10.5 Other Angles
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Transcript of 10.5 Other Angles
10.5 Other Angles10.5 Other Angles
In a CircleIn a Circle
Vertex on the circleVertex on the circle
When the vertex of the angle is When the vertex of the angle is on the circle, the measure of the on the circle, the measure of the angle is half the intercepted arc.angle is half the intercepted arc.
Formed by two chordsFormed by two chords Formed by chord and tangentFormed by chord and tangent
A
B
DC
m CD on AB = 53.73
mCBD = 26.86
E
F
H
G
m GF on EF = 130.06
mGFH = 65.02
Two Chords Chord/Tangent
What are the missing measures?
Angles inside the Angles inside the CircleCircle If two chords intersect inside the circle, then If two chords intersect inside the circle, then
the angle is the average of the intercepted the angle is the average of the intercepted arcs of the vertical angles. (half the sum)arcs of the vertical angles. (half the sum)
1
80 °
30°
What is the measure of Angle 1?
½( 80 + 30) = 55 °
Angles Outside the Angles Outside the CircleCircle If a tangent/ tangent, secant/secant or If a tangent/ tangent, secant/secant or
tangent/secant intersect outside a circle tangent/secant intersect outside a circle the angle formed is half the difference of the angle formed is half the difference of the intercepted arcs.the intercepted arcs.
Tangent/tangentSecant/secant Tangent/secant
aa a
b bb
Angle = ½ (a-b)
GUIDED PRACTICE for Example 1
Find the indicated measure.
SOLUTION
= 12 (210o) = 105om 1 = 2 (98o) = 196om TSR
SOLUTION
GUIDED PRACTICE
Find the value of the variable.
SOLUTION
The chords AC and CD intersect inside the circle.
Use Theorem 10.12.
Substitute.
Simplify.
= 12
(yo + 95o) 78o
= y 61
78° (mAB + mCD)= 12
156 = y +95
GUIDED PRACTICE
Find the value of the variable.
SOLUTION
The tangent JF and the secant JG intersect outside the circle.
Use Theorem 10.13.
Substitute.
Simplify.
= 12
(ao – 44o) 30o
= 104 a
m FJG (mFG – mKH)= 12
60 = a - 44
GUIDED PRACTICE
Find the value of the variable.
SOLUTION
Use Theorem 10.13.
Substitute.73.7o 12
[(xo) –(360 –x)o]
Solve for x.
xo 253.7
= 12
m TQR (mTUR – mTR)
CP
Congruent triangles (HL)Trig using 3-4-5
147.4 = x – 360 + x
507.4 = 2x
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 =On –half arc
Inside –half sumOutside---half difference
GeometryGeometry
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Sophomore MathSophomore Math
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