4.4.3 Inscribed Angles and Central Angles

Comparing inscribed angles and central angles, understanding the inscribed angle theorem and angle-arc relationships.

定义

An inscribed angle is an angle formed by two chords in a circle that share an endpoint on the circle. The vertex of an inscribed angle lies on the circle, and the two sides of the angle are chords of the circle. A central angle is an angle formed by two radii of a circle, with its vertex at the center of the circle. The inscribed angle theorem states that an inscribed angle is half the measure of the central angle that subtends the same arc. Mathematically, if an inscribed angle and a central angle subtend the same arc, then the inscribed angle measure equals \(\frac{1}{2}\) times the central angle measure. Additionally, all inscribed angles that subtend the same arc are congruent to each other.

核心公式

• \(\text{Inscribed Angle} = \frac{1}{2} \times \text{Central Angle (same arc)}\)
• \(\text{Inscribed Angle} = \frac{1}{2} \times \text{Arc Measure}\)
• \(\text{Central Angle} = \text{Arc Measure}\)
• \(\text{Angle inscribed in semicircle} = 90°\)
• \(\text{If two inscribed angles subtend the same arc, then } \angle A = \angle B\)

易错点

• ⚠️ Confusing the inscribed angle with the central angle and using the full arc measure instead of half when calculating an inscribed angle
• ⚠️ Forgetting that the inscribed angle theorem requires both angles to subtend the SAME arc; angles subtending different arcs will have different measures
• ⚠️ Incorrectly assuming that an inscribed angle equals the arc measure directly, rather than half the arc measure
• ⚠️ Failing to recognize that an angle inscribed in a semicircle (where the two endpoints of the diameter are the endpoints of the angle) is always a right angle (90°)