Download Theorem

Geometry – Inscribed and Other Angles
Inscribed Angle – an angle whose vertex lies on the circle and its sides are
chords of the circle.
A
A
C
C
A
B
B
B
All three of these inscribed angles intercept arc AB.
C
Geometry – Inscribed and Other Angles
Inscribed Angle – an angle whose vertex lies on the circle and its sides are
chords of the circle.
A
A
C
C
A
B
B
B
C
All three of these inscribed angles intercept arc AB.
Theorem : An inscribed angle is equal to half of its intercepted arc.
1
C  AB
2
Geometry – Inscribed and Other Angles
Inscribed Angle – an angle whose vertex lies on the circle and its sides are
chords of the circle.
A
A
C
1
200°
32°
C
A
2
B
40°
B
B
3
C
Theorem : An inscribed angle is equal to half of its intercepted arc.
EXAMPLE : Find the measure of angles 1 , 2 and 3.
C 
1
AB
2
Geometry – Inscribed and Other Angles
Inscribed Angle – an angle whose vertex lies on the circle and its sides are
chords of the circle.
A
A
C
1
200°
32°
C
A
2
B
40°
B
B
3
C
Theorem : An inscribed angle is equal to half of its intercepted arc.
EXAMPLE : Find the measure of angles 1 , 2 and 3.
200
2
1  100
1 
C 
1
AB
2
Geometry – Inscribed and Other Angles
Inscribed Angle – an angle whose vertex lies on the circle and its sides are
chords of the circle.
A
A
C
1
200°
32°
C
A
2
B
40°
B
B
3
C
Theorem : An inscribed angle is equal to half of its intercepted arc.
EXAMPLE : Find the measure of angles 1 , 2 and 3.
200
1 
2
1  100
40
2 
2
2  20
C 
1
AB
2
Geometry – Inscribed and Other Angles
Inscribed Angle – an angle whose vertex lies on the circle and its sides are
chords of the circle.
A
A
C
1
200°
32°
C
A
2
B
40°
B
B
3
C
Theorem : An inscribed angle is equal to half of its intercepted arc.
C 
EXAMPLE : Find the measure of angles 1 , 2 and 3.
200
1 
2
1  100
40
2 
2
2  20
32
3 
2
3  16
1
AB
2
Geometry – Inscribed and Other Angles
Inscribed Angle – an angle whose vertex lies on the circle and its sides are
chords of the circle.
A
A
C
86°
?
?
C
A
25°
18°
?
B
B
C
Theorem : An inscribed angle is equal to half of its intercepted arc.
EXAMPLE #2 : Find the measure of arc AB in each example.
B
C 
1
AB
2
Geometry – Inscribed and Other Angles
Inscribed Angle – an angle whose vertex lies on the circle and its sides are
chords of the circle.
A
A
C
86°
?
?
C
A
25°
18°
?
B
B
C
Theorem : An inscribed angle is equal to half of its intercepted arc.
C 
EXAMPLE #2 : Find the measure of arc AB in each example.
1
AB
2
1
862   2 AB
2
172  AB
B
86 
Take notice that the arc is two time
bigger than the angle.
1
AB
2
Geometry – Inscribed and Other Angles
Inscribed Angle – an angle whose vertex lies on the circle and its sides are
chords of the circle.
A
A
C
86°
?
?
C
A
25°
18°
?
B
B
C
Theorem : An inscribed angle is equal to half of its intercepted arc.
C 
EXAMPLE #2 : Find the measure of arc AB in each example.
1
AB
2
1
862   2 AB
2
172  AB
86 
B
2  C  AB
Take notice that the arc is two
times bigger than the angle.
1
AB
2
Geometry – Inscribed and Other Angles
Inscribed Angle – an angle whose vertex lies on the circle and its sides are
chords of the circle.
A
A
C
86°
?
36°
C
A
25°
18°
50°
B
B
B
C
Theorem : An inscribed angle is equal to half of its intercepted arc.
