This document provides examples and explanations of how to find angle measures formed by lines intersecting within and on circles. It includes examples of finding angle measures using tangent-secant angles, tangent-chord angles, and angles inside and on circles. Students are guided through step-by-step workings and are given practice problems to solve involving finding specific angle measures using the concepts taught.

1. UNIT 12.4 ANGLE MEASURESUNIT 12.4 ANGLE MEASURES

2. Warm Up
1. Identify each line or segment that intersects
F.
Find each measure.
2. m∠NMP
3. m∠NLP
chords: AE, CD
secant: AE
tangent: AB
110°
55°

3. Find the measures of angles formed by lines
that intersect circles.
Use angle measures to solve problems.
Objectives

5. Find each measure.
Example 1A: Using Tangent-Secant and
Tangent-Chord Angles
m∠EFH
= 65°

6. Find each measure.
Example 1B: Using Tangent-Secant and
Tangent-Chord Angles

7. Check It Out! Example 1a
Find each measure.
m∠STU
= 83°

8. Check It Out! Example 1b
Find each measure.

10. Example 2: Finding Angle Measures Inside
a Circle
Find each measure.
m∠AEB
= 126°

11. Check It Out! Example 2a
m∠ABD
Find each angle measure.

12. Check It Out! Example 2b
m∠RNM
Find each angle measure.
m∠RNM = 180° – ∠MNQ
m∠RNM = 180° – 158° = 22°

14. Example 3: Finding Measures Using
Tangents and Secants
= 40° = 63°
Find the value of x.

15. Check It Out! Example 3
Find the value of x.
50° = 83° – x
x = 33°

16. Example 4: Design Application
In the company logo shown, = 108°,
and = 12°. What is m∠FKH?

17. Check It Out! Example 4
Two of the six muscles that control eye
movement are attached to the eyeball and
intersect behind the eye. If mAEB = 225°, what
is m∠ACB?

19. Example 5: Finding Arc Measures
Find
Step 1 Find
If a tangent and a secant intersect on a
 at the pt. of tangency, then the
measure of the ∠ formed is half the
measure of its intercepted arc.
Substitute 180 – 113 for
m∠XVY and 68 for mWZ
Multiply both sides by 2.
Subtract 68 from both sides.
V
U

20. Example 5 Continued
Step 2 Find
Thm. 11-5-3
Substitute the given values.
Multiply both sides by 2.
Add 66 to both sides.
V
U

21. Check It Out! Example 5
Find mLP
Step 1 Find
Step 2 Find

22. Lesson Quiz: Part I
Find each measure.
1. m∠FGJ
2. m∠HJK 65°
41.5°

23. Lesson Quiz: Part II
3. An observer watches
people riding a Ferris
wheel that has 12
equally spaced cars.
Find x.
30°

24. Lesson Quiz: Part III
4. Find mCE.
12°

25. All rights belong to their respective owners.
Copyright Disclaimer Under Section 107 of
the Copyright Act 1976, allowance is made
for "fair use" for purposes such as criticism,
comment, news reporting, TEACHING,
scholarship, and research.
Fair use is a use permitted by copyright
statute that might otherwise be infringing.
Non-profit, EDUCATIONAL or personal use
tips the balance in favor of fair use.