Question

what is true regarding two adjacent arcs created by two intersecting diameters?

they always have equal measures.
the difference of their measures is \\(90^\circ\\).
the sum of their measures is \\(180^\circ\\).
their measures cannot be equal.

Explanation:

Identify the geometric setup

We have two intersecting diameters, \(RP\) and \(SQ\), intersecting at the center \(T\) of a circle.

Analyze adjacent central angles

The intersecting diameters form linear pairs of central angles. For example, \(\angle RTQ\) and \(\angle QTP\) are adjacent angles lying on the straight line \(RP\). Therefore, their measures must sum to \(180^\circ\):

$$m\angle RTQ + m\angle QTP = 180^\circ$$

Relate angles to arc measures

The measure of a central angle is equal to the measure of its intercepted arc:

$$m\overparen{RQ} = m\angle RTQ$$

$$m\overparen{QP} = m\angle QTP$$

Determine the relationship

Substituting the arc measures into the linear pair equation:

$$m\overparen{RQ} + m\overparen{QP} = 180^\circ$$

Thus, the sum of the measures of any two adjacent arcs created by two intersecting diameters is always \(180^\circ\).

Answer:

• They always have equal measures.
• The difference of their measures is 90°.
• The sum of their measures is 180°. (Correct answer)
• Their measures cannot be equal.