1. Section 9.5 Inscribed Angles
Circles
Section 9.5
Inscribed Angles

2. Warm Up Circles Crossword puzzle
Some answers are 2 words, do not leave blanks

3. Homework Answers p. 347
1. 8
2. 5
3. 9√2
4. 55
5. 80
6. 45
7. 24
8. 12
9. 10√5

4. Central Angle vertex at the center of a circle. angle equals arcB A
If AB = 100° then AOB = 100°

5. Inscribed Angle An angle whose vertex is on the edge of a circle.
C is a point on circle O
ACB is an inscribed angle O B C A
AB is the intercepted arc of inscribed angle ACB

6. Theorem 9.7
The measure of an inscribed angle is equal to HALF the intercepted arc A
If AB = 100° then ACB = 50° C O B

7. Example 1 Find x, y, z x° y° 90° x= ½ (80°) = 40° y= 2(55°) = 110°

8. Example 2 Find 1 BC = 70°,  1 = ½ (70°) = 35° B C
3. BAC = 280°, BC = 360°- 280°= 80°
 1 = ½ (80°) = 40°

9. Example 3 Find BC  1 = 40°, BC= 2(40°) = 80° B C

10. Theorem 9.7 – Corollary 1 If two inscribed angles intersect the same arc, then the angles are congruent.
1 intersects arc BC B C A 1 2
2 intersects arc BC
1 = 2

11. Theorem 9.7 – Corollary 2 An angle inscribed in a semicircle is a right angle.1
O is the center of the circle
arc AB is a semicircle
1 intersects arc AB
1 = ½ (AB) = 90°

12. Theorem 9.7 – Corollary 3 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. B O A Z 1 X 2 3 4
O is the center of the circle
BXAZ is a quadrilateral inscribed in circle O
1 and  2 are opposite angles
1 +  2 = 180°

13. Example 6 Find x, y, z 160° x= ½ (160°) = 80° a°+ 80° + 68° = 180°
Y intersects arcs 160° + 64°
y= ½(160° + 64°) = 112° y°
68°
80° x° z°
64°

14. Example 7 Find x, y, z O is the center
x and y are inscribed in the same semicircle
x = y = ½ (180°) = 90°
2 chords are marked congruent, therefore the 2 arcs are congruent.
The 2 arcs are inscribed in a semicircle
Therefore z = ½ (180°) = 90° O y° z°
68°

15. Example 8 Find x, y, z O is the center
AB is a semicircle = 180°
CB = 180°- 110° = 70°
y is a cental angle = arc
y = 70° C x°
110° z° y°
43° O A
x = ½ (CB) = ½(70°) = 35° B
z°= 2(43°) = 86°

16. Example 9 Find x, y, z A quadrilateral is inscribed in the circle 60°
110°
x° and 85° are opposite angles
Therefore they are supplementary
x= 180°- 85° = 95° z°
85°
y° and 110° are opposite angles
Therefore they are supplementary
y= 180°- 110° = 70° x°
y intersects arcs z° + 60°
y= ½(60° + z°)
70°= ½ (60° + z°)
140° = (60° + z°)
z° = 80°
70° y°

17. Theorem 9.8 The measure of an angle formed by a chord and a tangent is equal to
HALF its intercepted arc O B C A G F
vertex on the edge of the circle
If CB = 120° then FCB = 60°
ACB = _______
GCB= _______

18. Examples 10-12 CBD = ½(240°) = 120° BD = 360°- 240° = 120°O B C A D
240°
CBD = ½(240°) = 120°
BD = 360°- 240° = 120°
ABD = ½(120°) = 60°

19. Examples 13-15 PRT = ½(100°) = 50° PRQ = ½(170°) = 85° QRS = ½(QR)O s R Q
100°
170° P T
PRQ = ½(170°) = 85°
QRS = ½(QR)
QR = 360°- (170°+100°) = 90°
QRS = ½(90°) = 45°

20. Examples 16-18 AC is tangent to circle Z at B
BE is a diameter, therefore ABC =CBE= 90° Z B C A F
75° D E
EBD = 90°- 75° = 15°
DE = 2(15°) = 30°
DB = 2(75°) = 150°

21. Examples 19-21 AC is tangent to circle Z at B
BED = ½ (150°) = 75° Z B C A F
75° D E
30°
150°
15°
BDE is inscribed in a semicircle
BDE = 90°
BFE is a semicircle
BFE = 180°

22. Cool Down Complete exercises 1-6 on bottom of notesheet
Check your answers with Mrs. Baumher
Begin HW p , 19-21