Post on 22-Dec-2015
Answers to Warm Up Section 4.5 Find x:
1. 2. 3.
4. 5. 6.
xo
xo
70o
32o
xo
xo100ox 12
xo
45o
90o 140o 32o
40o 45o 12 2
Angles Formed byChords, Secants, and Tangents
Section 4.5
Standard: MM2G3 bd
Essential Question: How are properties of chords, tangents, and secants used to find angle measures?
Recall how an arc is related to its inscribed angle:
o40
ox
angle = ½ arc
Ex . Ex .
ox
ox2
ox
o64
arc = angle × 2angle = arc ÷ 2
x = 80 x = 32
The measure of the angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.
1X
Y
ZW
angle = ½ (arc 1 + arc 2)
11
2m mWY mXZ
Example 1:
If = 45o and = 75o, find m 1
S
1
R
P
Q
mRSmPQ
m1 = ½ (mRS + mPQ)m1 = ½(75o + 45o)m1 = ½(120o)m1 = 60o
75o45o
If and 80o , find
m1 = ½ (mRS + mPQ) 55o = ½(80o + x) 110o = (80o + x) 30o = x
Example 2:
S
R
P
Q
1 55m mRS .mPQ
80o55ox
x°
80°
20°100°
x°
Try these with your partner:3. 4.
xo = ½(80o + 20o)xo = ½(100o)xo = 50o
90o = ½(100o + xo)180o = (100o + xo )80o = xo
Case 2: Vertex outside the circle.The measure of an angle formed by 2 secants, 2 tangents, or a secant and a tangent is half the difference of the measures of the intercepted arcs.
angle = ½ (arc 1 – arc 2)
A
CB
D
E
Example 6: If mDC = 100o and mEB = 40o, find mA.
100o 40o
mA = ½ (mDC – mEB)xo = ½(100o – 40o)xo = ½ (60o)xo = 30o
xo
W
VX
Y
Z
Example 7: If mW = 65o and mXZ = 70o, find mXVY
mW = ½ (mXVY – mXZ) 65o = ½(xo – 70o) 130o = xo – 70o
200o = xo
70o 65o
xo
P
Q
RS
Example 8: If mQRS = 240o, find mQS and mP.
mP = ½ (mQRS – mQS)xo = ½(240o – 120o)xo = ½ (120o)xo = 60o
240o
mQS = 360o – 240o
= 120o
120o xo
x°
230°
80°
140°x°
120°
Try these with your partner:9. 10.
xo = ½(80o – 20o)xo = ½ (60o)xo = 30o
xo = ½(230o – 130o)xo = ½ (100o)xo = 50o
130o
20o
x°
150°
35°
30°
x°
100°
11. 12.
xo = ½(50o – 30o)xo = ½ (20o)xo = 10o
35o = ½(150o – xo) 70o = (150o – xo) -80o = – xo
80o = xo
50o
1
x°
The vertex of the angle is located at the center of the circle. So, the angle is a central angle and is equal to the measure of the intercepted arc.
m1 = xo
Summary: Measures of Angles Formed by Radii, Chords, Tangents and Secants
angle = arc
2
x°
2
x°
The vertex of the angle is a point on the circle.So, the measure of the angle is one half the measure of the intercepted arc.
angle = ½arc m2 = ½ xo
3 x°y°
The vertex of the angle is located in the interior of the circle and not at the center, so the measure of the angle is half the sum of the intercepted arcs.
angle = ½(arc1 + arc2) m3 = ½(xo + yo)
4
x°
y°
4
x°
y°
4
x°
y°
The vertex of the angle is located in the exterior of the circle and not at the center, so the measure of the angle is half the difference of the intercepted arcs.
angle = ½(arc1 – arc2) m4 = ½(xo – yo)
3 O
B
4 1
2
9 6
78
5
10 A
C
D
E
BE is a diameter of the circle with center O. AT is tangent to the circle at A. mAB = 80o, mBC = 20o, and mDE = 50o.
80o
20o
50o
100o
110o
3 O
B
4 1
2
9 6
78
5
10 A
C
D
E
80o
20o
50o
100o
110o
13. m1 = ½(80o) = 40o
14. m2 = ½(100o) = 50o
15. m3 = ½(80o+ 50o) = 65o
16. m4 = ½(100o – 50o) = 25o
17. m5 = ½(80o) = 40o 18. m6 = ½(180o) = 90o