Circle Angle Theorems: Formulas by Vertex Location

The provided image is a geometry reference sheet detailing the relationship between angles and intercepted arcs in circles. It categorizes these relationships based on the location of the angle's vertex (at the center, inside, on, or outside the circle).

Answer

The image outlines six fundamental circle theorems used to calculate angle measures or arc lengths. These include relationships for central angles, inscribed angles, angles formed by intersecting secants or tangents, and properties of cyclic quadrilaterals.

Explanation

1. Central Angle Theorem When the vertex is at the center of the circle (e.g., $\angle ABC$), the measure of the angle is exactly equal to the measure of its intercepted arc. $\angle ABC = m\text{arc}(AC)$ The central angle is always equivalent to the degree measure of the arc it subtends at the center.

2. Angles Outside the Circle (Secant-Secant or Secant-Tangent) When the vertex is outside (e.g., $\angle ACD$), the angle measure is half the difference of the intercepted arcs. $\angle ACD = \frac{1}{2}(\text{Far Arc} - \text{Near Arc})$ This formula subtracts the smaller interior arc from the larger exterior arc before halving the result. ⚠️ This step is required on exams to distinguish from the "Inside" addition rule.

3. Interior Intersecting Chords When the vertex $E$ is inside the circle (but not at the center), the angle is the average of the two intercepted arcs. $\angle AED = \frac{1}{2}(\text{arc}_1 + \text{arc}_2)$ The measure of an angle formed by two segments intersecting inside a circle is half the sum of the intercepted arcs.

4. Inscribed Angles or Tangent-Chord Angles When the vertex is on the circle (e.g., $\angle ABC$), the angle is exactly half the measure of the intercepted arc. $\angle ABC = \frac{1}{2}m\text{arc}(AC)$ Alternatively, the arc is twice the measure of the inscribed angle ($2 \times \angle = \text{arc}$).

5. Circumscribed Angles (Tangent-Tangent) When two tangents meet outside at vertex $B$, the angle and the minor arc are supplementary. $\angle ABC + \text{Minor Arc} = 180^{\circ}$ The external angle formed by two tangents plus the smaller intercepted arc always totals 180 degrees.

6. Inscribed Quadrilaterals (Cyclic Quadrilaterals) For a four-sided figure where all vertices lie on the circle, opposite angles must be supplementary. $\angle A + \angle C = 180^{\circ} \text{ and } \angle B + \angle D = 180^{\circ}$ Opposite angles in a cyclic quadrilateral always add up to a straight angle.

Final Answer

The formulas depend on the vertex location:

• Center: $\angle = \text{arc}$
• Inside: $\angle = \frac{1}{2}(\text{arc} + \text{arc})$
• On: $\angle = \frac{1}{2}(\text{arc})$
• Outside: $\angle = \frac{1}{2}(\text{arc}_{big} - \text{arc}_{little})$

Common Mistakes

• The Addition/Subtraction Swap: Students often add arcs for angles outside the circle or subtract them for angles inside. Remember: Inside = Add (think of the plus sign $+$ as an internal cross).
• Central vs. Inscribed: Forgetting that central angles are equal to the arc, while inscribed angles (on the rim) are half the arc. Always check if the vertex is at the center point or on the edge.