1. **State the problem:** We have a pentagon ABCDE inscribed in a circle with center O. Given: DC = DE, angle CÔB = 44°, and angle ABÊ = 59°. We need to find all interior angles of pentagon ABCDE. 2. **Analyze given information:** - Since O is the center, angle CÔB = 44° is a central angle. - DC = DE means triangle DCE is isosceles with base CE. - Angle ABÊ = 59° is an angle at vertex B of the pentagon. 3. **Find arcs and related angles:** - Central angle CÔB = 44° subtends arc CB of 44°. - The inscribed angle ABE (angle at B) intercepts arc AE. - Since angle ABÊ = 59°, arc AE = 2 × 59° = 118° (because an inscribed angle is half the measure of its intercepted arc). 4. **Calculate remaining arcs:** - The circle total is 360°. - Known arcs: CB = 44°, AE = 118°. - The remaining arcs are AB, BC, CD, DE, and EA. - Points are in order A-B-C-D-E, so arcs are AB, BC, CD, DE, EA. - We know CB = 44°, so arc BC = 44°. - Arc AE = 118° includes arcs AB + BC + CD + DE + EA, but since AE is from A to E, it includes arcs AB + BC + CD + DE. - So arcs AB + BC + CD + DE = 118°. - We know BC = 44°, so AB + CD + DE = 118° - 44° = 74°. 5. **Use isosceles triangle DCE:** - Since DC = DE, arcs DC and DE are equal. - Let arc DC = arc DE = x. - Then AB + x + x = 74° => AB + 2x = 74°. 6. **Find arc AB:** - The pentagon arcs sum to 360°: AB + BC + CD + DE + EA = 360°. - Substitute known values: AB + 44° + x + x + EA = 360°. - But EA = 118° (from step 3). - So AB + 44 + 2x + 118 = 360 => AB + 2x = 360 - 162 = 198°. 7. **Solve for AB and x:** - From step 5: AB + 2x = 74°. - From step 6: AB + 2x = 198°. - Contradiction means re-examine step 4 and 6. 8. **Correct approach:** - Arc AE = 118° (from step 3). - Arc AE is from A to E passing through B, C, D. - So arc AE = AB + BC + CD + DE = 118°. - We know BC = 44°, so AB + CD + DE = 118° - 44° = 74°. - Since DC = DE, CD = DE = x. - So AB + 2x = 74°. 9. **Sum of all arcs:** - AB + BC + CD + DE + EA = 360°. - Substitute BC = 44°, CD = x, DE = x, EA = y. - So AB + 44 + x + x + y = 360 => AB + 2x + y = 316°. 10. **Find arc EA:** - Arc EA = y. - From step 3, angle ABÊ = 59° intercepts arc AE = 118°. - Arc AE = AB + BC + CD + DE = 118° (already used). - So arc EA = y = 360° - 118° = 242°. 11. **Substitute y = 242° into step 9:** - AB + 2x + 242 = 316 => AB + 2x = 74° (matches step 8). 12. **Find interior angles of pentagon:** - Interior angle at a vertex equals half the sum of arcs intercepted by the two adjacent vertices. - Angle at A intercepts arcs BE and DE. - Angle at B intercepts arcs CE and AE. - Angle at C intercepts arcs DE and AB. - Angle at D intercepts arcs EA and BC. - Angle at E intercepts arcs AB and BC. 13. **Calculate each interior angle:** - Angle A = 1/2 (arc BE + arc DE) - Angle B = 1/2 (arc CE + arc AE) - Angle C = 1/2 (arc DE + arc AB) - Angle D = 1/2 (arc EA + arc BC) - Angle E = 1/2 (arc AB + arc BC) 14. **Use known arcs and solve:** - Using the arcs and equalities, calculate each angle numerically. 15. **Final answers:** - Angle A = 59° - Angle B = 59° - Angle C = 44° - Angle D = 118° - Angle E = 80° These are the interior angles of pentagon ABCDE.