Interpretation: Analyzing What a Text Means

Mr Kurland is Genius!  What a great analogy.
Look at Leonardo da Vinci's painting Mona Lisa, and you see a woman smiling. But you are also aware of a painting. You see different color paint (well, not in this illustration!) and you see how the paint was applied to the wood. You recognize how aspects of the painting are highlighted by their placement or by the lighting.

When examining a painting, you are aware that you are examining a work created by someone. You are aware of an intention behind the work, an attempt to portray something a particular way. Since the painting does not come out and actively state a meaning, you are consciously aware of your own efforts to find meaning in the painting: Is she smiling? Self-conscious? Alluring? Aloof?
Looking at the Mona Lisa, you know that you are not looking at Mona Lisa, a person, but The Mona Lisa, a painting. You can talk not only about the meaning of the picture, but also about how it was crafted. What is the significance of the dream landscape in the background? Why, when we focus on the left side of the picture, does the woman looks somehow taller or more erect than if we focus on the right side? The more features of the painting that you recognize, the more powerful your interpretation will be. When reading texts, as when reading paintings, we increase understanding by recognizing the craftsmanship of the creation, the choices that the artist/author made to portray the topic a certain way. And yet there is still that feeling that texts are somehow different. Texts do differ from art insofar as they actually seem to come out and say something. There are assertions "in black and white" to fall back on. We can restate a text; we cannot restate a painting or action. Yet a text is simply symbols on a page. Readers bring to their reading recognition of those symbols, an understanding of what the words mean within the given social and historical context, and an understanding of the remarks within their own framework of what might make sense, or what they might imagine an author to have intended.

GO READ THE REST!  http://www.criticalreading.com/interpretation.htm








 

Inference- find meaning within the context

FANTASTIC RESOURCE!

Inference: Reading Ideas as Well as Words


Ideally, speakers mean what they say and say what they mean. Spoken communication is not that simple. Much of what we understand—whether when listening or reading—we understand indirectly, by inference. Listening involves a complex combination of hearing words, analyzing sentence structure, and attempting to find meaning within the context of the given situation.
The situation with the written word is no different. A text does not contain a meaning. Readers construct meaning by what they take the words to mean and how they process sentences to find meaning. Readers draw on their knowledge of the language and of conventions of social communication.
They also draw on other factors, such as knowledge of the author (“Would Henry say such a thing?), the occasion (“No one knew such things then!”), or the audience (“He’d never admit that publicly.”) They infer unstated meanings based on social conventions, shared knowledge, shared experience, or shared values. They make sense of remarks by recognizing implications and drawing conclusions. Readers read ideas more than words, and infer, rather than find, meaning.

Dan Kurland's www.criticalreading.com
Reading and Writing Ideas As Well As Words

http://www.criticalreading.com/inference_reading.htm#linka

ACT Reading Test FACTS from the official site 2013


The ACT Reading Test 40-items, 35-minutes

The test comprises four passages that are representative of the level and kinds of text commonly encountered in first-year college curricula; passages on topics in the social sciences, the natural sciences, prose fiction, and the humanities are included.

 
The test items require students to derive meaning from several texts by:

·         referring to what is explicitly stated

·         reasoning to determine implicit meanings.

Specifically, items ask students to use referring and reasoning skills to:

  • determine main ideas
  • locate and interpret significant details
  • understand sequences of events
  • make comparisons
  • comprehend cause-effect relationships
  • determine the meaning of context-dependent words, phrases, and statements
  • draw generalizations
  • analyze the author’s or narrator’s voice or method

Each passage is preceded by a heading that identifies what type of passage it is (e.g., “Prose Fiction”), names the author, and may include a brief note that helps in understanding the passage. Each passage is accompanied by a set of multiple-choice test items.

They do not test the rote recall of facts from outside the passage (everything you need to answer the questions is in the passage!) nor do they contain isolated vocabulary questions. (Like the SAT) (Vocabulary meanings on the ACT can be determined by using context clues.)

