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Lesson 6 of 8

I’m Up and Down, and Round and Round · Lesson 6 of 8

Angles Subtended by an Arc

“One arc can create several angles, but geometry knows how they are related.”

Learning Objectives

• Define major and minor arcs. • Understand angle subtended by an arc at the centre and on the circle. • Prove the central-angle theorem. • Understand angles in the same arc segment. • Derive the result that an angle in a semicircle is 90°.

Definition
Arc

An arc is a connected portion of a circle between two endpoints on the circle.

Two points A and B divide a circle into two arcs. The shorter route from A to B is the minor arc, and the longer route is the major arc.

Definition
Minor Arc

The smaller arc between two endpoints, subtending a central angle less than 180°.

Definition
Major Arc

The larger arc between two endpoints, subtending a central angle greater than 180°.

Major and minor arcs of a circle A circle with centre O. Points A and B divide the circumference into an orange minor arc AB and a blue major arc ACB passing through point C. Major and minor arcs Points A and B divide the circumference into two arcs A B C O Minor arc AB The two arcs Minor arc AB The shorter path from A to B. Major arc ACB The longer path from A to B, passing through C. Same endpoints: A and B Together they form the whole circle. Orange shows the minor arc; blue shows the major arc through C.
Major and minor arcs

Angle Subtended at the Centre

The angle subtended by an arc AB at the centre is the angle swept by the radius as we move from OA to OB along that chosen arc. The minor arc gives the smaller central angle; the major arc gives the reflex central angle.

Angle Subtended at a Point on the Circle

Choose a point P on the circle that is not on the chosen arc AB. Join P to A and B. The angle APB is the angle subtended by the arc at P.

Theorem

The angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the remaining part of the circle.

Central-angle theoremLaTeX

Why the Theorem Works

The proof joins the point on the circle to the centre and uses isosceles triangles formed by radii. Base angles are equal, and the exterior-angle theorem doubles those base angles. Combining the resulting angle relations gives the central angle as twice the angle on the circle.

Central angle is twice the inscribed angle A circle with centre O, points A and B defining a highlighted arc, and point P on the remaining circle. The central angle AOB is marked 2 theta and the inscribed angle APB is marked theta. Central angle and inscribed angle Both angles subtend the same highlighted arc AB θ 2θ A B P O arc AB Angles subtending arc AB Angle at the circle ∠APB = θ Angle at the centre ∠AOB = 2θ Central-angle theorem ∠AOB = 2∠APB The two angles stand on the same arc. The angle at the centre is twice the angle at any point on the remaining circle.
Central angle double inscribed angle

Angles in the Same Segment

If P, Q and R are all points on the same remaining arc, then each angle APB, AQB and ARB is half the same central angle AOB. Therefore all these angles are equal.

Angles in the same arc segmentLaTeX

Angle in a Semicircle

Definition
Corollary

A corollary is a result that follows immediately from a theorem already proved.

If AB is a diameter, the arc from A to B not containing point P subtends a straight angle of 180° at the centre. The angle at P is therefore half of 180°.

Angle in a semicircleLaTeX
Worked Example: Arc Angle

Problem
An arc subtends 70° at the centre. What angle does it subtend at a point on the remaining part of the circle?

  1. 1.The central angle is twice the angle on the circle.
  2. 2.Let the angle on the circle be x.
  3. 3.70°=2x.
  4. 4.x=35°.
Worked Example: Diameter

Problem
AB is a diameter and C is any other point on the circle. Find ∠ACB.

  1. 1.The diameter subtends 180° at the centre.
  2. 2.The angle at the circumference is half of 180°.
  3. 3.Therefore ∠ACB=90°.

Practice Problems

Practice Problems
  1. An arc subtends 84° at the centre. Find the angle it subtends at a point on the remaining circle.
  2. An angle subtended by an arc at the circle is 42°. Find the central angle subtended by the same arc.
  3. AB is a diameter and P is on the circle. Find ∠APB and explain why.
  4. Points P and Q lie on the same arc segment determined by chord AB. If ∠APB=63°, find ∠AQB.
  5. Explain the difference between the angle subtended by a major arc and a minor arc at the centre.

Key Takeaways

Key Takeaways

• Two endpoints determine a major and a minor arc. • The central angle depends on the chosen arc. • The central angle is twice the angle subtended on the remaining circle. • Angles subtended by the same arc in the same segment are equal. • A diameter subtends a right angle at every point on the circle.

Coming Next

Next, we use equal subtended angles to identify concyclic points and study cyclic quadrilaterals.

Previous · Lesson 5

Distance of Chords from the Centre

Next · Lesson 7

Concyclicity and Cyclic Quadrilaterals