1. Let's start by stating the problem: We are examining properties of a circle focusing on chords, segments, angles, and tangents. 2. Recall that a chord is a segment with endpoints on the circle. A segment of a circle is the region bounded by a chord and the corresponding arc. 3. The tangent to a circle at a point is a line that touches the circle exactly at one point without crossing it. 4. **Properties of chords and segments:** - Equal chords subtend equal arcs. - A perpendicular dropped from the center of the circle to a chord bisects the chord. 5. **Angle properties related to chords and tangents:** - The angle subtended by a chord at the center is twice the angle subtended at the circumference. - The angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment. 6. For example, if there is a chord $AB$ and a tangent at point $A$, and angle $ABC = \theta$, then angle between the tangent at $A$ and chord $AB$ also equals $\theta$. 7. Using these properties allows solving various problems related to circle geometry involving lengths and angles. Answer summary: By understanding how chords, segments, angles, and tangents relate via equalities and bisections in circles, one can deduce many geometric facts and solve related problems.