1. 10.3 Arcs and Chords & 10.4 Inscribed Angles

2. Objectives
Recognize and use relationships between arcs, chords, and diameters
Find measures of inscribed angles
Find measures of angles of inscribed polygons

3. Arcs and Chords
Theorem 10.2: In a or  s, two minor arcs are  iff their corresponding chords are . B R A S

4. Diameters and Chords
Theorem 10.3: In a , if a diameter (or radius) is ┴ to a chord, then it bisects the chord and its arc.
If JK ┴ LM, then MO  LO and arc LK  arc MK.

5. More about Chords
Theorem 10.4: In a or in  s, two chords are  iff they are equidistant from the center.

6. Inscribed Angles
An inscribed angle is an angle that has its vertex on the circle and its sides are chords of the circle. C A B

7. Inscribed Angles
Theorem 10.5 (Inscribed Angle Theorem): The measure of an inscribed angle equals ½ the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle). C A B
mACB = ½m or
2 mACB =

8. Example 1:
In and Find the measures of the numbered angles.

9. Example 1: First determine Arc Addition Theorem Simplify.
Subtract 168 from each side.
Divide each side by 2.

10. Example 1:
So, m

11. Example 1:
Answer:

12. Your Turn:
In and Find the measures of the numbered angles.
Answer:

13. Inscribed Angles
Theorem 10.6: If two inscribed s intercept  arcs or the same arc, then the s are .
mDAC  mCBD

14. Example 2:
Given:
Prove:

15. Example 2: Proof: Statements Reasons 1. Given 1. 2.
2. If 2 chords are , corr. minor arcs are . 3.
3. Definition of intercepted arc 4.
4. Inscribed angles of arcs are . 5.
5. Right angles are congruent 6.
6. AAS

16. Your Turn:
Given:
Prove:

17. Your Turn: 1. Given 2. Inscribed angles of arcs are .
3. Vertical angles are congruent.
4. Radii of a circle are congruent.
5. ASA
Proof:
Statements Reasons 1. 2. 3. 4. 5.

18. Angles of Inscribed Polygons
Theorem 10.7: If an inscribed  intercepts a semicircle, then the  is a right .
i.e. If AC is a diameter of , then the mABC = 90°. o

19. Angles of Inscribed Polygons
Theorem 10.8: If a quadrilateral is inscribed in a , then its opposite s are supplementary.
i.e. Quadrilateral ABCD is inscribed in O, thus A and C are supplementary and B and D are supplementary. D A C B O

20. Example 3:
ALGEBRA Triangles TVU and TSU are inscribed in with Find the measure of each numbered angle if and

21. Example 3:
are right triangles since they intercept congruent arcs. Then the third angles of the triangles are also congruent, so
Angle Sum Theorem
Simplify.
Subtract 105 from each side.
Divide each side by 3.

22. Example 3: Use the value of x to find the measures of Given Given
Answer:

23. Your Turn:
ALGEBRA Triangles MNO and MPO are inscribed in with Find the measure of each numbered angle if and
Answer:

24. Example 4: Quadrilateral QRST is inscribed in If and find and
Draw a sketch of this situation.

25. Example 4: To find we need to know To find first find
Inscribed Angle Theorem
Sum of angles in circle = 360
Subtract 174 from each side.

26. Example 4: Inscribed Angle Theorem Substitution Divide each side by 2.
To find we need to know but first we must find
Inscribed Angle Theorem

27. Example 4: Sum of angles in circle = 360 Subtract 204 from each side.
Inscribed Angle Theorem
Divide each side by 2.
Answer:

28. Your Turn:
Quadrilateral BCDE is inscribed in If and find and
Answer:

29. Assignment
Geometry Pg. 540 #11 – 29 Pg. 549 #8 – 10, 13 – 16, 18,
Pre-AP Geometry Pg. 540 #11 – 33 Pg. 549 #8 – 10, 13 – 30