C 
1
AB
2
EXAMPLE #2 : Find the measure of arc AB in each example.
1
AB
2
1
862   2 AB
2
172  AB
86 
2  C  AB
2  25  50
2 18  36
Take notice that the arc is two
times bigger than the angle.
Geometry – Inscribed and Other Angles
Theorem : An angle formed by a tangent line and a chord is equal to half of its
intercepted arc.
A
1
B
1
1  AB
2
Geometry – Inscribed and Other Angles
Theorem : An angle formed by a tangent line and a chord is equal to half of its
intercepted arc.
A
1
1  AB
2
1
B
EXAMPLE : If arc AB = 65°, find the measure of angle 1.
Geometry – Inscribed and Other Angles
Theorem : An angle formed by a tangent line and a chord is equal to half of its
intercepted arc.
A
1
1  AB
2
1
B
EXAMPLE : If arc AB = 65°, find the measure of angle 1.
1
1  65 
2
1  32.5
Geometry – Inscribed and Other Angles
Theorem : An angle formed by a tangent line and a chord is equal to half of its
intercepted arc.
X
A
1
1  AB
2
1
B
EXAMPLE #2 : If arc AXB = 300°, find the measure of angle 1.
AB  360  AXB
AB  360  300
AB  60
Geometry – Inscribed and Other Angles
Theorem : An angle formed by a tangent line and a chord is equal to half of its
intercepted arc.
X
A
1
1  AB
2
1
B
EXAMPLE #2 : If arc AXB = 300°, find the measure of angle 1.
AB  360  AXB
AB  360  300
AB  60
1
1  60
2
1  30
Geometry – Inscribed and Other Angles
Theorem : An angle formed by two chords is equal to half of the sum of the
intercepted arcs
C
A
X
B
D
1
CXA  AC  BD 
2
Geometry – Inscribed and Other Angles
Theorem : An angle formed by two chords is equal to half of the sum of the
intercepted arcs
C
40°
A
X
B
42°
1
CXA  AC  BD 
2
D
EXAMPLE : Arc AC = 40° and arc BD = 42°.
Find the measure of angle CXA.
1
40  42
2
1
CXA  82
2
CXA  41
CXA 
Geometry – Inscribed and Other Angles
Theorem : An angle formed by two chords is equal to half of the sum of the
intercepted arcs
C
?
A
X
B
50°
1
CXA  AC  BD 
2
D
EXAMPLE # 2 : Angle CXA = 40° and arc BD = 50°.
Find the measure of arc CA.
Geometry – Inscribed and Other Angles
Theorem : An angle formed by two chords is equal to half of the sum of the
intercepted arcs
C
y
A
X
B
50°
1
CXA  AC  BD 
2
D
EXAMPLE # 2 : Angle CXA = 40° and arc BD = 50°.
Find the measure of arc CA.
1
40  50  y 
2
1
402   50  y   2
2
80  50  y
30  y
Geometry – Inscribed and Other Angles
Theorem : An angle formed by two secants is equal to half of the difference of the
intercepted arcs. ( a secant is a line that cuts through a circle )
D
C
X
A
B
1
X  BD  AC 
2
Geometry – Inscribed and Other Angles
Theorem : An angle formed by two secants is equal to half of the difference of the
intercepted arcs. ( a secant is a line that cuts through a circle )
D
C
75°
23°
A
X
1
X  BD  AC 
2
B
EXAMPLE : Arc BD = 75° and arc CA = 23°. Find the measure of angle “x” .
Geometry – Inscribed and Other Angles
Theorem : An angle formed by two secants is equal to half of the difference of the
intercepted arcs. ( a secant is a line that cuts through a circle )
D
C
75°
23°
A
X
1
X  BD  AC 
2
B
EXAMPLE : Arc BD = 75° and arc CA = 23°. Find the measure of angle “x” .
1
X  75  23
2
1
X  52
2
X  26