Question

What is the converse of the conjecture: If the endpoints of an arc are the same points as the points where an inscribed angle intercepts a circle, then its measure is twice that of the angle.

a. If the measure of an angle is twice that of an inscribed angle, their endpoints are the same.
b. If the measure of an inscribed angle is twice that of the angle, their endpoints differ.
c. If an arc intercepts a circle, its measure is twice that of an inscribed angle.
d. If an inscribed angle intercepts a circle, its measure is twice that of the arc.

Answer 1

b. If the measure of an inscribed angle is twice that of the angle, their endpoints differ. The converse of the conjecture is that if the measure of an inscribed angle is twice that of the angle, their endpoints differ.

Step-by-step explanation:

The converse of the given conjecture is option b. If the measure of an inscribed angle is twice that of the angle, their endpoints differ.

To understand why this is the correct converse, let's break it down:

1. Original Conjecture: If the endpoints of an arc are the same points as the points where an inscribed angle intercepts a circle, then its measure is twice that of the angle.
2. Converse: If the measure of an inscribed angle is twice that of the angle, their endpoints differ.

Option b correctly captures the idea that if the measure of the inscribed angle is twice that of the angle, it implies that their endpoints are different.

Answer 2

The correct converse of the given conjecture is Option A: If the measure of an angle is twice that of an inscribed angle, their endpoints are the same, as it correctly swaps the hypothesis and conclusion of the original statement.

Step-by-step explanation:

The converse of the original conjecture states: If the endpoints of an arc are the same points as the points where an inscribed angle intercepts a circle, then its measure is twice that of the angle. The correct converse would be: If the measure of an arc is twice that of an inscribed angle, then the endpoints of the arc are the same points as the points where the inscribed angle intercepts the circle. Therefore, looking at the options provided, the correct answer is Option A: If the measure of an angle is twice that of an inscribed angle, their endpoints are the same. This is because by the definition of a converse, the hypothesis and conclusion of the original statement are swapped.