Circle Geometry

Circle Geometry: Comprehensive Notes & Flashcards

Concept 1: Central Angles and Arcs

📝 Detailed Notes

• Definition: A central angle is an angle whose vertex is at the center of the circle, and whose sides are radii intersecting the circle.

• Key Theorem: The measure of a central angle is equal to the measure of its intercepted arc.
  \text{m}\angle\text{Central} = \text{m}\overarc{\text{Intercepted Arc}}

• When to Use It: Use this when you have an angle with its vertex at the very center of the circle, and you need to find either the angle's measure or the arc's measure.

💡 Flashcard Format

• Flashcard Front: What is the relationship between a central angle and its intercepted arc?

• Flashcard Back: The measure of a central angle is EQUAL to the measure of its intercepted arc. (m∠Central = mArc)

Concept 2: Inscribed Angles and Arcs

📝 Detailed Notes

• Definition: An inscribed angle is an angle whose vertex is on the circle itself, and whose sides are chords that intersect the circle at two other points.

• Key Theorem: The measure of an inscribed angle is half the measure of its intercepted arc.
  \text{m}\angle\text{Inscribed} = \frac{1}{2} \times \text{m}\overarc{\text{Intercepted Arc}}
  Conversely, the intercepted arc is twice the inscribed angle:
  \text{m}\overarc{\text{Intercepted Arc}} = 2 \times \text{m}\angle\text{Inscribed}

• When to Use It: Use this when you have an angle with its vertex on the circumference of the circle, and you need to relate its measure to the arc it "cuts off."

💡 Flashcard Format

• Flashcard Front: How do you find the measure of an inscribed angle given its intercepted arc?

• Flashcard Back: An inscribed angle is HALF the measure of its intercepted arc. (m∠Inscribed = 1/2 mArc)

Concept 3: Inscribed Quadrilaterals

📝 Detailed Notes

• Definition: An inscribed quadrilateral is a four-sided polygon (quadrilateral) where all four of its vertices lie on the circumference of a circle.

• Key Theorem: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. This means they add up to $180^{\circ}$.

  • For an inscribed quadrilateral ABCD, this means:

    • $\text{m}\angle A + \text{m}\angle C = 180^{\circ}$

    • $\text{m}\angle B + \text{m}\angle D = 180^{\circ}$

• When to Use It: Use this theorem whenever you encounter a quadrilateral with all its corners touching the circle, and you need to find missing angle measures.

💡 Flashcard Format

• Flashcard Front: What is the key property of an inscribed quadrilateral's angles?

• Flashcard Back: Its opposite angles are supplementary (add up to $180^{\circ}$).

Concept 4: Angles Formed by a Tangent and a Chord

📝 Detailed Notes

• Definition: This angle is formed when a tangent line (a line that touches the circle at exactly one point) and a chord (a line segment connecting two points on the circle) intersect at the point of tangency.

• Key Theorem: The measure of an angle formed by a tangent and a chord is half the measure of its intercepted arc.

  • This is very similar to the inscribed angle theorem!

  • $\text{m}\angle\text{Tangent-Chord} = \frac{1}{2} \times \text{m}\overarc{\text{Intercepted Arc}} $

• When to Use It: Look for a line that just touches the circle at one point, and a chord that starts from that same point. The angle between them relates to the arc "cut off" by the chord.

💡 Flashcard Format

• Flashcard Front: How do you find the measure of an angle formed by a tangent and a chord?

• Flashcard Back: It's HALF the measure of its intercepted arc.

Concept 5: Angles Formed by Two Intersecting Chords (Inside the Circle)

📝 Detailed Notes

• Definition: This angle is formed when two chords intersect inside the circle.

• Key Theorem: The measure of an angle formed by two intersecting chords inside a circle is half the sum of the measures of the two intercepted arcs.

  • There are two pairs of vertical angles formed. Each angle in a pair will be half the sum of the two arcs it intercepts (one directly in front, one directly behind).

