# Thread: Calculate X in a circle

1. ## Calculate X in a circle

I'm having some difficulties trying to solve this problem seen in the picture below. I just don't know where to begin. Any help would be very appreciated, thanks.

Calculate the angel X in the circle

2. ## Re: Calculate X in a circle

Originally Posted by gingerale
I'm having some difficulties trying to solve this problem seen in the picture below.
Calculate the angel X in the circle
.
Don't you know that $x+58=180~?$
Why is that true?

3. ## Re: Calculate X in a circle

So the opposite angel is a reflection that together is equal to 180? X = 122, and the angel that is unmarked would then be 77, all together 360. Makes sense. Why is the circle there then, just for confusing?

4. ## Re: Calculate X in a circle

Hello, gingerale!

Plato is absolutely correct.

Calculate the angel X in the circle.

Code:
              B
o * *
*           *
*               *
*                 o C

A o                   *
*         +         *
*                   *

*                 *
*               *
D o           *
* * *
We have cyclic quadrilateral $ABCD.$
Draw chords $AB, BC, CD, DA.$

$\text{We have: }\:\begin{Bmatrix}\angle A &=& \frac{1}{2}\overarc{BCD} \\ \angle C &=& \frac{1}{2}\overarc{DAB} \end{Bmatrix}$

$\text{Hence: }\:\angle A + \angle C \;=\;\tfrac{1}{2}\overarc{BCD} + \tfrac{1}{2}\overarc{DAB} \;=\; \tfrac{1}{2}(\overarc{BCD} + \overarc{DAB}) \;=\; \tfrac{1}{2}(360^o)$

$\text{Therefore: }\:\angle A + \angle C \;=\;180^o$

$\text{Opposite angles of a cyclic quadrilateral are supplementary.}$

5. ## Re: Calculate X in a circle

Generally speaking it is true that the measure of an angle, with vertex on a circle, is half the angle measure of the arc it subtends. Here, the angle opposite x has measure 58 degrees and so subtends an arc of angle 116 degrees. Then entire circle has 360 degrees so that leaves 360- 116= 144 degrees for the arc subtended by angle x.