3.) Same Arc

If two inscribed angles intercept the same arc:

Both angles $\angle ACB \space and \angle ADB$ point to arc AB.

Then:

∠ACB ≅ ∠ADB

Why?

Both equal half of arc AB.

5.) Central Angle vs Inscribed Angle

Problem 1 (Medium)

Draw

1. Draw a circle.
2. Mark center W.
3. Put A at the top.
4. Put B on the right.
5. Draw radii WA and WB.
6. Label ∠AWB = 120°.

Question:

Find:

a) m(arc AB)

b) m(∠ACB) where C is any point on the circle not on arc AB.

Problem 2 (Medium)

Draw

1. Circle W
2. Chord AB
3. Chord CD
4. Mark AB ≅ CD

Question:

What must be true?

A) Arc AB ≅ Arc CD

B) Arc AB = 2(Arc CD)

C) Central angles are supplementary

D) Tangents are perpendicular

Problem 3 (Medium-Hard)

Draw

1. Circle W
2. Arc AB = 110°
3. Point C on circle
4. Draw chords CA and CB

Question:

Find m∠ACB.

Problem 4 (Medium-Hard)

Draw

1. Circle W
2. Diameter AB
3. Point C on circle
4. Connect AC and BC

Question

Find ∠ACB.

Problem 5 (Medium-Hard)

Draw

1. Circle W
2. Arc AB = Arc CD
3. Central angles AWB and CWD

Question

If ∠AWB = 70°, find ∠CWD.

Problem 6 (Hard)

Draw

1. Circle W
2. Chord AB
3. Radius WM perpendicular to chord AB
4. M lies on AB

Given:

AB = 18

Question:

Find AM.

Problem 7 (Hard)

Draw

1. Circle W
2. Tangent line at A
3. Radius WA

Question

Find m∠ formed by tangent and radius.

Problem 8 (Hard)

Draw

1. Circle W
2. Arc AB = 80°
3. Arc BC = 60°

Question

Find central angle AWC.

Problem 9 (Hard)

Draw

1. Circle W
2. Arc AB = 100°
3. Points C and D on circle
4. Draw ∠ACB and ∠ADB intercepting arc AB

Question

Compare ∠ACB and ∠ADB.

Problem 10 (Hard-Plus)

Draw

1. Circle W
2. Chord AB = Chord CD
3. Arc AB = x + 10
4. Arc CD = 2x - 5

Question

Find x.