Post on 22-Jan-2016
description
transcript
1
Lesson 6.3 Inscribed Angles and their Intercepted Arcs
Objectives:
Using Inscribed Angles
Using Properties of Inscribed Angles.
Homework: Lesson 6.3/ 1-12Friday-Chapter 6 Quiz 2 on 6.1-6.3
2
Using Inscribed Angles
An INSCRIBED ANGLE is an angle whose vertex is on the circle and whose sides are chords of a circle.
Inscribed Angles & Intercepted Arcs
D
B A
C
∠ABC is an inscribed angle
3
Using Inscribed Angles
Measure of an Inscribed Angle
2
1
50°
100°
B
AC
50°
100°
B
AC
x°
2x°
B
AC
4
Using Inscribed Angles
Example 1:
63
Find the m and mPAQ .PQ
PQ = 2 * m PBQ = 2 * 63 = 126˚
PQ
5
Using Inscribed Angles
Find the measure of each arc or angle.
QSR
Example 2:
Q
R
= ½ 120 = 60˚
= 180˚
= ½(180 – 120)= ½ 60= 30˚
6
Using Inscribed Angles
Inscribed Angles Intercepting Arcs Conjecture
If two inscribed angles intercept the same arc or arcs of equal measure then the inscribed angles have equal measure.
mCAB = mCDB
P
A
BC
D
7
Using Inscribed Angles
Example 3:
70E D
A
FEDFmFind
14070*2 EFm
EDFm =360 – 140 = 220˚
m = 82˚
8
Using Properties of Inscribed Angles
Example 4:
41°
60°
P
C
DA
B
Find mCAB and m AD
mCAB = ½
mCAB = 30˚ADm = 2* 41˚ AD
CB
9
Using Properties of Inscribed Angles
Cyclic QuadrilateralA polygon whose vertices lie on the circle, i.e. a quadrilateral inscribed in a circle.
Quadrilateral ABFE is inscribed in Circle O.
O
A
B
F
E
10
Using Properties of Inscribed Angles
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
Cyclic Quadrilateral Conjecture
m A + m C = 180°∠ ∠
m B + m D = 180°∠ ∠
11
Using Properties of Inscribed Angles
Find the measure ofGDE
Opposite angles of an inscribed quadrilateral are supplementary
Intercepted arc of an inscribed angles = 2* angle measure
Example 5:
12
Find m A and ∠ m B∠ Opposite angles of an inscribed quadrilateral are supplementary
m A + 60° = 180°∠
m A = 120°∠
m B + 140° = 180°∠
m B = 40°∠
Example 6:
13
3x°
(y + 5)°
(2y - 3)°
Using Properties of Inscribed Angles
Find x and y Opposite angles of an inscribed quadrilateral are supplementary
Example 7:
14
Using Properties of Inscribed Angles
A polygon is circumscribed about a circle if and only if each side of the polygon is tangent to the circle.
Circumscribed Polygon
15
Using Properties of Inscribed Angles
A triangle inscribed in a circle is a right triangle if and only if the diameter is the
hypotenuse
Angles inscribed in a Semi-circle Conjecture
A has its vertex on the circle, and it intercepts half of the circle so thatmA = 90.
16
Find x.
3x°E
D
A
B
C
F
Angles inscribed in a semi-circle are right angles
Example 8:
17
Using Inscribed Angles
E DA
B
FFind mFDE
Example 9:146°
18
Using Properties of Inscribed Angles
Parallel (Secant) Lines Intercepted Arcs Conjecture
Parallel (secant) lines intercept congruent arcs.
A
B
X
Y
19
Using Properties of Inscribed Angles
Find x.
x122˚
189˚
360 – 189 – 122 = 49˚
x = 49/2 = 24.5˚
x
Example 10:
Tangent/Chord Conjecture
BDC2
1C mm
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.
B
C
D
BD
C
21
Example 11:
Using Tangent/Chord Conjecture
35o
xo
yoQ
L
K
J
Find x and y.
90 QJL m
90o
55x
55o
125180 x
35y
Triangle sum
Example 12:
23
Homework:
Lesson 6.3/ 1-12
24