To address the question concerning two adjacent arcs created by two intersecting diameters, let's break down the situation step-by-step.
1. Understanding Intersecting Diameters: In a circle, when two diameters intersect, they form four right angles at the point where they cross. Since the total degrees in a circle also equal 360 degrees, each of these four angles formed at the intersection point is 90 degrees.
2. Division of the Circle: The intersecting diameters essentially divide the circle into four equal quadrants. Each quadrant will then have an angular measure of 90 degrees.
3. Adjacent Arcs: When looking at any two adjacent arcs created by these intersecting diameters, focus on the fact that each of these arcs corresponds to one of those 90-degree angles at the intersection point.
4. Equality of the Measures: Given that each quadrant is 90 degrees, the arcs associated with these quadrants (two at a time, adjacent arcs) will likewise have the measure of 90 degrees each.
5. Correct Understanding of Options:
- "They always have equal measures." - This statement indeed reflects the true nature of the situation since each of the arcs created in between intersecting diameters is 90 degrees, making two adjacent arcs equal in measure.
- "The difference in their measures is [tex]\(90^{\circ}\)[/tex]." - This is incorrect since both measures are the same (both are 90 degrees), resulting in a difference of 0 degrees.
- "The sum of their measures is [tex]\(180^{\circ}\)[/tex]." - This is correct but slightly misinterpreted. While two 90-degree arcs do add up to 180 degrees, it is not specific to only their sum but emphasizes their individual equal measures more accurately.
- "Their measures cannot be equal." - This is entirely false given that the adjacent arcs do have equal measures.
So, the true statement regarding the measures of two adjacent arcs created by two intersecting diameters in a circle is:
They always have equal measures.
