Question

reflect: confidence check
drag the point to show how confident you are in your understanding of properties of angles in a circle.
summarize the properties of central angles, inscribed angles, angles in a semicircle, and angles subtended by the same arc.

Explanation:

• Central Angles: A central angle is an angle whose vertex is at the center of the circle, and its measure is equal to the measure of its intercepted arc. For example, if a central angle intercepts an arc of measure \( \theta \) (in degrees), the measure of the central angle is also \( \theta \). Mathematically, if the arc length is \( s \) and the radius is \( r \), the central angle \( \theta \) (in radians) is \( \theta=\frac{s}{r} \), and in degrees, \( \theta=\frac{s}{2\pi r}\times360^\circ \).
• Inscribed Angles: An inscribed angle is an angle whose vertex lies on the circle, and its sides contain chords of the circle. The measure of an inscribed angle is half the measure of its intercepted arc. If an inscribed angle intercepts an arc of measure \( \alpha \), the measure of the inscribed angle is \( \frac{\alpha}{2} \). For example, if the intercepted arc is a semicircle (measure \( 180^\circ \)), the inscribed angle is a right angle (\( 90^\circ \)).
• Angles in a Semicircle: An angle inscribed in a semicircle is a right angle. This is a special case of an inscribed angle where the intercepted arc is a semicircle (measure \( 180^\circ \)). Using the inscribed angle theorem, the measure of the inscribed angle is \( \frac{180^\circ}{2} = 90^\circ \), so it forms a right - angled triangle with the diameter of the semicircle as one of its sides.
• Angles Subtended by the Same Arc: Angles subtended by the same arc (or congruent arcs) are equal. If two inscribed angles intercept the same arc (or arcs of equal measure), then the measures of these inscribed angles are equal. Also, a central angle and an inscribed angle subtended by the same arc have a relationship where the central angle is twice the inscribed angle (as per the inscribed angle theorem).

Answer:

• Central Angles: Measure equals its intercepted arc's measure.
• Inscribed Angles: Measure is half its intercepted arc's measure.
• Angles in a Semicircle: Inscribed angle is \( 90^\circ \) (right angle).
• Angles Subtended by Same Arc: Angles (inscribed) subtended by same/congruent arcs are equal; central angle is twice the inscribed angle subtended by the same arc.