Warm-upSolve the equations4c = 180

½ (3x+42) = 27

8y = ½ (5y+55)

120 = ½ [(360-x) – x]

c= 45x=4y=5x=60

10.4 Other Angle Relationships in Circles

Theorem 10.12If 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

Find mGF

m∠FGD = ½ mGF 180∘ - 122∘ = ½ mGF

58∘

= ½ mGF

116∘

= mGF

D

Find m∠EFH

m∠EFH = ½ mFH m∠EFH = ½ (130)

= 65∘

D

Find mSR

m∠SRQ = ½ mSR 71∘ = ½ mSR

142∘

= ½ mSR

142∘

= mSR

Theorem 10.13If 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)

Find m∠AEB

m∠AEB = ½ (mAB + mCD

= ½ (139∘ + 113∘)= ½ (252∘)= 126∘

Find m∠RNM

m∠MNQ = ½ (mMQ + mRP= ½ (91∘ + 225∘)= 158∘

m∠RNM = 180∘ - ∠MNQ= 180∘ - 158∘

= 22∘

Find m∠ABD

m∠ABD = ½ (mEC + mAD)

= ½ (37∘ + 65∘)= ½ (102∘)= 51∘

Theorem 10.14If 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.

Find the value of x.

40∘ 63∘

Find the value of x.

33∘

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

48∘