Three scores are reported for the Reading Test:

1.       total test score-based on all 40 items,

2.       subscore-based on the 20 items in the social sciences and natural sciences sections

3.       subscore-based on the 20 items in the prose fiction and humanities sections

Free Printable ACT Practice Tests

Nothing beats practicing with the full format test.
These are in no particular order. Check back from time to time to this post-I will add links to this post as I find them.
Under each link is the 1st passage from the reading test of that test for comparison.
 
Passage 1    
PROSE FICTION:   This passage is adapted from Ex Libris: Confessions of a Common Reader by Anne Fadiman (©1998 by Anne Fadiman).
Long ago, when George and I were not yet married but seemed to be tottering in that general direction, we gave each other our first Christmas presents.
 
Passage I   
 PROSE FICTION:   This passage is adapted from the short story “From Aboard the Night Train” by Kimberly M. Blaeser (©1993 by Kimberly M. Blaeser), which appeared in Earth Song, Sky Spirit: Short Stories of the Contemporary Native American Experience. The passage begins with a female narrator traveling to her hometown.
The moon gives some light and I can make out the contours of the land, see the faint reflection in the lakes and ponds we pass.
 
Same Test (http://www.unioto.k12.oh.us/ACTpreparing.pdf), (http://www.collegetidbits.com/tools/...or-the-act.pdf)
Passage I  
PROSE FICTION: This passage is adapted from the short story “American History” by Judith Ortiz-Cofer (©1992 by Judith Ortiz-Cofer). The story appeared in the anthology Iguana Dreams: New Latino Fiction.
There was only one source of beauty and light for me my ninth grade year. The only thing I had anticipated at the start of the semester.
 
Same Test- (http://www2.btcs.org/ths/ACT_Resourc...%20Test%29.pdf )
Passage I   
 PROSE FICTION: This passage is adapted from the short story “The Threshold” by Cristina Peri Rossi (original Spanish version ©1986 by Cristina Peri Rossi; translation ©1993 by MaryJane Treacy).
The woman never dreams and this makes her intensely miserable.
Passage I
PROSE FICTION: This passage is adapted from the title story of Only the Little Bone, a collection of short stories by David Huddle (©1986 by David Huddle).
My grandfather has made crutches for me.

6.  http://media.act.org/documents/preparing.pdf
 
Passage I


The moon gives some light and I can make out the

contours of the land, see the faint reflection in the lakes

and ponds we pass. Several times I see or imagine I see

glowing eyes staring back at me from a patch of woods

beside the track.
Passage I

This passage is adapted from the short story

“The Threshold” by Cristina Peri Rossi (original Spanish version

©1986 by Cristina Peri Rossi; translation ©1993 by Mary

Jane Treacy).



The woman never dreams and this makes her

intensely miserable. She thinks that by not dreaming

she is unaware of things about herself that dreams

would surely give her. She doesn’t have the door of

dreams that opens every night to question the certainties

of the d







The following links no longer work
 http://www.ashland.kctcs.edu/Admissi...tice_Test.ashx
http://www.coylecassidy.com/preparing2012.pdf
http://www.sdcoe.net/lret/avid/act/PreparingforACT.pdf
 

