with a central angle here.)
Ok. Let's go back to our problem, and look at the inscribed angles of the two arcs we are dealing with.
I marked in the figure the two angles of the two arcs we are dealing with. |dw:1434115587886:dw|
Let's look at angle EFG first. Its arc is arc EHG. Arc EHG measures x. What is the measure of angle EFG?
half of x?
Exactly. It is half of the arc.
Now let's look at the other arc and its corresponding inscribed angle. Arc EFG measures 360 - x. What is the measure of the inscribed angle EHG?
that would be half of 360- x so 1/2(360-x)
Exactly. Let me write that down.
We are trying to prove that opposite angles are supplementary. Angles EFG and EHG are opposite angles, so let's add their measures.
Now let's simplify the right side.
We can factor out 1/2 |dw:1434116259666:dw|
Now we combine the x's in the parentheses. x - x = 0, so we get: |dw:1434116311737:dw|
Finally we get: |dw:1434116353183:dw|
The definition of supplementary angles is: Two angles are supplementary if their measures add up top 180 degrees. We have just shown that the two opposite angles have measures that add up to 180 degrees, so the opposite angles are supplementary.