Question

In circle J below, what is the value of x?

Answer 1

central angles have the same measure as the arc they cut...so since the angle LJK cuts arc LK, and mLK = 156, mLJK = 156 as well

so \[156=3x-6\]
solve for x

Answer 2

x=54

Answer 3

looks good to me :)

Answer 4

These seem so simple once you spell it all out to me lol.

Answer 5

lol...geometry's not too bad once you gather a few rules

Answer 6

I have a couple more like these. Care to help?

Answer 7

sure...i've got time

Answer 8

ok...this one is fun :) maybe i'm just crazy... :P

Answer 9

It's okay I find math fun sometimes too. X)

Answer 10

couple things to add to your toolbox...

we already know that central angles measure the same as the arc they cut...but these aren't central angles

if it's *not* a central angle (but with vertex on the circle), then the angle is *half* the measure of the arc...still not what we have...

so what *DO* we have?!?

Answer 11

i'm glad you asked...

Answer 12

we know that mXLY = 148...so we can find mYLZ (what is it?)

also, we know that angle YLZ = angle XLW (why?)

Answer 13

Ehh I'm not sure.. This unit is really confusing to me.

Answer 14

well angle YLZ =angle XLW because they're vertical angles right?

Answer 15

exactly...

and we can find mYLZ since it is 180 - mXLY since they form a straight line...

Answer 16

but actually, i just noticed there's an easier way

Answer 17

So YLZ and XLW= 32

Answer 18

yes...but it turns out we don't need those measures...we need another tool...

notice that angle XLY = angle WLZ as they are also vertical anthese angles are not central angles and they don't have vertices on the circle...but the measure of these angles is the average of the measure of the arcs they cut...

Answer 19

probably a little confusing, so let me explain better...

Answer 20

mXLY = mWLZ = 148

(mXY + mWZ)/2 = 148

Answer 21

are you fine with that, or do you need it clearer?

Answer 22

Maybe a little clearer

Answer 23

So the angles marked A have the same measure because they are vertical angles, yes?

Answer 24

Mhmm

Answer 25

sorry...screwed up the numbers

Answer 26

let's say P is 36 and Q is 200, their average\[\frac{36+200}{2}=\frac{236}{2}=118\]
is the measure of angle A

Answer 27

Okay

Answer 28

thsis is true for any interior anglese...and if you think about it, it leads to the fact about central angles and angles whose vertex is on the circle too...

Answer 29

more comfortable now?

Answer 30

Yeah

Answer 31

good...so how does this help?

we have an expression for arcs XY and WZ...we can take the average of them...this will equal the angle XLY whch we know is 148.

Answer 32

\[\frac{(7x-9) + 3(2x+15)}{2}=148\]
solve for x

Answer 33

20?

Answer 34

with confidence :)

Answer 35

20!

Answer 36

Haha I have one more.

Answer 37

bring it on!

Answer 38

i know it's one of two answers...need to dust off a couple cobwebs...1 sec :)

Answer 39

Mmkay

Answer 40

got it...

so here we have an external angle. the idea is very similar, but whereas, we add and divide by two for internal angles, we *subtract* and divide by 2 for external angles

Answer 41

meaning
\[\frac{mAD-mBC}{2}=mP\]

Answer 42

30!

Answer 43

so remember:
(1/2)(sum of arcs) = internal angle
(1/2)(difference of arcs) = external angle

Answer 44

i like the confidence :)

correct!

Answer 45

Thank you so much!!

Answer 46

you are quite welcome...anytime