**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:

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