Download 10.5 Other Angles

10.5 Other Angles
In a Circle
Vertex on the circle
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When the vertex of the angle is on the
circle, the measure of the angle is half
the intercepted arc.
Formed by two chords
Formed by chord and tangent
Chord/Tangent
Two Chords
What are the missing measures?
m CD on
AB = 53.73
C
m GF on
EF = 130.06
D
G
A
E
B
F
mGFH = 65.02
mCBD = 26.86
Angles inside the Circle
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If two chords intersect inside the circle, then
the angle is the average of the intercepted arcs
of the vertical angles. (half the sum)
80 °
What is the measure of
Angle 1?
½( 80 + 30)
1
30°
= 55 °
Angles Outside the Circle
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If a tangent/ tangent, secant/secant or
tangent/secant intersect outside a circle the
angle formed is half the difference of the
intercepted arcs.
a
b
Tangent/tangent
a
b
Secant/secant
a
b
Tangent/secant
Angle = ½ (a-b)
GUIDED PRACTICE
for Example 1
Find the indicated measure.
SOLUTION
1
m 1 = 2 (210o) = 105o
SOLUTION
o
(98
)
2
m TSR =
= 196o
GUIDED PRACTICE
Find the value of the variable.
SOLUTION
The chords AC and CD intersect inside
the circle.
1 (mAB + mCD)
Use Theorem 10.12.
78° =
2
78o =
1 o
(y + 95o)
2
Substitute.
156 = y +95
Simplify.
61 = y
GUIDED PRACTICE
Find the value of the variable.
SOLUTION
The tangent JF and the secant JG
intersect outside the circle.
1 (mFG – mKH)
Use Theorem 10.13.
m FJG =
2
30o =
1 o
(a – 44o)
2
Substitute.
60 = a - 44
Simplify.
a = 104
CP
GUIDED PRACTICE
Find the value of the variable.
SOLUTION
Congruent triangles (HL)
Trig using 3-4-5
m
TQR = 1 (mTUR – mTR)
2
Use Theorem 10.13.
1
[(xo) –(360 –x)o] Substitute.
2
147.4 = x – 360 + x
Solve for x.
73.7o
507.4 = 2x
xo
253.7
Review of all angles with circles
• Central angle = the intercepted arc.
• Vertex on circle = half the intercepted arc.
(chords sharing common endpoint or
a chord and a tangent intersecting on
circle.)
• Vertex inside circle = half the sum
• Vertex outside circle = half the difference
Central =
Inside –half sum
On –half arc
Outside---half difference
Geometry
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Page 683 (1-6, 9-14,16-18,22-25,3439)
Sophomore Math
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Page 683 (1-16, 22)