To understand the relationship between two adjacent arcs created by intersecting diameters, let's first visualize the scenario.
1. Intersecting Diameters in a Circle:
- When two diameters intersect, they do so at the center of the circle, creating four equal-sized angles of [tex]\(90^{\circ}\)[/tex] each.
- Since any diameter splits the circle into two equal halves, each diameter cuts the circle into two [tex]\(180^{\circ}\)[/tex] arcs.
2. Creation of Adjacent Arcs:
- The intersection of two diameters will thus create four arcs around the circle.
- These arcs are adjacent to one another, and each arc corresponds to a [tex]\(90^{\circ}\)[/tex] segment of the circle.
3. Properties of the Arcs:
- Because each angle at the intersection is [tex]\(90^{\circ}\)[/tex], each adjacent arc also measures [tex]\(90^{\circ}\)[/tex], as it represents one-quarter of the entire circle, which is [tex]\(360^{\circ}\)[/tex].
Therefore, each of the four arcs measures [tex]\(90^{\circ}\)[/tex]. Given that these arcs are adjacent to each other and are created by the same pair of intersecting diameters, all adjacent arcs will measure the same, specifically [tex]\(90^{\circ}\)[/tex].
So, the correct answer is:
"They always have equal measures."
