Answer :
To analyze the properties of two adjacent arcs created by two intersecting diameters of a circle, let's break down the geometric principles involved.
1. Two intersecting diameters of a circle divide the circle into four equal arcs. Each diameter passes through the center of the circle and extends to the circumference, forming four 90-degree angles at the point of intersection, since each quadrant of the circle is one-fourth of the total 360 degrees.
2. These intersecting diameters form eight arcs along the circumference: four major arcs and four minor arcs. The adjacent arcs created around the point of intersection will each measure 90 degrees, totaling 360 degrees for the entire circumference.
Now, let's examine each statement:
1. They always have equal measures.
- Adjacent arcs created by the intersecting diameters will each measure 90 degrees. Since these adjacent arcs are formed by the angles from the intersecting diameters and each subtended angle is 90 degrees, we conclude that this statement is true.
2. The difference of their measures is [tex]$90^{\circ}$[/tex].
- Since each adjacent arc measures 90 degrees, the difference between any two adjacent arcs is \(90^\circ - 90^\circ = 0^\circ\). This statement is false.
3. The sum of their measures is [tex]$180^{\circ}$[/tex].
- If two adjacent arcs each measure 90 degrees, their sum is \(90^\circ + 90^\circ = 180^\circ\). This statement is true.
4. Their measures cannot be equal.
- As previously established, the arcs created by intersecting diameters each measure 90 degrees. Since their measures can and do equal 90 degrees, this statement is false.
Based on these analyses:
- The first statement is true.
- The second statement is false.
- The third statement is true.
- The fourth statement is false.
Thus, the true statements regarding two adjacent arcs created by two intersecting diameters are that they always have equal measures, and the sum of their measures is 180 degrees. The complete answer is:
1. \(1\) (True)
2. \(0\) (False)
3. \(1\) (True)
4. [tex]\(0\)[/tex] (False)
1. Two intersecting diameters of a circle divide the circle into four equal arcs. Each diameter passes through the center of the circle and extends to the circumference, forming four 90-degree angles at the point of intersection, since each quadrant of the circle is one-fourth of the total 360 degrees.
2. These intersecting diameters form eight arcs along the circumference: four major arcs and four minor arcs. The adjacent arcs created around the point of intersection will each measure 90 degrees, totaling 360 degrees for the entire circumference.
Now, let's examine each statement:
1. They always have equal measures.
- Adjacent arcs created by the intersecting diameters will each measure 90 degrees. Since these adjacent arcs are formed by the angles from the intersecting diameters and each subtended angle is 90 degrees, we conclude that this statement is true.
2. The difference of their measures is [tex]$90^{\circ}$[/tex].
- Since each adjacent arc measures 90 degrees, the difference between any two adjacent arcs is \(90^\circ - 90^\circ = 0^\circ\). This statement is false.
3. The sum of their measures is [tex]$180^{\circ}$[/tex].
- If two adjacent arcs each measure 90 degrees, their sum is \(90^\circ + 90^\circ = 180^\circ\). This statement is true.
4. Their measures cannot be equal.
- As previously established, the arcs created by intersecting diameters each measure 90 degrees. Since their measures can and do equal 90 degrees, this statement is false.
Based on these analyses:
- The first statement is true.
- The second statement is false.
- The third statement is true.
- The fourth statement is false.
Thus, the true statements regarding two adjacent arcs created by two intersecting diameters are that they always have equal measures, and the sum of their measures is 180 degrees. The complete answer is:
1. \(1\) (True)
2. \(0\) (False)
3. \(1\) (True)
4. [tex]\(0\)[/tex] (False)
