Chapter 10.3 Inscribed Angles
I) Vocabulary
A) Inscribed Angles - an angle whose vertex is on a circle and whose sides contain chords of the circle.
B) Intercepted arc - the arc that lies in the interior of an inscribed angle and has endpoints on the angle.
C) If all of the vertices of a polygon lie on a circle, the polygon is inscribed in the circle and the circle is circumscribed about the polygon.
II) Theorems:
A) Measure of an inscribed angle - if an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
Example: if inscribed angle ABC of circle D is 20º, what is the measure of arc AC?
20º times 2 = 40º so the measure of arc AC is 40º.
B) If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
Example: If angle A and angle B are both inscribed angles and both intercept arc CD, then angle A = angle B
C) If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of a n inscribed triangle is a diameter of the circle, then the triangle is a right triangle is a right triangle and the angle opposite the diameter is the right angle.
D) A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
Example: Given quadrilateral ABCD is inscribed in circle E, angle A = 115º, angle B = angle D = y, and angle C = x, solve for x and y.
measure of angle A + measure of angle C = 180º
115 º+ x = 180 º
x = 65º
measure of angle B + measure of angle D = 180º
y + y = 180º
2y = 180º
y = 90º
Example: Given quadrilateral ABCD is inscribed in Circle E, angle A = 26y, angle B = 3x, angle C = 2x, and angle D = 21y, solve for both x and y.
measure of angle A + measure of angle C = 180º
26y + 2x = 180
measure of angle B + measure of angle D = 180º
3x + 21y = 180
Now solve both equations for x:
2x = -26y + 180
x = -13y + 90º
3x = -21y + 180
x = -7y + 60
Since x is equal to itself, by substitution:
-13y + 90 = -7y + 60
90 = 6y + 60
30 = 6y
y = 5
Now substitute y = 5,
x = (-7)(5) + 60 = -35 + 60 = 25
check to make sure both answers work!
Chapter 10.4 Other Angle Relationships in Circles
I) Theorems:
A) 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 = (1/2)measure of arc AB.
Example: Given circle P with angle ACB that intercepts arc AB, the measure of arc AB = 50º, what is the measure of angle ACB?
m∠ACB = (1/2) measure of arc AB
m∠ACB = (1/2) (50)
m∠ACB = 25º
B) If two chords intersect in the interior of a circle, then the measure of each angle in one half the sum of the measures of the arcs intercepted by angle and its vertical angle.
m∠1 = (1/2)(measure arc CD + measure of arc AB)
Example: If Chord AB and chord CD intersect circle P at point E, the measure of arc AD = 25 degrees and the measure of arc CB = 75º what is the m∠AED?
measure of angle AED = (1/2)(25 + 75 )
measure of angle AED = (1/2)(100)
measure of angle AED = 50º
C) If a tangent and a secant, two tangents, or two secants intersect 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.
Example: Given Secant PAB and tangent PC of circle O, the measure of arc BC = 200º and the measure of arc AC = 30º, what is the measure of angle APC?
measure of angle APC = (1/2)(measure of arc BC - measure of arc AC)
measure of angle APC = (1/2)(200 - 30)
measure of angle APC = (1/2)(170)
measure of angle APC = 85º
Example: Given secant PAB and secant PCD of circle O, the measure of arc BD = 105º and measure of arc AC = 51º, what is the measure of angle APC ?
measure of angle APC = (1/2)(105 - 51 )
measure of angle APC = (1/2)(54)
measure of angle APC = 27º
Example: Given tangent PA and tangent PC, the measure of angle APC = 40º, what are the measures of the major arc AC and the minor arc AC?
m∠APC = (1/2)(major arc AC - minor arc AC)
if we let the minor arc AC = x, then the major arc AC = 360 - x
m∠APC = (1/2)((360 -x) - x)
m∠APC = (1/2)(360 - 2x)
40 = 180 - x
x = 220º
so measure of major arc AC = 220º and the measure of minor arc AC = 140º
