11-Inscribed Angles - Kuta Software - Free Printable

I will solve this geometry worksheet on inscribed angles step by step. The worksheet has two sections: identifying inscribed angles and finding the measure of arcs or angles.

Let me begin by analyzing the first section, which asks to state if each angle is an inscribed angle and, if so, name the angle and the intercepted arc.

Section 1: Identifying Inscribed Angles

An inscribed angle is an angle formed by two chords in a circle that have a common endpoint on the circle. The vertex of the angle must lie on the circle, and the sides of the angle must be chords (segments connecting two points on the circle).

I'll examine each diagram one by one.

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#### Problem 1:
- Diagram shows a circle with points A, B, and C.
- Angle ∠BAC has its vertex at point A, which is on the circle.
- The sides AB and AC are chords of the circle.
- Therefore, ∠BAC is an inscribed angle.
- The intercepted arc is arc BC (the arc opposite the angle, not containing point A).

✔ Answer for 1: Yes, it is an inscribed angle. Angle: ∠BAC, Intercepted arc: arc BC.

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#### Problem 2:
- Diagram shows a circle with points L, K, and M.
- Angle ∠LKM has its vertex at point K, which is on the circle.
- The sides KL and KM are chords.
- So, ∠LKM is an inscribed angle.
- The intercepted arc is arc LM.

✔ Answer for 2: Yes, it is an inscribed angle. Angle: ∠LKM, Intercepted arc: arc LM.

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#### Problem 3:
- Diagram shows a circle with points X, Y, and W.
- Angle ∠XYW has its vertex at point Y, which is on the circle.
- The sides XY and YW are chords.
- Thus, ∠XYW is an inscribed angle.
- The intercepted arc is arc XW.

✔ Answer for 3: Yes, it is an inscribed angle. Angle: ∠XYW, Intercepted arc: arc XW.

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#### Problem 4:
- Diagram shows a circle with points J, K, and M.
- Angle ∠JKM has its vertex at point K, which is on the circle.
- The sides JK and KM are chords.
- So, ∠JKM is an inscribed angle.
- The intercepted arc is arc JM.

✔ Answer for 4: Yes, it is an inscribed angle. Angle: ∠JKM, Intercepted arc: arc JM.

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Section 2: Finding the Measure of the Arc or Angle Indicated

This section requires applying the Inscribed Angle Theorem, which states that the measure of an inscribed angle is half the measure of its intercepted arc.

I’ll now solve problems 5 through 8.

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#### Problem 5:
- We are given arc BC = 80°.
- We need to find angle ∠BAC, which is an inscribed angle intercepting arc BC.
- By the Inscribed Angle Theorem: m∠BAC = ½ × m(arc BC) = ½ × 80° = 40°.

✔ Answer for 5: 40°

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#### Problem 6:
- We are given angle ∠XWV = 42°.
- This is an inscribed angle intercepting arc XV.
- So, m(arc XV) = 2 × m∠XWV = 2 × 42° = 84°.

✔ Answer for 6: 84°

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#### Problem 7:
- We are given angle ∠EFD = 35°.
- This is an inscribed angle intercepting arc ED.
- So, m(arc ED) = 2 × m∠EFD = 2 × 35° = 70°.

✔ Answer for 7: 70°

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#### Problem 8:
- We are given two angles: ∠GBC = 49° and ∠GCB = 70°.
- We need to find angle ∠BGC.
- Note: Points G, B, and C are on the circle, so triangle GBC is inscribed in the circle.
- In triangle GBC, the sum of interior angles is 180°.
- So, m∠BGC = 180° - m∠GBC - m∠GCB = 180° - 49° - 70° = 61°.

✔ Answer for 8: 61°

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Final Answers:

Section 1:
1) Yes, ∠BAC, arc BC
2) Yes, ∠LKM, arc LM
3) Yes, ∠XYW, arc XW
4) Yes, ∠JKM, arc JM

Section 2:
5) 40°
6) 84°
7) 70°
8) 61°

This completes the solution to the worksheet.