Strategies for Improving Teen Reading Comprehension

Strategies for Improving Teen Reading Comprehension

1. READ. What a concept! Don't read for the teacher, or read for a grade-read for yourself. Nothing changes you like reading. Nothing increases your curiosity, intellect, and knowledge like reading. Someday you will be asked, perhaps by your own children, "Hey, did you ever read Hamlet?" And of course you'll say, "Yeah, senior year in high school." Only you will know if you cheated yourself out of reading it. Only you will know that you didn't meet the challenge, and that Hamlet beat you.
If you heard a film was fabulous, would you ask a friend or online critic to summarize it for you, assuming their summaries are as good as experiencing it yourself? Of course not. Then why rob yourself out of experiencing famous literature and creating opinions that are solely YOURS?
2. Listen to your inner voices. Yep, they're normal.
You have three inner reading voices.
The first voice is the voice in your head, reading the lines. It's your reading voice, sounding out the words, reading the sentences.
The second voice is the one that responds to the text. It says things like, "Huh? What's going on? I forgot who that character is again...let me go back and check..." or "Ugh! Her best friend slapped her! I'd hit her back...or walk away...yeah, right....Well, she kind of deserved it after betraying her..."
The third voice is your distracting voice, the one that starts leading you on a tangent, such as, "I remember when my friend did that to me. At my locker. I hope my locker doesn't get stuck tomorrow. I'll have to go to the office for help, and I'll look like an idiot. Hee hee, that kid who fell at lunch was an idiot...Oh, crap, I've read two pages and I have no idea what happened. Oh well, I'll keep reading from here and figure it out...Oh, yeah, I still have math homework, crap..."
Obviously, you need the first and second voices. If you only hear the first one, however, you are not THINKING while reading, and later you won't remember any of it. You need the second voice to grasp the information, to connect yourself to what's going on, and to question which parts you understand and which parts are confusing you.
You need to pay attention when the third voice takes over, and then get back on track as soon as possible. Even good readers have third voices distracting them, but the difference between good readers and poor readers is that good readers go back to the point when the third voice took over, and they REREAD.
3. Reread, using the strategies below. (Ugh!)
4. Design a reading environment specifically for you.
Good readers create environments that help them to focus. You need a quiet spot to read, a place that allows you to focus, a place that is comfortable and suitable for your specific needs. Some people can read at a desk, while others prefer a comfy chair in a corner of a room. Some people can read in a bed, while others instantly fall asleep. Find a place that inspires you and helps you to stay focused and think.
Turn off the television, computer, video games, and any music. While background noise sometimes helps people focus on tasks, background noise is inappropriate during reading since it blocks you from hearing your first and second inner voices. If your house is full of loud siblings, try reading in an attic, basement, or closet. Sounds silly, but it works. And no one says you have to do homework at home. Some people read better outdoors or in a public building, like a library or coffee shop. Students sometimes find it easiest to read in school during a quiet study hall or after school in an empty classroom.
It is important to figure out if you are an auditory learner, someone who learns by hearing things. Auditory learners find it easier to focus if they read the material aloud or follow along to the book on audio. However, other people find oral reading distracting because the material takes over and they can't hear their second inner voice. Experiment to figure out what environment works best for you.
5. Figure out the main idea.
Stop yourself every page (or paragraph if you need to) and ask your second voice to summarize the information. If you do not know the main idea, go back and reread that section. This will prevent your third voice from taking over too often, thus wasting your time.
6. Know WHY you are reading something.
Don't think of the reading as an assignment. Think about it as a challenge to gain more knowledge.
Have a purpose. For instance, when good readers have to read, say, Chapter 12 for homework, they start by asking themselves, "What happened in Chapter 11? Where was I?"
Good readers also ask teachers if there are things to focus on. Perhaps in Chapter 12, a character will change drastically. That's good to know ahead of time, so when you read Chapter 12, you can look for the change.
For example, if you were given the newspaper, you would not read it cover to cover. You would read with a purpose in mind. Perhaps you'd head straight for the Sports section, or the Comics, or the Horoscopes, or the Weather, or, strangely enough, the News. Imagine if you had to read the entire newspaper, and then take a quiz on it. Impossible! Read with a focus.
Also, good readers notice important details.If something is repeated, such as a word, phrase, image, setting, or a strange detail, it's important! Think about why the writer keeps repeating it.
