**I. Exterior Intersection Angles Problem:** Given angles and arcs around circle points A, B, C, D, E with secants/tangents intersecting outside. 1. m\angle BQC = ? (angle formed outside by secants BQ and QC) 2. m\angle EPD = ? (angle formed outside by secants EP and PD) 3. m\angle ED = ? (angle at E and D's relation to arcs) 4. m\angle CDE = ? 5. m\angle CBE = ? 6. m\angle AB = ? (arc measure) **Step 1: State exterior angle theorem** Angle formed by two secants/tangents intersecting outside circle equals half the difference of intercepted arcs. $$m\angle = \frac{1}{2} |m\text{arc major} - m\text{arc minor}|$$ **1. Find m\angle BQC** - Secants intersect at Q. - Arcs intercepted: arc AB = 86°, arc BC = 50° - So, angle BQC = \frac{1}{2}|86-50| = \frac{1}{2} \times 36 = 18° **2. Find m\angle EPD** - Angle at P formed by secants EP and PD. - Major arc ED = 138°, minor arc CD = 125° (assumed from given info adjacent to points C and D). - angle EPD = \frac{1}{2}|138-125| = \frac{1}{2} \times 13 = 6.5° **3. Find m\angle ED** - This seems incomplete, but likely arc ED = 138° (given). **4. Find m\angle CDE** - Angle inside circle formed by chord CD and DE. - Formula for angle inside circle formed by chords: angle = half sum of intercepted arcs. - intercept arcs: arc CE (not given here), approximate using 86° + 50° + 125° = 261° total so rest arc is 360-261=99^ - angle CDE = \frac{1}{2}(arc CE + arc BD) approximates to - Given incomplete info, we rely on provided arcs. **5. m\angle CBE** - Angle formed at B, tangent with arc C and E. - Using tangent-secant or tangent-chord theorem: - angle = \frac{1}{2} arc CE = \frac{1}{2}(arc C + arc E) [assuming arc C = 125°, arc E unknown] - With incomplete info, leave answer incomplete. **6. m\angle AB** (arc measure AB given 86°) **II. Interior Intersection Angles Problem:** Angles formed inside the circle at V by chords and secants. 1. m\angle WVZ = ? angle formed at V by chords WV and VZ - Angle formed inside circle equals half sum of intercepted arcs - arcs WZ, XY given indirectly - given angle at Z 130°, at V 62° Calculate: **1. m\angle WVZ** - Using inscribed angle theorem: angle = \frac{1}{2} intercepted arc WX (assumed) - Given angle = 62° (possibly). **2. m\angle WVX** - Angle inside circle formed by chords WV and VX - equals half sum of arcs WX + VX **3. m\angle YZ** - Given arc likely 130° **4. m\angle XY** - Given chord angle 62° possibly **5. m\angle WX** - Combination using above arcs **6. m\angle WZY** - Angle at Z formed by chords WZ and ZY --- Since details insufficient for exact numeric answers for some angles due to missing arcs or instructions, the main recognized steps follow the theorems: 1. Exterior angle formed by two secants/tangents outside circle = half difference of intercepted arcs. 2. Interior angle formed by chords/secants inside circle = half sum of intercepted arcs. **Final answers (where derivable):** 1. m\angle BQC = 18° 2. m\angle EPD = 6.5° 3. m\arc ED = 138° 6. m\arc AB = 86° Other angle measures require additional arc measures. ---