Kite DCFE is inscribed in circle A shown below.
If the measure of is 232°, what is the measure of ∠ DEF? (5 points)

- anonymous

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- anonymous

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- mathstudent55

The measure of what is 232 deg? The info is missing in your post.

- anonymous

I know that a circle = 360 and that 360 - 232 = 128 but 128 is the arc of angle FDE not DEF? I also know that the inscribed angles theorem means that angles = 1/2 the value of their intercepted arcs.

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- anonymous

Oh oops hold on

- anonymous

The measure of DEF @mathstudent55

- mathstudent55

Angle DEF?

- anonymous

Yes.

- mathstudent55

Arc DEF?

- anonymous

No angle DEF

- mathstudent55

Wait. We are looking for the measure of angle DEF. That much I see in your post.
What measures 232 deg? Is it arc DEF or something else?

- mathstudent55

|dw:1434041246519:dw|

- anonymous

The question is this: If the measure of ARC DE is 232°, what is the measure of ∠ DEF?
I have shared an imagine. This is as much as I know: I know that a circle = 360 and that 360 - 232 = 128 but 128 is the arc of angle FDE not DEF? I also know that the inscribed angles theorem means that angles = 1/2 the value of their intercepted arcs.

- mathstudent55

|dw:1434041331681:dw|

- anonymous

Naw, its the other way from DE. Sorry for the confusion

- mathstudent55

Arc DE cannot be 232 deg because arc FE is congruent to arc FE, and just that would total more than 360 deg.
Is it measure of arc DEF that is 232?

- anonymous

Yes, sorry

- anonymous

@mathstudent55 Yes, arc DEF

- mathstudent55

Ok. Remember a kite is symmetric, so these arcs are congruent. See pic below.
|dw:1434041802642:dw|

- mathstudent55

If arc DEF measures 232, arcs DE and FE measure half that much.

- anonymous

Why's that?

- mathstudent55

Also, because of symmetry of a kite, the angles below are congruent.
|dw:1434041934430:dw|

- anonymous

Ahhhh I see

- anonymous

Okay

- mathstudent55

Arc measures below:
|dw:1434042008754:dw|

- mathstudent55

Angle DEF measures twice angle DEC.
An inscribed angle measure half of its tended arc.

- anonymous

Wow

- anonymous

But wait

- anonymous

64+64+112 = 244

- anonymous

not 232

- mathstudent55

OK. Now that I know what the 232 deg refers to, let's start from the beginning.

- mathstudent55

You start like this.
|dw:1434042759323:dw|

- mathstudent55

You were correct to subtract 232 from 360.
You get this:
|dw:1434042831738:dw|

- mathstudent55

|dw:1434042874732:dw|

- mathstudent55

Angle DEF is the inscribed angle of an arc of measure 128, so angle DEF measures half of 128, which is 64.

- anonymous

OH MY GOODNESS! Okay I get everything now, I didnt realize that that was DEF I thought ANGLE DEF was from point D to E to F clockwise I get it now

- anonymous

xD

- anonymous

@mathstudent55 Thanks, you get a medal :) well-deserved

- mathstudent55

You're welcome.

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