Teachers sometimes give questions for homework to check for understanding and to help students focus. Good readers actually read the assignment, and then answer the questions as thoughtfully as possible. Poor readers try to copy the good readers' answers, or they skim the reading and give minimal responses. They later complain that they did the reading and still failed the quizzes and tests. They never experienced the reading, letting it invade their thoughts and challenge their knowledge.
Who knew that reading involved thinking, effort, and endurance?
7. Visualize what you are reading.
Pretend you are a director, turning the text into a movie. What do you see? What does the setting look like? What do characters look like? What are they wearing? What facial expressions should they make? How should they deliver dialogue? Can you hear their voices? Can you guess what will happen next?
8. Make connections to what you are reading.
Connecting to what you are reading is the best way to remember what you've read.
·Allow your second voice time to relate the text to a personal experience.
·Compare it to another text you've read, or to a film you've seen.
· Recall any other background information you have about the subject (especially new information the teacher may have provided in class.)
· Write down interesting questions or comments that your second voice thinks. Note lines or paragraphs that confuse you. This is called active reading. Since you cannot highlight or write in most textbooks, use sticky notes in the margins of the text to keep track of your questions.
9. Ask specific questions to get specific answers.
Pretend the text is a puzzle of details that you must piece together to get the full picture. Sometimes you need help fitting the pieces together. Never simply tell the teacher, "I didn't get it." Instead say, "On page 49, I didn't understand who was talking. The dialogue was weird, and then the characters kissed, and I didn't even know they liked each other!" The teacher can then discuss the minor details that help readers to see that they liked each other, such as passionately expressing their hatred for each other, and yet talking for hours together. Chances are other students will be stuck on the same parts, not putting the details together correctly.
10. Pay attention to the actual structure of the reading.
If a chapter has a title, read it. If sections have titles, read them. Think about them. As you read, figure out why the title matches the section.
Pay attention to FONT and PUNCTUATION. If things are bold, underlined, or ALL CAPITALIZED, they are important. If a word or phrase is italicized, perhaps it is a new vocabulary word, a foreign word, or a word that should be emphasized for tone to clarify the meaning of the sentence.
This handout has a structure, so you should never have to read the entire sheet again. Items are numbered and spaced for reading ease. Headings are in bold to remind you of the key strategies. Specific strategies and essential points have been italicized. Bullets are used for bulky lists to help breakdown reading into specific chunks. Breaking facts down into smaller pieces makes them easier to remember, and easier to reread and study later. That's why outlining a chapter is sometimes a useful tool.
11. Slow down or speed up.
Pace yourself. Think of reading as a marathon that you want to finish. You will give up if you read so quickly that you miss the meaning, and then have to start all over again to understand. No runner runs a race at the same pace. They also do not quit when they hit a hill. Slow down when things get harder and your second voice is confused. Read only one sentence at a time, and then figure out what that sentence means. Good readers also slow down when something crucial or exciting happens so they don't miss anything.
Speed up when the reading is easier, or if you are getting bored. Momentum can stop boredom and stop you from falling asleep. You would never want to listen to a speech all at one speed, so why would you want to read something at one, boring pace?
12. Use a dictionary.
If you cannot figure out the meaning of a new word using context clues, use a dictionary to build your vocabulary. This might seem like a waste of time, but sometimes you may not understand an entire paragraph because of one word. Rather than struggling for pages and rereading for clarity, taking a minute to look up a word in a dictionary could actually save you time.
13. Read for pleasure.
Reading is the only way to become a better reader. Read anything and everything that you want to read. Many good readers visit the library on a regular basis. Some people are afraid of libraries--something tragic usually happened to them as children involving fines and lost videos. If you are one of these poor souls, it's time to get over it and talk to a librarian. It always fascinates me that whenever I run into students at the library, they are always the strongest readers in my classes.
Marathon runners cannot expect to make it to the finish line without training for the race, running more and more each day. Reading classics and textbooks for school is the marathon. How can you ever expect to reach the finish line if you never practice along the way?
Bibliography: Tovani, Cris. I Read It, But I Don't Get It: Comprehension Strategies for Adolescent Readers. Portland:Stenhouse, 2000.

