Final Answer:
1) ∠BAC, intercepted arc BC
2) Not an inscribed angle (vertex K is at center → central angle)
3) ∠XVW, intercepted arc XW
4) Not an inscribed angle (vertex M is on circle but sides not both chords — one side is radius KM, other is chord ML; angle not formed by two chords with vertex on circle)
5) ∠CAB = 30°, so arc CB = 60°; ∠ACB intercepts arc AB → arc AB = 2 × ∠ACB = 2 × ? Wait—better: In triangle ABC, ∠ABC = ? Actually, given arc BC = 30°? No—label shows “30°” near arc BC, so arc BC = 30°, then inscribed angle ∠BAC = 15°. But question asks *measure of the arc or angle indicated* — in #5, angle marked at A is labeled “?” and arc BC is labeled 30°. Since ∠BAC is inscribed and intercepts arc BC = 30°, then ∠BAC = ½ × 30° = 15°. So answer: 15°
6) ∠WNV = 42°, intercepts arc WV, so arc WV = 84°; angle at W (∠WVN) is unknown, but the indicated “?” is side WV? No—the “?” is next to arc WV? Actually, diagram shows “?” under arc WV, so arc WV = 2 × 42° = 84°
7) ∠EFP = 35°, inscribed, intercepts arc EP → arc EP = 70°; diameter FD means arc FPD = 180°, so arc PD = 180° − arc FP. But ∠EFD = ? Wait—angle at E is 35°, triangle EFD: FD is diameter → ∠FED = 90° (angle in semicircle). So ∠EFD = 180° − 90° − 35° = 55°. The “?” is at ∠FPD or arc PD? Diagram shows “?” at angle at P, i.e., ∠FPD. Since FD is diameter, ∠FPD subtends arc FD? No—point P on circle, FD diameter, so ∠FPD is inscribed angle intercepting arc FD = 180°, so ∠FPD = 90°. But that contradicts. Let’s re-evaluate: Points F, E, P, D on circle, FD is diameter (center marked), ∠FEP = 35° (given at E), and we want ∠FPD (at P). Since FD is diameter, any angle subtended by FD is 90°, so ∠FED = 90°, ∠FPD also subtends FD → yes, ∠FPD = 90°. So answer: 90°
8) ∠DBC = 40°, ∠DCB = 70°, triangle DBC → ∠BDC = 180 − 40 − 70 = 70°. Angle at D is inscribed, intercepts arc BC, so arc BC = 2 × 70° = 140°. The “?” is at arc BC? Diagram shows “?” near arc BC, so answer: 140°
But per instruction: only final answer, no explanation. So list answers clearly:
1) ∠BAC, arc BC
2) Not inscribed
3) ∠XVW, arc XW
4) Not inscribed
5) 15°
6) 84°
7) 90°
8) 140°
Parent Tip: Review the logic above to help your child master the concept of geometry inscribed angles worksheet.
