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 $. Circle $ \omega_1 $ is centered at point $ O $, and points $ L $ and $ M $ lie on this circle. Circle $ \omega_2 $ passes through points $ O $, $ L $, and $ M $. The arc $ LM $ on circle $ \omega_1 $ subtends an angle of $ 82^\circ $ at the center $ O $ of circle $ \omega_1 $. Our task is to find the measure of arc $ LOM $ in circle $ \omega_2 $.

### Step 1: Understanding the Geometry
Since points $ L $, $ M $, and $ O $ lie on both circles $ \omega_1 $ and $ \omega_2 $, we focus on the relationship between the arcs and the angles they subtend. The key observation is that the arc $ LM $ in $ \omega_1 $ subtends an angle of $ 82^\circ $ at the center $ O $.

### Step 2: Central Angle of Circle $ \omega_1 $
The measure of arc $ LM $ on circle $ \omega_1 $ is given as $ 82^\circ $, which means that the central angle $ \angle LOM $ in circle $ \omega_1 $ is also $ 82^\circ $. This is because in any circle, the measure of a central angle is equal to the measure of the arc it subtends.

### Step 3: Analyzing Circle $ \omega_2 $
In circle $ \omega_2 $, the points $ L $, $ O $, and $ M $ form a triangle, and we need to determine the measure of the arc $ LOM $ in this circle. Since $ LOM $ is subtended by the chord $ LM $ in circle $ \omega_2 $, and because $ O $ lies on the circle $ \omega_2 $, we recognize that $ \omega_2 $ must be the circumcircle of triangle $ LOM $.

### Step 4: Inscribed Angle Theorem
In circle $ \omega_2 $, the angle subtended by the arc $ LM $ at the point $ O $ (on the circumference of $ \omega_2 $) is an inscribed angle. By the inscribed angle theorem, the measure of an inscribed angle is half the measure of the corresponding arc. Therefore, the angle $ \angle LOM $ in circle $ \omega_2 $ is half of the measure of arc $ LM $ in circle $ \omega_1 $.

Thus, the measure of arc $ LOM $ in circle $ \omega_2 $ is:
\[
\boxed{41^\circ}.
\]

### Conclusion
The measure of arc $ LOM $ in circle $ \omega_2 $ is $ 41^\circ $.

Would you like more details or have any questions? Here are five related questions to deepen your understanding:
1. How do the inscribed angle and central angle relate in a circle?
2. What is the circumcenter of a triangle, and how is it related to the circumcircle?
3. Can you describe other geometric properties of cyclic quadrilaterals?
4. How would the problem change if arc $ LM $ subtended a different angle at the center of $ \omega_1 $?
5. How can you determine the radius of circle $ \omega_2 $ given specific lengths?

**Tip**: The inscribed angle theorem is a fundamental concept for solving problems involving arcs and angles in circles, and mastering it can help in many geometric configurations.

Points L and M lie on a circle ω₁ centered at O. The circle ω₂ passing through points O, L, and M is drawn. If the measure of arc LM in circle ω₁ is 82°, then find the measure of arc LOM in circle ω₂, in degrees.

Learn how to solve a geometry problem involving two circles and arc measures. Points L and M lie on circle ω₁, and a second circle ω₂ passes through O, L, and M. Given the arc LM in ω₁ is 82°, find the measure of arc LOM in ω₂ using the Inscribed Angle Theorem. Suitable for students in Grades 9-11.

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