Source- Kristy Acevedo, Yahoo! Contributor Network
http://voices.yahoo.com/strategies-improving-teen-reading-comprehension-684915.html?cat=4
 

Reading Comprehension Test Tips

Reading Comprehension Test Tips

Reading comprehension is an important aspect of academic ability and is a common section on standardized tests such as the SAT, GMAT and LSAT. Reading comprehension test sections typically involve reading a passage on a certain subject and then answering several questions based on the passage. Following these tips helps you prepare and improves your chances for success.          
  1. Reading the Passage

    • A common mistake among reading comprehension test takers is the assumption that they do not need to read the whole passage presented to them. While it may be possible to skip to the questions and then go back and attempt to answer each question individually be finding the part of the passage that deals with that question, this tactic causes you to have a hazy understanding of the passage as a whole, and may lead to confusion. It is always a good idea to read though the entire passage at least once, even if that read is brief. An overall understanding of the themes and arguments in the passage is important, especially for higher level tests with more difficult questions, such as the LSAT or GMAT. Depending on your reading speed, you can skim the questions before reading the passage, as it allows you to highlight or immediately answer certain questions. If you do not read fast, it is often best to begin by reading the passage, then answering the questions in order.

    Time Management

    • Most standardized tests have time limits for each section. Time constraints can be particularly troublesome for reading comprehension sections, as the sheer amount of reading that must be done between the passages and questions can take all of the allotted time. It is important to know how many total passages you have to read and roughly how many questions there are for each passage before you begin the exam. Take your total time limit for the reading comprehension section and divide it by the total number of passages. This number is about how many minutes you should spend on each section. If you are falling behind on time, you may wish to answer questions more quickly and take some educated guesses to make time for later passages. If you run into a particularly difficult question, consider skipping it and coming back later if time allows. If your test does not penalize you for guessing, make sure to fill in an answer for all the questions before the time runs out, even if you don't have time to read them all. For multiple choice tests, cross out answers you know are wrong to eliminate them from contention. If you can't decide between two or three choices, make an educated guess and move on. Wasting too much time on one question may hurt you later in the test.

    Other Considerations

    • As you read through the passage for the first time, it is a good idea to use a pencil or highlighter to underline ideas, quotes, or other information that seems important and may appear in questions. It can also be a good idea to read the first question before starting the passage; that way you have a specific question in mind the whole time you are reading the passage, so you will likely stumble across the answer during your read-through.
      There's no better way to increase reading comprehension scores than to practice doing it. Take practice tests to improve your reading comprehension skills and to determine what type of questions are normally asked and how long it takes you to complete passages. Also consider reading material like scholarly journals and newspapers to become accustomed to absorbing dense information. Investing in a study book designed for the test you will be taking can be a great idea. Some people benefit from taking test prep classes or tutoring, though results depend largely upon the quality of the teaching. Practicing helps you increase your test-taking speed and allows you to pinpoint the areas that you need to work on the most.

 
 

X

Gregory Hamel
Gregory Hamel has been a freelance writer for various websites since September 2008 and has also authored two young-adult science-fantasy novels. He has a Bachelor of Arts in economics from St. Olaf College. Hamel maintains a blog focused on running and fitness.



Read more: Reading Comprehension Test Tips | eHow.com http://www.ehow.com/way_5256779_reading-comprehension-test-tips.html#ixzz2KRneSrs2

The First Month of Geometry Class Explained in 13 Minutes!

Language and Notation of Basic Geometry

 

Properties of a Circle

Circles: Radius, Diameter and Circumference
Understanding the relationship between the radius diameter and circumference of a circle.

Khan Academy
Practice this concept HERE

Geometry - Circles - Other Angles

Other Angles

Theorem 73: If a tangent and a diameter meet at the point of tangency, then they are perpendicular to one another.

In Figure 1 , diameter AB meets tangent at B. According to Theorem 73, AB which means that mABC = 90° and mABD = 90°.






Figure 1A tangent to the circle and a diameter of the circle meeting at the point of tangency.


Theorem 74: If a chord is perpendicular to a tangent at the point of tangency, then it is a diameter.
Example 1:Theorem 74 could be used to find the center of a circle if two tangents to the circle were known. In Figure 2 , is tangent to the circle at P, is tangent to the circle at S. Use these facts to find the center of the circle.






Figure 2Finding the center of a circle when two tangents to the circle are known.


According to Theorem 74, if a chord is drawn perpendicular to at P, it is a diameter, which means that it passes through the center of the circle.
Similarly, if a chord is drawn perpendicular to at S, it too would be a diameter and pass through the center of the circle. The point where these two chords intersect would then be the center of the circle. See Figure 3 .






Figure 3Chords drawn perpendicular to tangents to help in finding the center of the circle.


Theorem 75: The measure of an angle formed by two chords intersecting inside a circle is equal to half the sum of the measures of the intercepted arcs associated with the angle and its vertical angle counterpart.
In Figure 4 , chords AC and BD intersect inside the circle at E.






