1. Warm-up Solve the equations 4c = 180 ½ (3x+42) = 27 8y = ½ (5y+55) 120 = ½ [(360-x) – x] c= 45 x=4 y=5 x=60

2. 10.4 Other Angle Relationships in Circles

3. Theorem 10.12 If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. m ∠ 1 = ½ mAB m ∠ 2 = ½ mBDA D 2 1

4. Find mGF m ∠ FGD = ½ mGF 180 ∘ - 122 ∘ = ½ mGF 58 ∘ = ½ mGF 116 ∘ = mGF D

5. Find m ∠ EFH m ∠ EFH = ½ mFH m ∠ EFH = ½ (130) = 65 ∘ D

6. Find mSR m ∠ SRQ = ½ mSR 71 ∘ = ½ mSR 142 ∘ = ½ mSR 142 ∘ = mSR

7. Theorem 10.13 If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 2 m ∠ 1 = ½ (mCD + mAB), m ∠ 2 = ½(mBD + mAC)

8. Find m ∠ AEB m ∠ AEB = ½ (mAB + mCD = ½ (139 ∘ + 113 ∘ ) = ½ (252 ∘ ) = 126 ∘

9. Find m ∠ RNM m ∠ MNQ = ½ (mMQ + mRP = ½ (91 ∘ + 225 ∘ ) = 158 ∘ m ∠ RNM= 180 ∘ - ∠ MNQ = 180 ∘ - 158 ∘ = 22 ∘

10. Find m ∠ ABD m ∠ ABD = ½ (mEC + mAD) = ½ (37 ∘ + 65 ∘ ) = ½ (102 ∘ ) = 51 ∘

11. Theorem 10.14 If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

12. Find the value of x. 40 ∘ 63 ∘

13. Find the value of x. 33 ∘

14. In the company logo shown, mFH = 108 ∘, and mLJ = 12 ∘. What is m ∠ FKH 48 ∘