Quadrilateral EFGH is inscribed inside a circle as shown below. Write a proof showing that angles H and F are supplementary.

- anonymous

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- anonymous

##### 1 Attachment

- anonymous

@ganeshie8

- mathstudent55

Do you need a two-column proof or just an outline of the proof?

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## More answers

- anonymous

just outline should be fine

- mathstudent55

Ok.
Here's the idea for this proof.
A full circle is an arc of measure 360 deg.

- mathstudent55

|dw:1434114713677:dw|

- mathstudent55

Let's look at two arcs on the figure.

- anonymous

so would we have to figure they both have to equal 180 right?

- mathstudent55

|dw:1434114801663:dw|

- mathstudent55

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.

- mathstudent55

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.

- mathstudent55

so if one arc is x, the other arc must be 360 - x.
Ok so far?

- mathstudent55

|dw:1434114997252:dw|

- anonymous

so we would do 360 minus x for both angles?

- mathstudent55

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?

- anonymous

oh yeah I get it now.

- anonymous

the arc is two times the measure of the angle correct?

- mathstudent55

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.)

- mathstudent55

|dw:1434115526244:dw|

- mathstudent55

Ok.
Let's go back to our problem, and look at the inscribed angles of the two arcs we are dealing with.

- mathstudent55

I marked in the figure the two angles of the two arcs we are dealing with.
|dw:1434115587886:dw|

- mathstudent55

Let's look at angle EFG first.
Its arc is arc EHG. Arc EHG measures x.
What is the measure of angle EFG?

- anonymous

half of x?

- mathstudent55

Exactly. It is half of the arc.

- mathstudent55

|dw:1434115776669:dw|

- mathstudent55

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?

- anonymous

that would be half of 360- x so 1/2(360-x)

- anonymous

right?

- mathstudent55

Exactly.
Let me write that down.

- mathstudent55

|dw:1434116074608:dw|

- mathstudent55

We are trying to prove that opposite angles are supplementary.
Angles EFG and EHG are opposite angles, so let's add their measures.

- mathstudent55

|dw:1434116159157:dw|
Ok?

- anonymous

okay

- mathstudent55

Now let's simplify the right side.

- mathstudent55

We can factor out 1/2
|dw:1434116259666:dw|

- mathstudent55

Now we combine the x's in the parentheses.
x - x = 0, so we get:
|dw:1434116311737:dw|

- mathstudent55

Finally we get:
|dw:1434116353183:dw|

- mathstudent55

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.

- mathstudent55

gtg, bye

- anonymous

thank you!

- mathstudent55

yw

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