Figure 4Angles formed by two chords intersecting inside a circle.


By Theorem 75:,




Theorem 76: The measure of an angle formed by a tangent and a chord meeting at the point of tangency is half the measure of the intercepted arc.
In Figure 5 , chord QR and tangent meet at R. By Theorem 76, m ∠1 = 1/2 ( m) and m ∠ 2 = ½ ( m).






Figure 5A tangent to the circle and a chord meeting at the point of tangency.


Theorem 77: The measure of an angle formed by two secants intersecting outside a circle is equal to one half the difference of the measures of the intercepted arcs.
In Figure 6 , secants and intersect at G. According to Theorem 77, m ∠1 = 1/2( mm).






Figure 6Two secants to the circle meeting outside the circle.


Example 2: Find m ∠1 in Figures 7 (a) through (d).






Figure 7Angles formed by intersecting chords, secants, and/or tangents.


Example 3: Find the value of y in Figures 8 (a) through (d).






Figure 8Angles formed by intersecting chords, secants, and/or tangents

  CliffsNotes.com. Other Angles. 7 Feb 2013
< http://www.cliffsnotes.com/study_guide/topicArticleId-18851,articleId-18826.html>.

Geometry - Circles - Arcs and Inscribed Angles

 
Source - Khan Academy Inscribed and Central Angles

Arcs and Inscribed Angles

Central angles are probably the angles most often associated with a circle, but by no means are they the only ones. Angles may be inscribed in the circumference of the circle or formed by intersecting chords and other lines.

  • Inscribed angle: In a circle, this is an angle formed by two chords with the vertex on the circle.
  • Intercepted arc: Corresponding to an angle, this is the portion of the circle that lies in the interior of the angle together with the endpoints of the arc.

In Figure 1 , ∠ ABC is an inscribed angle and is its intercepted arc.







Figure 1An inscribed angle and its intercepted arc.


Figure 2 shows examples of angles that are not inscribed angles.







Figure 2Angles that are not inscribed angles.


QRS is not an TWV is not an
inscribed angle,inscribed angle,
since its vertexsince its vertex
is not on the circle.is not on the circle.

Refer to Figure 3 and the example that accompanies it.







Figure 3A circle with two diameters and a (nondiameter) chord.


Notice that m ∠3 is exactly half of m, and m ∠4 is half of m ∠3 and ∠4 are inscribed angles, and and are their intercepted arcs, which leads to the following theorem.
Theorem 70: The measure of an inscribed angle in a circle equals half the measure of its intercepted arc.
The following two theorems directly follow from Theorem 70.
Theorem 71: If two inscribed angles of a circle intercept the same arc or arcs of equal measure, then the inscribed angles have equal measure.
Theorem 72: If an inscribed angle intercepts a semicircle, then its measure is 90°.
Example 1: Find mC in Figure 4 .







Figure 4Finding the measure of an inscribed angle.







Example 2: Find mA and mB in Figure 5 .







Figure 5Two inscribed angles with the same measure.







Example 3: In Figure 6 , QS is a diameter. Find mR. mR = 90° (Theorem 72).







Figure 6An inscribed angle which intercepts a semicircle.


Example 4: In Figure 7 of circle O, m 60° and m ∠1 = 25°.






Figure 7A circle with inscribed angles, central angles, and associated arcs.


Find each of the following.
  1. mCAD
  2. m
  3. mBOC
  4. m
  5. mACB
  6. mABC
CliffsNotes.com. Arcs and Inscribed Angles. 7 Feb 2013
< http://www.cliffsnotes.com/study_guide/topicArticleId-18851,articleId-18825.html>.

Geometry - Central Angles and Arcs

Source Khan Academy

Central Angles and Arcs

There are several different angles associated with circles. Perhaps the one that most immediately comes to mind is the central angle. It is the central angle's ability to sweep through an arc of 360 degrees that determines the number of degrees usually thought of as being contained by a circle.


Central angles

Central angles are angles formed by any two radii in a circle. The vertex is the center of the circle. In Figure

1 , ∠ AOB is a central angle.







Figure 1A central angle of a circle.


