1. 2x = 84 + 32 Solve for x. ANSWER 58 2. x = ( 360 – 120) 1 2 ANSWER 120 3. 180 – x = (( 2x + 4) + 28). 1 2 ANSWER 82 4. One-half the measure of an angle and its supplement is equal to the measure of the angle. Find the measure of the angle. ANSWER 120º

Apply angle relationships in circles. Target Apply angle relationships in circles. You will… Find the measures of angles inside, on, or outside a circle.

Vertex Inside the Circle Vocabulary Angles and their Arcs angle = sum of arcs 12 Vertex Inside the Circle Central Angle – Theorem 10.12 – 50º Extending the radii allow you to think of angles whose vertex is inside the circle the same way.

Vocabulary Angles and their Arcs angle = arc 12 Vertex On the Circle Theorem 10.7 – Inscribed Angle Theorem 10.11 –

Vocabulary Angles and their Arcs angle = difference of arcs 12 Vertex Outside the Circle Theorem 10.13 –

EXAMPLE 1 Find angle and arc measures Line m is tangent to the circle. Find the measure of the red angle or arc. SOLUTIONS = 12 (130o) b. m KJL = 2 (125o) a. m 1 = 250o = 65o

GUIDED PRACTICE for Example 1 Find the indicated measure. SOLUTIONS = 12 (210o) m XY = 2 (80o) m 1 m RST = 2 (98o) = 196o = 160o = 105o

Find an angle measure inside a circle EXAMPLE 2 Find an angle measure inside a circle Find the value of x. SOLUTION The chords JL and KM intersect inside the circle. xo = 12 (mJM + mLK) Use Theorem 10.12. xo = 12 (130o + 156o) Substitute. xo = 143 Simplify.

Find an angle measure outside a circle EXAMPLE 3 Find an angle measure outside a circle Find the value of x. SOLUTION The tangent CD and the secant CB intersect outside the circle. m BCD (mAD – mBD) = 12 Use Theorem 10.13. = 12 (178o – 76o) xo Substitute. = 51 x Simplify.

EXAMPLE 4 Solve a real-world problem SCIENCE The Northern Lights are bright flashes of colored light between 50 and 200 miles above Earth. Suppose a flash occurs 150 miles above Earth. What is the measure of arc BD, the portion of Earth from which the flash is visible? (Earth’s radius is approximately 4000 miles.)

Solve a real-world problem EXAMPLE 4 Solve a real-world problem SOLUTION Because CB and CD are tangents, CB AB and CD AD Also,BC DC and CA CA . So, ABC ADC by the Hypotenuse-Leg Congruence Theorem, and BCA DCA.Solve right CBA to find that m BCA 74.5°. = 12 m BCD (mDEB – mBD) Use Theorem 10.13. 149o 12 [(360o – xo) –xo] Substitute. xo 31 Solve for x. ANSWER The measure of the arc from which the flash is visible is about 31o.

GUIDED PRACTICE for Examples 2, 3, and 4 Find the value of the variable. 5. 6. y = 61o ANSWERS = 104o a xo 253.7o