10.4 Other Angle Relationships in Circles

10.4 Other Angle Relationships in Circles Geometry Mrs. Spitz Spring 2005

Objectives/Assignment • Use angles formed by tangents and chords to solve problems in geometry. • Use angles formed by lines that intersect a circle to solve problems. • Assignment: pp. 624-625 #2-35

Using Tangents and Chords • You know that measure of an angle inscribed in a circle is half the measure of its intercepted arc. This is true even if one side of the angle is tangent to the circle. mADB = ½m

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= ½m m2= ½m

Ex. 1: Finding Angle and Arc Measures • Line m is tangent to the circle. Find the measure of the red angle or arc. • Solution: m1= ½ m1= ½ (150°) m1= 75° 150°

Ex. 1: Finding Angle and Arc Measures • Line m is tangent to the circle. Find the measure of the red angle or arc. • Solution: m = 2(130°) m = 260° 130°

Ex. 2: Finding an Angle Measure • In the diagram below, is tangent to the circle. Find mCBD • Solution: mCBD = ½ m 5x = ½(9x + 20) 10x = 9x +20 x = 20  mCBD = 5(20°) = 100° (9x + 20)° 5x° D

Lines Intersecting Inside or Outside a Circle • If two lines intersect a circle, there are three (3) places where the lines can intersect. on the circle

Inside the circle

Outside the circle

Lines Intersecting • You know how to find angle and arc measures when lines intersect ON THE CIRCLE. • You can use the following theorems to find the measures when the lines intersect INSIDE or OUTSIDE the circle.

m1 = ½ m + m m2 = ½ m + m 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.

m1 = ½ m( - m ) Theorem 10.14 • If a tangent and a secant, two tangents or two secants intercept 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.

Theorem 10.14 • If a tangent and a secant, two tangents or two secants intercept 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. m2 = ½ m( - m )

Theorem 10.14 • If a tangent and a secant, two tangents or two secants intercept 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. 3 m3 = ½ m( - m )

Ex. 3: Finding the Measure of an Angle Formed by Two Chords 106° • Find the value of x • Solution: x° = ½ (m +m x° = ½ (106° + 174°) x = 140 x° 174° Apply Theorem 10.13 Substitute values Simplify

mGHF = ½ m( - m ) Ex. 4: Using Theorem 10.14 200° • Find the value of x Solution: 72° = ½ (200° - x°) 144 = 200 - x° - 56 = -x 56 = x x° 72° Apply Theorem 10.14 Substitute values. Multiply each side by 2. Subtract 200 from both sides. Divide by -1 to eliminate negatives.

mGHF = ½ m( - m ) Ex. 4: Using Theorem 10.14 Because and make a whole circle, m =360°-92°=268° x° 92° • Find the value of x Solution: = ½ (268 - 92) = ½ (176) = 88 Apply Theorem 10.14 Substitute values. Subtract Multiply

Ex. 5: Describing the View from Mount Rainier • You are on top of Mount Rainier on a clear day. You are about 2.73 miles above sea level. Find the measure of the arc that represents the part of Earth you can see.

Ex. 5: Describing the View from Mount Rainier • and are tangent to the Earth. You can solve right ∆BCA to see that mCBA  87.9°. So, mCBD  175.8°. Let m = x° using Trig Ratios

175.8  ½[(360 – x) – x] 175.8  ½(360 – 2x) 175.8  180 – x x  4.2 Apply Theorem 10.14. Simplify. Distributive Property. Solve for x. From the peak, you can see an arc about 4°.

Reminder: • Quiz after section 10.5 • Deficiencies go out the week of April 25-28; so all work is due no later than the end of this week.

10.4 Other Angle Relationships in Circles