Applying the circle theorems, we have:
4. y = 120°
x = 45°
5. x = 90°
y = 32°
6. x = 40°
y = 20°
7. x = 116°
y = 98°
8. x = 100°
y = 60°
9. x = 54°
y = 75°
What is the Angle of Intersecting Secants Theorem?
The angle formed outside a circle by the intersection of two secants equals half the positive difference of the measures of the intercepted arcs, based on the angle of intersecting secants theorem.
What is the Angle of Intersecting Chords Theorem?
The angle formed when two chords of a circle intersect, equals the half the sum of the measures of the intercepted arcs, based on the angle of intersecting chords theorem.
4. y = 360 - 95 - 30 - 115 = 120°
x = 1/2(120 - 30) [angle of intersecting secants theorem]
x = 45°
5. x = 90° [based on the tangent theorem]
y = 180 - 90 - 58 [triangle sum theorem]
y = 32°
6. x = 360 - 160 - 160 [central angle theorem]
x = 40°
y = 1/2(40) [inscribed angle theorem]
y = 20°
7. x = 180 - 2(32)
x = 116°
y = 1/2[132 + 2(32)] [angle of intersecting chords theorem]
y = 98°
8. x = 180 - 80 [opposite angles of an inscribed quadrilateral are supplementary]
x = 100°
y = 1/2[(2(80) - 90) + 50] [inscribed angle theorem]
y = 1/2[70 + 50]
y = 60°
9. 40 = 1/2(134 - x) [angle of intersecting secants theorem]
2(40) = 134 - x
80 = 134 - x
80 - 134 = - x
-54 = -x
x = 54°
y = 360 - 134 - 54 - 97
y = 75°
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