Let's analyze the scenario of two diameters intersecting each other in a circle to answer the given question step-by-step.
### Step 1: Understanding the Intersecting Diameters
When two diameters intersect each other in a circle, they divide the circle into four equal parts. Each part is an arc formed by the endpoints of these intersecting diameters.
### Step 2: Total Degrees in a Circle
A circle has a total of 360 degrees. When this circle is split into four parts by the intersecting diameters, each part (each arc) will have an equal measure.
### Step 3: Measure of Each Arc
Since the circle is divided into four equal parts:
[tex]\[ \text{Measure of each arc} = \frac{360^{\circ}}{4} = 90^{\circ} \][/tex]
### Step 4: Properties of Adjacent Arcs
Considering two adjacent arcs:
- Equal Measures: Each arc has a measure of [tex]\(90^{\circ}\)[/tex], so they always have equal measures.
- Difference of Their Measures: Both arcs are [tex]\(90^{\circ}\)[/tex]. The difference between their measures is
[tex]\[ 90^{\circ} - 90^{\circ} = 0^{\circ} \][/tex]
### Step 5: Specific Statement Analysis
- Sum of Their Measures: The sum of the measures of two adjacent arcs is
[tex]\[ 90^{\circ} + 90^{\circ} = 180^{\circ} \][/tex]
### Step 6: Interpretation of the Statements
Let's check each of the given statements:
1. They always have equal measures.
- This is true because each arc has a measure of [tex]\(90^{\circ}\)[/tex].
2. The difference of their measures is [tex]\(90^{\circ}\)[/tex].
- This is incorrect because the difference is [tex]\(0^{\circ}\)[/tex].
3. The sum of their measures is [tex]\(180^{\circ}\)[/tex].
- This is true because [tex]\(90^{\circ} + 90^{\circ}=180^{\circ}\)[/tex].
4. Their measures cannot be equal.
- This is incorrect because their measures are equal, [tex]\(90^{\circ}\)[/tex].
### Conclusion
From the analysis, two statements are true:
- They always have equal measures.
- The sum of their measures is [tex]\(180^{\circ}\)[/tex].
Thus, the final step-by-step solution confirms the accuracy of the statements regarding the two adjacent arcs created by two intersecting diameters.
