1. The set of all points in the plane that are the same distance away from a specific point, called the center, is a **circle**. 2. The distance from the center of the circle to its outer rim is the **radius**. 3. A line segment whose endpoints are on a circle is a **chord**. 4. The longest chord of a circle is the **diameter**. 5. A portion of a circle is an **arc**. 6. A line that touches the circle at exactly one point is a **tangent**. 7. A line that touches the circle at exactly two points is a **secant**. 8. An arc that measures 180° is a **semicircle**. 9. An arc that measures greater than 180° is a **major arc**. 10. An arc that measures less than 180° is a **minor arc**. II. Without the figure, typical identifications are: 11. HF - likely a chord or segment 12. AF - likely a radius or chord 13. DG - likely a chord 14. BD - likely a chord 15. DB - same as BD, chord 16. JB - likely chord or secant 17. Point A - center or point on circle 18. Point H - point on circle or tangent point 19. HDF - likely an angle or arc 20. oA - center of circle (often denoted as O or oA) III. 21-22. Given angle at F inside triangle FGE with inscribed circle, opposite arc measures 265°. Step 1. The angle at F is half the measure of the intercepted arc: $$x = \frac{265^\circ}{2} = 132.5^\circ$$ Answer: $$\boxed{132.5^\circ}$$ 23-24. Angles at R = 53° and L = 51°, angle at T is x. Step 1. By circle theorems, angles subtended by the same arc relate. Step 2. Use angle sum or inscribed angle properties (more specific figure needed). Assuming angle at T subtends arc between R and L, $$x = 180^\circ - (53^\circ + 51^\circ) = 76^\circ$$ Answer: $$\boxed{76^\circ}$$ 25-26. For angle at E marked x, Given arcs measure 190° and 50°. Step 1. The measure of angle at E is half the difference of the intercepted arcs: $$x = \frac{190^\circ - 50^\circ}{2} = \frac{140^\circ}{2} = 70^\circ$$ Answer: $$\boxed{70^\circ}$$ 27-28. Angle at O marked x, arc marked 105°. Step 1. Angle at center intercepts the arc directly: $$x = 105^\circ$$ Answer: $$\boxed{105^\circ}$$ 29-30. Angles at A is 55°, angle at O is $2x+3$. Step 1. Assuming angle at O is central angle, angle at A is inscribed angle intercepting the same arc. Step 2. The central angle is twice the inscribed angle: $$2x + 3 = 2 \times 55 = 110$$ Step 3. Solve for x: $$2x + 3 = 110$$ $$2x = 107$$ $$x = \frac{107}{2} = 53.5$$ Answer: $$\boxed{53.5}$$