5-Minute Check

10.5 Other Angle Relationships in Circles February 26, 2013

10.5 Objective(s) Students will determine measures of angles inside or outside a circle. Why? So you can determine the part of the Earth seen from a hot air balloon, as seen in Ex. 25. Mastery is 80% or better on 5-min checks and Indy work.

Concept Development 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 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

Skill Dev- Ex. 1: Finding Angle 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°

Skill Dev-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.

Theorem 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 + mm  2 = ½ m + m

Theorem 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  1 = ½ m( - m )

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

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

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

CFU- 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 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.

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 Ex. 5: Describing the View from Mount Rainier

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

Exit Slips What was the Objective for today? Students will determine measures of angles inside or outside a circle. Why? So you can determine the part of the Earth seen from a hot air balloon, as seen in Ex. 25.

Homework # 3-21 page 683