1. Warm-Up
For each circle C, find the value of x. Assume that segments that appear to be tangent are tangent.
1) 2)

2. Inscribed Angles

3. Homework
Inscribed Angles PAGE 4

4. Using Inscribed Angles
Inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle.

5. Determine whether each angle is an inscribed angle
Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle. 1.
YES; CL C L O T

6. Determine whether each angle is an inscribed angle
Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle.
NO; QVR 2. Q V K R S

7. Measure of an Inscribed Angle Theorem
If an angle is inscribed in a circle, then its measure is one half the measure of its intercepted arc.
mADB = ½m

8. Finding Measures of Arcs and Inscribed Angles
PROBLEM 1
Finding Measures of Arcs and Inscribed Angles
Find the measure of the blue arc or angle.
m = 2mQRS =
2(90°) = 180°

9. Finding Measures of Arcs and Inscribed Angles
PROBLEM 2
Finding Measures of Arcs and Inscribed Angles
Find the measure of the blue arc or angle.
m = 2mZYX =
2(115°) = 230°

10. Finding Measures of Arcs and Inscribed Angles
PROBLEM 3
Finding Measures of Arcs and Inscribed Angles
Find the measure of the blue arc or angle.
100°
m = ½ m
½ (100°) = 50°

11. PROBLEM 4
Find the measure of each arc or angle. Q R

12. Two Inscribed Angles Theorem
If two inscribed angles have the same intercepted arc, then the angles are equal.

13. PROBLEM 5
Find mCAB and m

14. Finding the Measure of an Angle
PROBLEM 6
Finding the Measure of an Angle
It is given that mE = 75°. What is mF?
E and F both intercept , so E  F. So, mF = mE = 75°
75°

15. Definitions
A polygon whose vertices are touching a circle is called inscribed.
If a Circle is drawn around a polygon, it is circumscribed about the circle.

16. Circumscribed Polygon
A polygon is circumscribed about a circle if and only if each side of the polygon is tangent to the circle.

17. A circle can be circumscribed around a quadrilateral if and only if its opposite angles are supplementary. B A D C

18. y = 70 z = 95 110 + y =180 z + 85 = 180 Find y and z. PROBLEM 7 z 110

19. Theorem: If a right triangle is inscribed in a circle then the hypotenuse is the diameter of the circle.
180
diameter

20. In K, m<GNH = 4x – 14. Find the value of x.
PROBLEM 8
4x – 14 = 90 H K
x = 26 N G

21. x = 11 6x – 5 + 3x – 4 + 90 = 180 or 6x – 5 + 3x – 4 = 90 H K N G
In K, m<1 = 6x – 5 and m<2 = 3x – 4. Find the value of x.
PROBLEM 9
6x – 5 + 3x – = 180 or
6x – 5 + 3x – 4 = 90 H 2 K 1 N G
x = 11

22. Find the value of each variable.
PROBLEM 10
Find the value of each variable.
AB is a diameter. So, C is a right angle and mC = 90°
2x° = 90°
x = 45
2x°

23. Find the value of each variable.
PROBLEM 11
Find the value of each variable.
DEFG is inscribed in a circle, so opposite angles are supplementary.
mD + mF = 180°
z + 80 = 180
z = 100 z°
120°
80° y°

24. Find the value of each variable.
PROBLEM 12
Find the value of each variable.
DEFG is inscribed in a circle, so opposite angles are supplementary.
mE + mG = 180°
y = 180
y = 60 z°
120°
80° y°

25. Using an Inscribed Quadrilateral
PROBLEM 13
Using an Inscribed Quadrilateral
In the diagram, ABCD is inscribed in circle P. Find the measure of each angle.
ABCD is inscribed in a circle, so opposite angles are supplementary.
3x + 3y = 180
5x + 2y = 180
2y°
3y°
3x°
5x°
To solve this system of linear equations, you can solve the first equation for y to get y = 60 – x. Substitute this expression into the second equation.

26. Using an Inscribed Quadrilateral
PROBLEM 13 continue
Using an Inscribed Quadrilateral
5x + 2y = 180.
5x + 2 (60 – x) = 180
5x – 2x = 180
3x = 60
x = 20
y = 60 – 20 = 40
Write the second equation.
Substitute 60 – x for y.
Distributive Property.
Subtract 120 from both sides.
Divide each side by 3.
Substitute and solve for y.
x = 20 and y = 40, so mA = 80°, mB = 60°, mC = 100°, and mD = 120°