**LONG ANSWER**

Determine the measure of ےECB, ےBDC, ےBAD, and ےDBE, where E is the centre of the circle.

So first of all we always start with a table that has STATEMENT on one side and REASON on the other.

It's obvious at this point that BE and CE are radii, therefor triangle BCE is an isosceles triangle. This also means that ےEBC and ے BCE are the same.

**NOTE:**CLICK ON BUBBLESHARE BOX TO SEE FULL EFFECTS. =)

We now know the exact amount of ےBCD and according to cyclic quadrilateral theorem ے BCD's and ےBAD's sum should equal 180°. So in order to find ےBAD, we subtract 30° and 40° from 180° and come up with 110°. Therefore ےBAD is 110°.

Another thing we looked at were the two triangles that are subtended by the same arc; triangle BCE and triangle BCD. Since all of us know of the central inscribed angle theorem ( AKA the star trek theorem ) we can easily point out that ےBDC is half the amount of ےBEC. As a result ےBDC = 100°/2 which equals 50°.

We only have one angle left! Angle DBE! So if we look closely at triangle BCD you'll realize that we only have one angle left to find, ےDBE. We also know that every the angles of a triangle sum up to be 180° and by subtracting every angle that we have we'll be able to come up with the value of ےDBE! :D

Which in this case in 20°! =)

Once you've found your information and proofs, your table (including "Statement" and "reason") should look something like this:

**MULTIPLE CHOICE**

In the diagram below, R, S, T, U are points on the circle with RS = RU = UT and SU = ST. Find the measure of ےUST.

a) 31°

b) 35°

c) 36°

d) 40°

e) none of these

The first thing you should always do is mark down what is given to you. Have a wishful thinking colour, a colour used for given facts and a colour used for things you have proven! ( it comes in handy :D )

I'll use purple as the facts that are given, blue for what we're looking for and orange for the facts i prove. =)

There's no particular order to do this in, but what i looked at first was the isosceles triangles. We can easily point out that ےRSU and ےRUS are the same and ےSUT = ےSTU.

Now if we look at RS, RU and UT they are all the same length. Which means that the angles subtended by this chord length will all be the same. SOOO.. ےRST = ےRUT = ےUST.

In order to find the exact degree that angle is we make three equations. (this is like going back to substitution in linear equations). Here are the three equations (let ےSRT = w):

**1. w + z = 180**

2. 3y + z = 180 --> z = 180 - 3y

3. w + 2y = 180 --> w = 180 - 2y

2. 3y + z = 180 --> z = 180 - 3y

3. w + 2y = 180 --> w = 180 - 2y

If we substitude 2 and 3 into 1, we can find out what Y is, which is ےUST.

**180 - 2y + 180 - 3y = 180**

180 + 180 - 180 = 2y + 3y

180 = 5y

36 = y

180 + 180 - 180 = 2y + 3y

180 = 5y

36 = y

And there you have it!

**c) 36°**is the correct answer. =)

That's all for tonight! Sorry for the late entry, i had to go and watch my brother's christmas concert! haha. =) Well, no homework tonight, just study! Test tomorrow. Good luck guys!

**HAPPY HOLIDAYS!**=]

And the next scribe will bee..... -Zeph! Enjoy. =)

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