Angle Measures and Segment Lengths in Circles Objectives: 1) To find the measures of s formed by chords, secants, & tangents. 2) To find the lengths of segments associated with circles.

Secants Secant – A line that intersects a circle in exactly 2 points. F B A E Secant – A line that intersects a circle in exactly 2 points. EF or AB are secants AB is a chord

Theorem. The measure of an  formed by 2 lines that intersect inside a circle is half of the arcs measures, intercepted by the lines. Arcs are across from angles. m1 = ½(x + y) x° 1 y° Measure of intercepted arcs

Theorem. The measure of an  formed by 2 lines that intersect outside a circle is m1 = ½(x - y) Smaller Arc 3 cases: Larger Arc 1 1 Tangent & a Secant 2 Secants: y° y° 1 y° x° 2 Tangents x° x°

Ex.1 & 2: Find the mx. Find the measure of arc x. mx = ½(x - y) 92° 104° 68° 94° 268° 112° mx = ½(x - y) mx = ½(268 - 92) mx = ½(176) mx = 88° m1 = ½(x + y) 94 = ½(112 + x) 188 = (112 + x) 76° = x

Lengths of Secants, Tangents, & Chords Tangent & Secant y a c t z x b z d w y a•b = c•d t2 = y(y + z) w(w + x) = y(y + z)

Ex. 3 & 4 Find the length of g. Find length of x. t2 = y(y + z) 8 15 g 3 x 7 5 t2 = y(y + z) 152 = 8(8 + g) 225 = 64 + 8g 161 = 8g 20.125 = g a•b = c•d (3)•(7) = (x)•(5) 21 = 5x 4.2 = x

Ex.5: 2 Secants Find the length of x. w(w + x) = y(y + z) 20 14 w(w + x) = y(y + z) 14(14 + 20) = 16(16 + x) (34)(14) = 256 + 16x 476 = 256 + 16x 220 = 16x 3.75 = x 16 x

Ex.6: A little bit of everything! Find the measures of the missing variables Solve for k first. w(w + x) = y(y + z) 9(9 + 12) = 8(8 + k) 186 = 64 + 8k k = 15.6 12 k 175° 9 8 60° Next solve for r t2 = y(y + z) r2 = 8(8 + 15.6) r2 = 189 r = 13.7 a° r Lastly solve for ma m1 = ½(x - y) ma = ½(175 – 60) ma = 57.5°

What have we learned?? When dealing with angle measures formed by intersecting secants or tangents you either add or subtract the intercepted arcs depending on where the lines intersect. There are 3 formulas to solve for segments lengths inside of circles, it depends on which segments you are dealing with: Secants, Chords, or Tangents.