  • $\text{m}\angle\text{Inside} = \frac{1}{2} (\text{m}\overarc{\text{Arc 1}} + \text{m}\overarc{\text{Arc 2}})$

• When to Use It: When you see two lines crossing inside the circle, and both lines are chords.

💡 Flashcard Format

• Flashcard Front: How do you find the measure of an angle formed by two chords intersecting inside a circle?

• Flashcard Back: It's HALF the SUM of the two intercepted arcs.

Concept 6: Angles Formed by Secants and/or Tangents (Outside the Circle)

📝 Detailed Notes

• Definition: This category covers angles formed when two secants, two tangents, or one secant and one tangent intersect outside the circle.

• Key Theorem: The measure of an angle formed by two secants, two tangents, or a secant and a tangent intersecting outside a circle is half the difference of the measures of the two intercepted arcs (the farther arc minus the closer arc).

  • $\text{m}\angle\text{Outside} = \frac{1}{2} (\text{m}\overarc{\text{Farther Arc}} - \text{m}\overarc{\text{Closer Arc}})$

• When to Use It: When you see two lines (secants or tangents) meeting at a point outside the circle, and they both cut off arcs on the circle.

💡 Flashcard Format

• Flashcard Front: How do you find the measure of an angle formed by lines intersecting outside a circle?

• Flashcard Back: It's HALF the DIFFERENCE of the two intercepted arcs (farther arc - closer arc).

Concept 7: Intersecting Chords Theorem (Segment Lengths)

📝 Detailed Notes

• Definition: This theorem applies when two chords intersect inside a circle.

• Key Theorem: If two chords intersect inside a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord.

  • If chord AB and chord CD intersect at point P inside the circle, then:
    AP \times PB = CP \times PD

• When to Use It: Use this when you have two lines crossing inside the circle, and both lines are chords. You'll typically be given three segment lengths and need to find the fourth.

💡 Flashcard Format

• Flashcard Front: What is the relationship between the segments of two chords intersecting inside a circle?

• Flashcard Back: The product of the segments of one chord equals the product of the segments of the other. (Part × Part = Part × Part)

Concept 8: Secant-Secant Product Theorem (Segment Lengths)

📝 Detailed Notes

• Definition: This theorem applies when two secant segments are drawn to a circle from an external point.

• Key Theorem: If two secant segments are drawn to a circle from the same external point, then the product of the measure of one secant segment and its external segment is equal to the product of the measure of the other secant segment and its external segment.

  • If secant segment PA and secant segment PC are drawn from external point P, intersecting the circle at B and D respectively (so P-B-A and P-D-C), then:
    PA \times PB = PC \times PD

  • Remember: The "whole" segment (PA or PC) includes the "external" part (PB or PD).

• When to Use It: When you have two lines originating from the same point outside the circle, and both lines pass through the circle at two points (i.e., they are secants). You'll be given three segment lengths and need to find the fourth.

💡 Flashcard Format

• Flashcard Front: What is the relationship between the segments of two secants drawn from the same external point?

• Flashcard Back: Whole Secant × External Part = Whole Secant × External Part.

Concept 9: Secant-Tangent Product Theorem (Segment Lengths)

📝 Detailed Notes

• Definition: This theorem applies when a secant segment and a tangent segment are drawn to a circle from the same external point.

• Key Theorem: If a tangent segment and a secant segment are drawn to a circle from the same external point, then the square of the length of the tangent segment is equal to the product of the measure of the secant segment and its external segment.

  • If tangent segment PT touches the circle at T, and secant segment PA intersects the circle at B (so P-B-A), then:
    PT^2 = PA \times PB

  • Remember: The tangent segment is squared. The secant part is "whole times external."

• When to Use It: When you have one line touching the circle at one point (tangent) and another line passing through the circle at two points (secant), both originating from the same external point.

💡 Flashcard Format

• Flashcard Front: What is the relationship between a tangent segment and a secant segment drawn from the same external point?

• Flashcard Back: Tangent² = Whole Secant × External Part.