Arcs and Inscribed Angles

Unit 12: Circles · ~15–30 min

What's an inscribed angle?

An inscribed angle is an angle whose vertex sits on the circle itself, with both sides drawn as chords. That's the key difference from last lesson's central angle, whose vertex sits at the center. Same circle, same idea of intercepting an arc — but a very different relationship between the angle and the arc.

The Inscribed Angle Theorem

• Inscribed Angle Theorem: the measure of an inscribed angle is half the measure of its intercepted arc.
• Corollary — same arc: two inscribed angles that intercept the same arc are congruent to each other.
• Corollary — semicircle: an angle inscribed in a semicircle (its intercepted arc is a semicircle) is always a right angle.
• Corollary — cyclic quadrilateral: if a quadrilateral is inscribed in a circle, its opposite angles are supplementary.

Guided practice

• 1.Inscribed angle ∠PQR intercepts an arc of 130°. Find m∠PQR.
  Show answer

  m∠PQR = ½(130°) = 65°.
• 2.Inscribed angle ∠DEF measures 40°. Find its intercepted arc DF.
  Show answer

  Since the angle is half the arc, the arc is double the angle: arc DF = 2(40°) = 80°.
• 3.Triangle ABC is inscribed in a circle so that segment AC is a diameter. What must m∠ABC be, and why?
  Show answer

  90°. Since AC is a diameter, it cuts off a semicircle (180°), and ∠ABC is inscribed in that semicircle, so by the corollary it's a right angle.
• 4.Inscribed angles ∠ABC and ∠ADC both intercept arc AC. If m∠ABC = 58°, find m∠ADC.
  Show answer

  58°. Inscribed angles that intercept the same arc are congruent, regardless of where their vertices sit on the circle.
• 5.Cyclic quadrilateral has m∠R = 72°. What is the measure of the angle opposite ∠R, and what would change if the quadrilateral were not inscribed in a circle?
  Show answer

  The opposite angle is 180° − 72° = 108°, because opposite angles of a cyclic quadrilateral are supplementary. If the quadrilateral weren't inscribed in a circle, that guarantee disappears — a general quadrilateral's opposite angles don't have to sum to 180°.