Draw two diameters. Then, fix one and rotate the other. What happens when the diameters are perpendicular?
Sometimes
Practice makes perfect
Let's consider a circle and two chords such that they intersect at the center of the circle.
As we can see, the two chords divide the circle in four intercepted arcs — namely, AB, BC, CD, and DA. Because the chords intersect at the center of the circle, we have that AB≅ CD and BC ≅ DA.
In the diagram above we can see that m BC ≠ m AB. However, let's fix BD and move AB to see whether the measures of these arcs become the same.
As we can see, the four arc measures are equal only when the angle between the chords is 90^(∘) — that is, when the chords are perpendicular. In conclusion, the measures of the intercepting arcs are sometimes equal to each other.
