Download 10.4 Inscribed Angles and Polygons

10.4 Inscribed Angles and Polygons
April 27, 2010
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10.4 Inscribed Angles and Polygons
April 27, 2010
Bell Ringers ‐ April 26
1. The measure of the interior angles of a quadrilateral are 80º, 100º, 55º, and 5xº. Find the value of x.
2. Two supplementary angles have measures 6xº and 12xº. Find the measure of both angles.
3. Solve 80 = 1
(360 – 2x)
2
5. Find the value of x in Q.
Explain your reasoning.
3x + 2
4. Find the measure of the red arc.
34
6 .
6
.
128º
34
.
22
10.4 Inscribed Angles and Polygons
April 27, 2010
10.4 Use Inscribed Angles and Polygons
inscribed angle : An angle whose vertex is on a circle and whose sides contain chords of the circle
intercepted arc : the arc that lies in the interior of an inscribed angle and has its endpoints on the angle
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Thm 10.7 Measure of an Inscribed Angle
If an angle is inscribed in a circle, then its measure is A
m∠ABC = .
B
C
B
Ex 1 Find each measure.
E
a) m ADC
16º
.
A
C
.
b) m∠ABE
D
B
c) m AC
70º
A
.
C
10.4 Inscribed Angles and Polygons
April 27, 2010
Thm 10.8 If two inscribed angles intercept the same arc, then,
A
.
C
B
D
Ex 2
Find the measures.
a) m∠ABE
38º
B
C
42º
b) m∠ACE
.
A
c) m∠ADE e) m∠BEC
D
f) m∠CAD
g) m∠CED
d) m∠BAC
90º
E
Guided Practice
B
.
1. Find the indicated measure in C.
a. m∠B
b. m AB
E
.C
A
2. Find the measure of the angles.
a. m∠ZYW
b. m∠WXZ
D
Y
X
.
W
Z
154º
Homework
Pages 676–679 # 1, 2, 4–7, 9–12, 16–18
100º
35º
10.4 Inscribed Angles and Polygons
April 27, 2010
10.4 Inscribed Angles and Polygons (Day 2)
Inscribed polygon – a polygon with all vertices on a circle. The circle is called circumscribed.
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Thm 10.9 If a right triangle is inscribed in a circle
then the hypotenuse is a diameter.
If m∠B = 90o (and ΔABC is a right triangle)
then AC is the diameter
Conversely, if one side of an inscribed
triangle is a diameter
then the triangle is B
A
.
C
right Δ
If AC is a diameter,
then ΔABC is a right Δ
(m B = 90o)
1
10.4 Inscribed Angles and Polygons
April 27, 2010
Thm 10.10 A quadrilateral can be inscribed in a circle iff its opposite angles are supplementary.
A
iff = if and
B
ABCD is inscribed in
a circle
only if
.
C
D
m∠A + m∠C = 180
m∠B ‐+ m∠D = 180
EX 3 Find the measure of each variable.
B
a)
m∠C = 90o
.
3xº C
3x = 90
x=30
A
b)
S
xº
yº
m∠S + m∠U = 180
T
x + 85 = 180
x=95o
.
80º
85º
R
U
m∠T + m∠R = 180
y + 80 = 180
y = 100o
O
7xº
c)
N 17yº
17y + 19y = 180
y = 5
.
19yº
P
5xº
M
5x + 7x = 180
x = 15
Homework
Pages 676 – 679 # 13–15, 19–25, 28–36