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°.
○ the sum of their measures is 180°.
○ their measures cannot be equal.

Explanation:

1. Recall the property of a circle: A diameter divides the circle into two semicircles, each with a measure of \(180^\circ\). When two diameters intersect at the center (point \(T\) in the diagram), they create adjacent arcs that together form a semicircle or a straight angle.
2. Analyze each option:

• Option 1: "They always have equal measures." This is not true because the diameters can be at any angle (not necessarily perpendicular or equal - angled in a non - symmetric way), so the adjacent arcs don't have to be equal. For example, if one diameter is horizontal and the other is at a \(30^\circ\) angle to the horizontal, the adjacent arcs will have measures \(30^\circ\) and \(150^\circ\) (not equal).
• Option 2: "The difference of their measures is \(90^\circ\)." There is no reason for the difference to be specifically \(90^\circ\). The angles between the diameters can vary, so the difference between the arc measures can be any value (not fixed at \(90^\circ\)).
• Option 3: "The sum of their measures is \(180^\circ\)." Since the two adjacent arcs are formed by two intersecting diameters, they lie on a straight line (a semicircle) at the center. The measure of a semicircle is \(180^\circ\), so the sum of the measures of two adjacent arcs formed by intersecting diameters is \(180^\circ\).
• Option 4: "Their measures cannot be equal." This is false. If the two diameters are perpendicular (e.g., one horizontal and one vertical), the adjacent arcs will each have a measure of \(90^\circ\) (so they can be equal).

Answer:

The sum of their measures is \(180^\circ\) (the third option: "The sum of their measures is \(180^\circ\)").