Points $L$ and $M$ lie on a circle $\omega_1$ centered at $O$. The circle $\omega_2$ passing through points $O,$ $L,$ and $M$ is drawn. If the measure of arc $LM$ in circle $\omega_1$ is $82^\circ,$ then find the measure of arc $LOM$ in circle $\omega_2$, in degrees.

[asy]
unitsize(1.5 cm);

pair A, B, O;

B = dir(-20);
A = dir(60);
O = (0,0);

draw(Circle((0,0),1));
draw(circumcircle(A,B,O));

dot("$L$", A, NE);
dot("$M$", B, SE);
dot("$O$", O, W);
label("$\omega_1$", dir(135), NW);
label("$\omega_2$", circumcenter(A,B,O) + circumradius(A,B,O)*dir(30), NE);
[/asy]


We are given two circles, $\omega_1$ and $\omega_2$, with some information about arcs and points on these circles. Specifically, $\omega_1$ is centered at point $O$, and points $L$ and $M$ lie on $\omega_1$. The circle $\omega_2$ passes through points $O$, $L$, and $M$. The arc $LM$ on $\omega_1$ subtends an angle of $82^\circ$. We are asked to find the measure of arc $LOM$ in circle $\omega_2$.

### Step 1: Relationship Between Circles $\omega_1$ and $\omega_2$
Circle $\omega_1$ is centered at $O$, meaning $O$ is the center of the circle, and points $L$ and $M$ lie on this circle. The angle subtended by the arc $LM$ at the center $O$ of circle $\omega_1$ is $82^\circ$, because the arc measure of $LM$ directly corresponds to this central angle.

Circle $\omega_2$ passes through the three points $O$, $L$, and $M$, forming a triangle with these points. We need to find the measure of arc $LOM$ on this circle $\omega_2$.

### Step 2: Angle Subtended by Arc $LM$ in $\omega_2$
In circle $\omega_2$, the points $L$, $M$, and $O$ form a triangle. The key observation here is that $\omega_2$ is the circumcircle of triangle $OLM$. The angle subtended by arc $LM$ in circle $\omega_2$ at point $O$ is the same as the angle subtended by the arc at the other points of the triangle. This is because $O$, $L$, and $M$ form an isosceles triangle with vertex angle $82^\circ$.

### Step 3: Measure of Arc $LOM$
Since the angle subtended by arc $LM$ at the center \(O\

Points $L$ and $M$ lie on a circle $\omega_1$ centered at $O$. The circle $\omega_2$ passing through points $O,$ $L,$ and $M$ is drawn. If the measure of arc $LM$ in circle $\omega_1$ is $82^\circ,$ then find the measure of arc $LOM$ in circle $\omega_2$, in degrees.

Solve a geometry problem involving two circles where you are asked to find the measure of arc LOM in a circle passing through points O, L, and M. This problem requires understanding circle geometry, central angles, and circumcircles. Ideal for students in Grades 9-12.

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