Transcript 10 - 4 Inscribed Angles - Ms. Fowls' Math Classes

Inscribed Angle: An angle that has its vertex on the
circle and its sides contained in chords of the circle
B
Vertex B is on the circle
Arc ADC is the arc
intercepted by angle ABC.
AB and BC are
chords of the circle
A
C
Theorem 10.5 Inscribed Angle Theorem
If an angle is inscribed in a circle, then the
measure of the angle equals one-half the
measure of its intercepted arc (or the measure
of the intercepted arc is twice the measure of
the inscribed angle.
1
m ÐABC = ( mADC )
2
2( m ÐABC ) = mADC
Example 1 p. 579
Use circle O on pg 579 & review the example
shown. mAB = 140,mBC=100, mAD=mDC. Find
the measures of angle 4 & 5.
mAB = 140 therefore, measure of angle 4 is ½ of
140. Measure of angle 4 = 70
mBC = 100 therefore, measure of angle 5 is ½ of
100. Measure of angle 5 = 50
Theorem 10.6: If two inscribed angles of a circle (or congruent circles)
intercept congruent arcs or the same arc, then the angles are congruent.
A
B
B
C
A
D
C
D
F
ÐDAC @ ÐDBC ÐFAE @ ÐCBD
E
Review the Proof in Ex. 2 p. 580
Try check your progress #2
Answer: Statements (Reasons)
1) RT bisects SU (Given)
2) SV = VU (def of segment bisector)
3) Angle SRT intercepts arc ST. Angle SUT intercepts arc
ST. (def of intercepted arc)
4) Angle SRT = angle SUT (inscribed angles of same arc
are congruent)
5) Angle RVS = angle UVT (vertical angles are congruent
6) Triangle RVS = Triangle UVT (AAS)
Theorem 10.7
If the inscribed angle of a triangle intercepts a
semicircle, the angle is a right angle.
A
D
B
C
Arc ADC is a semicircle, so the
measure of angle ABC is 90.
Refer to Circle F & given
info in Ex. 4 on pg.581
Find the measure of angle 3 & angle 4.
Answer
Since Arc AD = Arc BD, then angle 3 & angle 4
are also equal. Therefore, each are 45 since Arc
ADE is a semicircle so angle B is 90 leaving the
other two angles (3 & 4) are complementary
(add to 90).
Refer to Circle V on pg. 582
Quadrilateral WXYZ is inscribed in circle V. If the
measure of angle W = 95 and measure of angle Z is
60, find the measure of angle X and Y.
Arc WXY = 120 & Arc ZYX = 190
This means that arc WZY = 360-120 = 240 & XWZ =
360-190 = 170
Angle Y is ½ of 170 = 85
Angle X is ½ of 240 = 120