mm2g3b understand and use properties of central, inscribed, and related angles
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MM2G3bMM2G3bUnderstand and use Understand and use
properties of central, properties of central, inscribed, and inscribed, and related anglesrelated angles
Theorem 6.9Theorem 6.9
The measure of an The measure of an inscribed angle is one inscribed angle is one half the measure of its half the measure of its
intercepted arcintercepted arc
EXAMPLE 1
a. m T mQRb.
Find the indicated measure in P.
SOLUTION
12
12
M T = mRS = (48o) = 24oa.
mQR = 180o mTQ = 180o 100o = 80o. So, mQR = 80o. – –
mTQ = 2m R = 2 50o = 100o. Because TQR is a semicircle,b.
Theorem 6.10
If two inscribed angles of If two inscribed angles of a circle intercept the a circle intercept the same arc, then the same arc, then the
angles are congruentangles are congruent
EXAMPLE 2
Find mRS and m STR. What do you notice about STR and RUS?
SOLUTION
From Theorem 6.9, you know that mRS = 2m RUS = 2 (31o) = 62o.
Also, m STR = mRS = (62o) = 31o. So, STR RUS.12
12
EXAMPLE 3
SOLUTION
Notice that JKM and JLM intercept the same arc, and so JKM JLM by Theorem 6.10. Also, KJLand KML intercept the same arc, so they must also be congruent. Only choice C contains both pairs of angles.
GUIDED PRACTICE
Find the measure of the red arc or angle.
1.
SOLUTION
m G = mHF = (90o) = 45o12
12
a.
GUIDED PRACTICE
Find the measure of the red arc or angle.
2.
SOLUTION
mTV = 2m U = 2 38o = 76o. b.
GUIDED PRACTICE
Find the measure of the red arc or angle.
3.
SOLUTION
ZYN ZXN
ZXN 72°
Notice that ZYN and ZXN intercept the same arc, and so ZYN by Theorem 6.10. Also, KJL and KML intercept the same arc, so they must also be congruent.
ZXN
Theorem 6.11Theorem 6.11
If a right triangle is If a right triangle is inscribed in a circle (all inscribed in a circle (all
vertices lie on the circle) vertices lie on the circle) then the hypotenuse is then the hypotenuse is
the diameter of the the diameter of the circle.circle.
EXAMPLE 4
PhotographyYour camera has a 90o field of vision and you want to photograph the front of a statue. You move to a spot where the statue is the only thing captured in your picture, as shown. You want to change your position. Where else can you stand so that the statue is perfectly framed in this way?
EXAMPLE 4
SOLUTION
From Theorem 6.11, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a diameter of the circle. So, draw the circle that has the front of the statue as a diameter. The statue fits perfectly within your camera’s 90o field of vision from any point on the semicircle in front of the statue.
Theorem 6.12Theorem 6.12
A quadrilateral can be A quadrilateral can be inscribed in a circle if inscribed in a circle if
and only if its opposite and only if its opposite angles are angles are
supplementary.supplementary.
EXAMPLE 5
Find the value of each variable.
a.
SOLUTION
PQRS is inscribed in a circle, so opposite angles are supplementary.
a.
m P + m R = 180o
75o + yo = 180o
y = 105
m Q + m S = 180o
80o + xo = 180o
x = 100
EXAMPLE 5
b. JKLM is inscribed in a circle, so opposite angles are supplementary.
m J + m L = 180o
2ao + 2ao = 180o
a = 45
m K + m M = 180o
4bo + 2bo = 180o
b = 30
4a = 180 6b = 180
Find the value of each variable.
b.
SOLUTION
GUIDED PRACTICE
4.
Find the value of each variable.
SOLUTION
y = 112 x = 98
GUIDED PRACTICE
5.
Find the value of each variable.
SOLUTION
c = 62 x = 10
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