When chords intersect in a circle, the vertical angles formed intercept congruent arcs.
always
sometimes
never
Sometimes.
To determine whether the statement "When chords intersect in a circle, the vertical angles formed intercept congruent arcs" is always, sometimes, or never true, we need to examine the relationship between the vertical angles formed by the intersecting chords and the arcs intercepted by those angles.
Vertical angles are formed when two lines intersect. In the case of intersecting chords in a circle, the vertical angles are formed at the point of intersection inside the circle.
Now, let's consider the arcs intercepted by these vertical angles. An arc is a curve on the circle between two points. When a chord intersects another chord inside a circle, it divides each chord into two segments. These two segments of each chord intercept arcs on the circle.
To determine whether these intercepted arcs are congruent in all cases, we need to consider different scenarios:
1. When the intersecting chords are diameters of the circle:
In this case, the chords intersect at the center of the circle. When considering the vertical angles formed, we can observe that they are subtended by a semi-circle (180 degrees). Since a semi-circle is always the same size, the intercepted arcs are always congruent. Therefore, the statement is always true in this scenario.
2. When the intersecting chords are not diameters of the circle:
In this case, the chords intersect at a point inside the circle. The vertical angles formed by the intersecting chords can vary in size. The arcs intercepted by these angles will also vary in length since they are dependent on the size of the vertical angles. Therefore, the statement is sometimes true but not always.
In conclusion, the statement "When chords intersect in a circle, the vertical angles formed intercept congruent arcs" is sometimes true but not always. It is true when the intersecting chords are diameters of the circle, but it is not true when the intersecting chords are not diameters.