Alternate Segment Theorem: Statement, Proof & Examples

In the study of mathematics, a circle is understood to be a flat, round shape where every point along its edge maintains an equal distance from a central point. 

• The segment of a circle refers to the area that lies between a chord and the arc of the circle it subtends.
• When chords are drawn in a circle, they will form two segments, namely the major segment and the minor segment.
• Alternate segment theorem determines that the angle between a tangent and a chord is equal to the angle formed by the same chord in the alternate segment.
• Various theorems in circular geometry elucidate the properties and relationships of these segments. 
• In this article, we will explore the alternate segment theorem, including its proof and practical application, and use it to solve difficult problems. 

Key Terms: Alternate Segment Theorem, Segment, Circle, Alternate Segment, Major Segment, Minor Segment, Alternate Segment Theorem Proof, Quadrilateral

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Alternate Segment Theorem Statement

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The Alternate Segment Theorem is a principle within circle geometry. It states that in any given circle, the angle created where a tangent meets a chord at the point of tangency is congruent to the angle produced in the corresponding alternate segment. This theorem is also commonly referred to as the Tangent-Chord Theorem.

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Alternate Segment Theorem Proof

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Consider a circle where a tangent touches it at point A. From this point of contact, a chord AB is extended, forming an angle “α” with the tangent.

• Let’s say this chord AB creates an angle β at a point C on the circle’s circumference, as depicted in the diagram. 
• It is understood that the angle ∠ACB, which equals ∠β, is the alternate angle corresponding to the angle formed between tangent A and chord AB in the opposite segment.

∠ACB = ∠β 

Proof: Let’s consider a scenario where point A lies on the perimeter of a circle, with O as the circle’s center.

• Imagine a tangent PQ that intersects the circle at point A.
• These tangents form an angle α with chord AB.
• Now, let’s say angle ∠ACB, denoted as ∠β, is situated in the alternate segment relative to the angle between tangent A and chord AB.
• Our goal is to establish that ∠α equals ∠β

∠α =∠β

• Given that OA and OB are both radii of the circle, they are equal in length.

OA =OB

• Consequently, angles ∠OAB and ∠OBA are equal, as they are opposite equal sides.

∠OAB = ∠OBA

• Since triangle OAB is isosceles,
• ∠AOB is calculated as 180° minus the sum of ∠OAB and ∠OBA.
• This simplifies to ∠AOB being 180° minus twice ∠OAB, which we’ll refer to as equation (1).

∠AOB = 180° – ∠OAB – ∠OBA

∠AOB = 180° – 2∠OAB …(1)

• Given that PQ is a tangent,
• ∠OAQ is a right angle, 90°.

∠OAQ = 90°

• Thus, α can be expressed as 90° minus ∠OAB, which we’ll call equation (2).

α= 90° – ∠OAB …(2)

• Combining equations (1) and (2), we deduce that
• ∠AOB is twice α.

∠AOB = 2α

• It’s known that the central angle of a circle is double any angle at its circumference.
• So, ∠AOB is also twice ∠ACB.
• This leads us to conclude that ∠ACB is half of ∠AOB.

∠AOB = 2∠ACB

∠ACB= (½)∠AOB

• Substituting ∠AOB with twice α in this relationship, we find that
• ∠β is half of twice α which simplifies to ∠β being equal to α.

∠β= (½) 2α

∠β= ∠α

• Hence, the Alternate Segment Theorem is validated.

Alternate Segment Theorem Proof

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Alternate Segment Theorem Quadrilateral

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The alternate segment theorem also finds its application in determining the corresponding segments within a quadrilateral.

• Imagine the scenario depicted in the provided illustration,
• Utilizing the Alternate Segment Theorem, it can be established that:

∠p = ∠r

• The objective now is to demonstrate that

∠ s = ∠q

• Given that LM is a straight tangent line, it follows that:

∠p +∠s =180°------(3)

• And since ∠r and ∠q are opposite angles within the cyclic quadrilateral, it holds that:

∠q +∠r = 180°-----(4)

• By equating expressions (3) and (4), we arrive at:

∠p +∠s =∠q +∠r

• This leads to the conclusion that

∠p =∠r

• and consequently,

∠s =∠q

• Thus, the theorem’s validity is confirmed within the context of quadrilaterals.

Alternate Segment Theorem Quadrilateral

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Things to Remember

• The alternate segment theorem applies to circles and the angles created by tangents and chords.
• It equates the angle between the tangent and chord to the angle in the alternate segment.
• The theorem is valid regardless of the size of the circle.
• It can be applied to find missing angles in cyclic quadrilaterals.
• The proof relies on the properties of tangents, chords, and isosceles triangles.
• Remember that the angle at the center is twice the angle at the circumference.
• The alternate segment theorem is a testament to the uniformity and symmetry inherent in circles.
• It is a fundamental theorem with applications in various geometric problems.