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Warm-Up Exercises
1. The measure of the interior angles of a quadrilateral
are 80º, 100º, 55º, and 5xº. Find the value of x.
ANSWER
25
2. Two supplementary angles have measures 6xº and
12xº. Find each angle measure. ANSWER 60º; 120º
1
3. Solve 3x = ( 4x + 12).
2
ANSWER
6
1
4. Solve 80 = ( 360 – 2x).
2
ANSWER
100
Homework Check 10.3 - With your textbook closed
copy your solutions from your homework for
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Warm-Up Exercises
Target
Apply angle relationships in circles.
You will…
Use inscribed angles of circles.
Warm-Up Exercises
Vocabulary
• inscribed angle – an angle whose vertex is on the circle
and whose sides contain chords of the circle
• intercepted arc – the arc that lies in the interior of an
inscribed angle and has endpoints on the angle
• Theorem 10.7 –
• Theorem 10.8 – inscribed
angles that intercept the
same arc are congruent.
Warm-Up Exercises
Vocabulary
• inscribed polygon – a polygon whose vertices all lie on
a circle; the circle is called a circumscribed circle
• Theorem 10.9 – an inscribed
angle is a right angle if and only
if it intercepts a semicircle
• Theorem 10.10 – a quadrilateral
can be inscribed in a circle if and
only if its opposite angles are
supplementary.
Warm-Up1Exercises
EXAMPLE
Use inscribed angles
Find the indicated measure in
a.
m
T
b.
P.
mQR
SOLUTION
a. m
1
T = 1 mRS =
(48o) = 24o
2
2
b. mTQ = 2m R = 2 50o = 100o. Because TQR is a semicircle,
mQR = 180o– mTQ = 180o – 100o = 80o. So, mQR = 80o.
Warm-Up2Exercises
EXAMPLE
Find the measure of an intercepted arc
Find mRS and m
and
RUS?
STR. What do you notice about
STR
SOLUTION
From Theorem 10.7, you know that mRS = 2m
= 2 (31o) = 62o.
Also, m
STR = 1 mRS = 1 (62o) = 31o. So,
2
2
RUS
STR
RUS.
Warm-Up3Exercises
EXAMPLE
Standardized Test Practice
SOLUTION
Notice that JKM and
JLM intercept the same arc, and
so JKM
JLM by Theorem 10.8. Also,
KJL and
KML intercept the same arc, so they must also be
congruent. Only choice C contains both pairs of angles.
So, by Theorem 10.8, the correct answer is C.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1, 2 and 3
Find the measure of the red arc or angle.
1.
2.
SOLUTIONS
m G = 1 mHF
2
1
=
(90o)
2
= 45o
3.
mTV = 2m U
= 2 38o
= 76o
m
ZYN
ZXN
ZXN = 72°
Warm-Up4Exercises
EXAMPLE
Use a circumscribed circle
Photography
Your 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?
Warm-Up4Exercises
EXAMPLE
Use a circumscribed circle
SOLUTION
From Theorem 10.9, 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.
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 4
4. What If ? In Example 4, explain how to find locations
if you want to frame the front and left side of the
statue in your picture.
SOLUTION
Make the diameter of your circle the diagonal of the
rectangular base.
Warm-Up5Exercises
EXAMPLE
Use Theorem 10.10
Find the value of each variable.
SOLUTION
PQRS is inscribed in a circle, so opposite angles
are supplementary.
m
P+m
R = 180o
75o + yo = 180o
y = 105
m
Q+m
S = 180o
80o + xo = 180o
x = 100
Warm-Up5Exercises
EXAMPLE
Use Theorem 10.10
Find the value of each variable.
SOLUTION
JKLM is inscribed in a circle, so opposite angles
are supplementary.
m
J+m
L = 180o
m
K+m
M = 180o
2ao + 2ao = 180o
4bo + 2bo = 180o
4a = 180
6b = 180
a = 45
b = 30
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 5
Find the value of each variable.
5.
6.
SOLUTIONS
x = 98
c = 62
y = 112
x = 10