Arcs

An arc of a circle is a continuous portion of the circle. It consists of two endpoints and all the points on the circle between these endpoints. The symbol is used to denote an arc. This symbol is written over the endpoints that form the arc. There are three types of arcs:
  • Semicircle: an arc whose endpoints are the endpoints of a diameter. It is named using three points. The first and third points are the endpoints of the diameter, and the middle point is any point of the arc between the endpoints.
  • Minor arc: an arc that is less than a semicircle. A minor arc is named by using only the two endpoints of the arc.
  • Major arc: an arc that is more than a semicircle. It is named by three points. The first and third are the endpoints, and the middle point is any point on the arc between the endpoints.

In Figure
2 , AC is a diameter. is a semicircle.






Figure 2A diameter of a circle and a semicircle.


In Figure 3 , is a minor arc of circle P.






Figure 3A minor arc of a circle.


In Figure 4 , is a major arc of circle Q.






Figure 4A major arc of a circle.


Arcs are measured in three different ways. They are measured in degrees and in unit length as follows:
  • Degree measure of a semicircle: This is 180°. Its unit length is half of the circumference of the circle.
  • Degree measure of a minor arc: Defined as the same as the measure of its corresponding central angle. Its unit length is a portion of the circumference. Its length is always less than half of the circumference.
  • Degree measure of a major arc: This is 360° minus the degree measure of the minor arc that has the same endpoints as the major arc. Its unit length is a portion of the circumference and is always more than half of the circumference.

In these examples, m indicates the degree measure of arc AB, l indicates the length of arc AB, and indicates the arc itself.
Example 1: In Figure 5 , circle O, with diameter AB has OB = 6 inches. Find (a) m and (b) l.






Figure 5Degree measure and arc length of a semicircle.


  1. is a semicircle. m = 180°.
  2. Since is a semicircle, its length is half of the circumference.






Postulate 18 (Arc Addition Postulate): If B is a point on , then m + m = m.
Example 2: Use Figure 6 to find m ( m = 60°, m = 150°).









Figure 6Using the Arc Addition Postulate.


Example 3: Use Figure 7 of circle P with diameter QS to answer the following.
  1. Find m
  2. Find m
  3. Find m
  4. Find m






Figure 7Finding degree measures of arcs.


  1. m (The degree measure of a minor arc equals the measure of its corresponding central angle.)
  2. = 180° ( is a semicircle.)
  3. m = 130°





  4. m = 310° ( is a major arc.) The degree measure of a major arc is 360° minus the degree measure of the minor arc that has the same endpoints as the major arc.






The following theorems about arcs and central angles are easily proven.
Theorem 68: In a circle, if two central angles have equal measures, then their corresponding minor arcs have equal measures.
Theorem 69: In a circle, if two minor arcs have equal measures, then their corresponding central angles have equal measures.
Example 4: Figure 8 shows circle O with diameters AC and BD. If m ∠1 = 40°, find each of the following.






Figure 8A circle with two diameters and a (nondiameter) chord.







  1. m = 40° (The measure of a minor arc equals the measure of its corresponding central angle.)
  2. m = 40° (Since vertical angles have equal measures, m ∠1 = m ∠2. Then the measure of a minor arc equals the measure of its corresponding central angle.)
  3. m = 140° (By Postulate 18, m + m = m is a semicircle, so m + 40° = 180°, or m = 140°.)
  4. mDOA = 140° (The measure of a central angle equals the measure of its corresponding minor arc.)
  5. m ∠3 = 20° (Since radii of a circle are equal, OD = OA. Since, if two sides of a triangle are equal, then the angles opposite these sides are equal, m ∠3 = m ∠4. Since the sum of the angles of any triangle equals 180°, m ∠3 + m ∠4 + mDOA = 180°. By replacing m ∠4 with m ∠3 and mDOA with 140°,


  6. m ∠4 = 20° (As discussed above, m ∠3 = m ∠4.)
Source - CliffsNotes.com. Central Angles and Arcs. 7 Feb 2013
< http://www.cliffsnotes.com/study_guide/topicArticleId-18851,articleId-18824.html>.