The angle in a major segment is proven to be acute by applying the Inscribed Angle Theorem, which establishes that an inscribed angle is precisely half the measure of its central angle. Since the central angle subtending the minor arc (which corresponds to the major segment) is invariably less than 180 degrees, the inscribed angle in the major segment will always measure less than 90 degrees, thus making it acute.
Understanding the Geometry
To effectively prove that the angle in a major segment is acute, it's crucial to first understand the key geometric terms involved:
- Major Segment: This is the larger region of a circle cut off by a chord. It includes the major arc and the chord.
- Major Arc: The longer arc created when a circle is divided by a chord.
- Inscribed Angle (Angle in a Segment): An angle formed by two chords in a circle that have a common endpoint on the circle's circumference. The vertex of the angle lies on the circle.
- Central Angle: An angle whose vertex is at the center of the circle and whose sides are radii intersecting the circle at two points.
- Acute Angle: An angle measuring less than 90 degrees.
The Inscribed Angle Theorem: The Foundation of the Proof
The proof relies fundamentally on the Inscribed Angle Theorem, a cornerstone of circle geometry. This theorem states:
The measure of an inscribed angle is half the measure of its central angle that subtends the same arc.
Let's consider a visual representation to clarify the components:
| Component | Description |
|---|---|
| Chord AB | A line segment connecting two points on the circle. This chord divides the circle into a major segment and a minor segment. |
| Major Arc AB | The larger arc of the circle defined by points A and B. |
| Minor Arc AB | The smaller arc of the circle defined by points A and B. |
| Point C | Any point chosen on the major arc AB. The angle formed at C (∠ACB) is the angle in the major segment. |
| Center O | The central point of the circle. |
| Central Angle ∠AOB | The angle formed at the center O by radii OA and OB. This specific central angle subtends the minor arc AB. (Note: There's also a reflex central angle subtending the major arc, but for this proof, we focus on the central angle subtending the minor arc.) |
| Inscribed Angle ∠ACB | The angle formed by the chords AC and BC, with its vertex C on the major arc. This angle subtends the minor arc AB. |
Step-by-Step Proof
Here's how to formally prove that the angle in a major segment is always acute:
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Identify the Angle in the Major Segment:
- Let a circle have its center at O.
- Draw a chord AB. This chord divides the circle into a minor segment and a major segment.
- Let C be any point on the major arc (the arc that forms the major segment).
- The angle we want to prove as acute is the inscribed angle ∠ACB.
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Identify the Corresponding Central Angle:
- The inscribed angle ∠ACB subtends the minor arc AB.
- The central angle that also subtends the same minor arc AB is ∠AOB (where O is the center of the circle).
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Apply the Inscribed Angle Theorem:
- According to the Inscribed Angle Theorem, the measure of the inscribed angle ∠ACB is half the measure of the central angle ∠AOB.
- Therefore, we can write the relationship as: ∠ACB = 1/2 * ∠AOB.
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Determine the Range of the Central Angle:
- The central angle ∠AOB subtends the minor arc AB.
- By definition, a minor arc is always less than a semicircle. Consequently, the central angle subtending a minor arc is always less than 180 degrees.
- So, 0° < ∠AOB < 180°. This means ∠AOB can be acute, right, or obtuse, but never 180° or more.
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Calculate the Range of the Inscribed Angle:
- Since ∠ACB = 1/2 * ∠AOB, we can divide the range of ∠AOB by two:
- 1/2 * 0° < 1/2 * ∠AOB < 1/2 * 180°
- 0° < ∠ACB < 90°
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Conclude the Nature of the Angle:
- An angle that measures greater than 0° and less than 90° is, by definition, an acute angle.
- Therefore, the angle ∠ACB, which is the angle in the major segment, is always acute.
This proof demonstrates how the fundamental properties of circles and the Inscribed Angle Theorem work together to establish this consistent geometric characteristic.
