Re: Calculate X in a circle

Quote:

Originally Posted by

**gingerale**

.

Don't you know that $\displaystyle x+58=180~?$

Why is that true?

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? :)

Re: Calculate X in a circle

Hello, gingerale!

Plato is absolutely correct.

Code:

` B `

o * *

* *

* *

* o C

A o *

* + *

* *

* *

* *

D o *

* * *

We have cyclic quadrilateral $\displaystyle ABCD.$

Draw chords $\displaystyle AB, BC, CD, DA.$

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

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

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

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

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.