# Calculate X in a circle

• Jul 7th 2012, 04:45 AM
gingerale
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

http://img854.imageshack.us/img854/9758/cirkel.jpg
• Jul 7th 2012, 05:12 AM
Plato
Re: Calculate X in a circle
Quote:

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
http://img854.imageshack.us/img854/9758/cirkel.jpg

.
Don't you know that $\displaystyle x+58=180~?$
Why is that true?
• Jul 7th 2012, 07:55 AM
gingerale
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? :)
• Jul 7th 2012, 08:25 AM
Soroban
Re: Calculate X in a circle
Hello, gingerale!

Plato is absolutely correct.

Quote:

Calculate the angel X in the circle.

http://img854.imageshack.us/img854/9758/cirkel.jpg

Code:

              B               o * *           *          *         *              *       *                o C     A o                  *       *        +        *       *                  *       *                *         *              *         D o          *               * * *
We have cyclic quadrilateral $\displaystyle ABCD.$
Draw chords $\displaystyle AB, BC, CD, DA.$

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

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

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

$\displaystyle \text{Opposite angles of a cyclic quadrilateral are supplementary.}$
• Jul 7th 2012, 12:25 PM
HallsofIvy
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.