Question

Find all of the missing angle measures. Remember you cannot assume right angles or diameters. Also think about how many degrees are in a triangle. Angle 1: Angle 2: Angle 3: Angle 4: Angle 5: Angle 6: Angle 7: Angle 8: Angle 9: Angle 10: Angle 11: Angle 12: Angle 13: Angle 14: Angle 15:

Answer 1

See text below or attached figure

Explanation:

Given arcs

AC=70

CR=18

therefore AR = 88

RB=80

BE=130

therefor EA = 360-(70+18+80+130) = 360-298 = 62

angles will be denoted (1) for angle 1, etc.

We ASSUME

ARD is a straight line

PFRB is a straight line

FCE is a straight line

Using inscribed angle theorem, angles subtended by chords/arcs equal to half the arc central angle.

Therefore

(4)=80/2=40

(13)=130/2=65

(12)=62/2=31

(11)=70/2=35

(5) = (70+18)/2 = 44

Consider triangle AEG,

(7)=(13)+(11)=65+35=100 [exterior angle]

Consider triangle EGB,

(10)=180-100-31 = 49 [sum of angles of a triangle]

Consider triangle AEH,

(3) = 180-(4)-(13)-(11) = 180-40-65-35 = 40 [sum of angles of a triangle]

From cyclic quadrilateral ARBE,

ARB+AEB=180 =>

ARB=180-AEB=180-(35+49) = 96

By the intersecting secants theorem,

(2) = (130-18)/2 = 56 [secants FE, FB]

(1) = (130+62 - (18+70))/2 = 104/2 = 52 [secants PA,PB]

(8) = (130+62 -80)/2 = 112/2 = 56

ARD is straight line (see assumptions above)

(9) = 180-96 = 84 [sum of angles on a line]

ARP = (9) = 84 [vertically opposite angles]

Consider triangle ARP

(14) = 180-52-84 = 44

Consider tangent PA

(15) = 180-(44+40+65) = 31 [sum of angles of a triangle]

Consider triangle ABD

(6) = 180 - (40+44+56) = 40 [sum of angles of a triangle]

This completes the search for all sixteen angles, as shown in the diagram, or in the text above.