as u like it
Orlando, the youngest son of the recently deceased Sir Rowland de Bois, describes his unfortunate state of affairs to Adam, Sir Rowland’s loyal former servant. Upon his father’s death, Orlando was bequeathed a mere 1,000crowns, a paltry sum for a young man of his social background. His only hope for advancement is if his brother, Oliver, honors their father’s wish and provides him with a decent education. Oliver, as the eldest son, inherited virtually everything in his father’s estate, yet he not only neglects this charge but actively disobeys it. Although he arranges for his other brother, Jaques, to attend school, Oliver refuses to allow Orlando any education whatsoever, leaving the young man to lament that his upbringing is little different from the treatment of a piece of livestock. Orlando has long borne this ill treatment, but he admits to Adam that he feels rising within himself a great resentment against his servile condition and vows that he will no longer endure it.

Oliver enters, and the hostility between the brothers soon boils over into violence. Orlando claims that the system that allows the eldest son to inherit the bulk of a father’s estate does not reduce the ancestral blood in the other sons. Oliver, offended by his brother’s insolence, assails Orlando, while Orlando seizes Oliver by the throat. Adam tries to intervene, seeking peace in the name of their father, but the brothers do not heed him. Orlando, undoubtedly the stronger of the two, refuses to unhand his brother until Oliver promises to treat him like a gentleman, or else give him his due portion of their father’s estate so that he may pursue a gentlemanly -lifestyle on his own. Oliver hastily agrees to give Orlando part of his small inheritance and, in a rage, dismisses Orlando and Adam, whom he chastises as an “old dog” (I.i.69).
Oliver bids his servant Denis to summon Charles, the court wrestler, who has been waiting to speak to him. Oliver asks Charles for the news at court, and Charles reports that Duke Senior has been usurped by his younger brother, Duke Frederick, and has fled with a number of loyal lords to the Forest of Ardenne. Because the noblemen have forfeited their land and wealth by going into voluntary exile, Duke Frederick allows them to wander unmolested. When Oliver asks if Senior’s daughter, Rosalind, has been banished, Charles says that the girl remains at court. Not only does Duke Frederick love Rosalind as though she is his own daughter, but the duke’s daughter, Celia, has a great friendship with her cousin and cannot bear to be parted from her. Charles asserts that two ladies never loved as Celia and Rosalind do. Charles then admits his real reason for coming to see Oliver: he has heard rumors that Orlando plans to disguise himself in order to enter a wrestling match at the royal court. Because Charles’s reputation depends upon the brutal defeat of all of his opponents, he worries that he will harm Orlando. He begs Oliver to intervene on his brother’s behalf, but Oliver replies that Orlando is a conniving and deceitful scoundrel. He convinces Charles that Orlando will use poison or some other trick in order to bring down the wrestler. Charles threatens to repay Orlando in kind, and Oliver, pleased with Charles’s promise, plots a way to deliver his brother to the wrestling ring.
Analysis
Shakespeare begins his play with a pair of dueling brothers, an amendment of his source material—Thomas Lodge’s popular prose romance, Rosalynde—that allows him to establish, with great economy, the corrupt nature of so-called civilized life. Oliver’s mistreatment of his brother spurs Orlando to journey into the curative Forest of Ardenne as surely as Frederick’s actions did his own brother Duke Senior, which immediately locates the play in the pastoral tradition: those wounded by life at court seek the restorative powers of the country. But fraternal hostilities are also deeply biblical and resonate with the story of Cain’s murder of Abel, an act that confirmed mankind’s delivery from paradise into a world of malignity and harm. The injustice of Oliver’s refusal to educate or otherwise share his fortune with Orlando seems all the more outrageous because it is perfectly legal. The practice of primogeniture stipulated that the eldest son inherits the whole of his father’s estate so that estates would not fragment into smaller parcels. Primogeniture was not mandated by law in Shakespeare’s England, but it was a firmly entrenched part of traditional English custom. With such a system governing society, inequality, greed, and animosity become unfortunate inevitabilities, and many younger sons in Shakespeare’s time would have shared Orlando’s resentment.
In this opening scene, Shakespeare begins to muse on another theme common in pastoral literature: the origins of gentleness. As scholar Jean E. Howard makes clear in her introduction to the play, “gentleness” refers to both nobility and a virtuous nature (p. 1591). Elizabethans were supremely interested in whether this quality could be achieved or whether one had to be born with it, and Orlando shows himself to be a man of the times. Though Oliver has denied him all forms of education and noble living, Orlando nonetheless has a desire for gentleness. As he assails Oliver, he claims that his “gentleman-like qualities” have been obscured, but feels confident that he could develop them still (I.i.59). Of course, Oliver’s behavior suggests that gentleness has little to do with being born into nobility. Though he has the vast majority of his father’s estate at his fingertips, he proves lacking in the generosity and grace that would make him a true gentleman. The audience, then, looks optimistically to Orlando, who vows to go find his fortune on his own.
The episode with the wrestler Charles is important for several reasons. First, it provides further evidence of the prejudices that rule court society. Charles visits Oliver because he worries about defeating Orlando. Although Charles is paid to be a brute, he fears that pummeling a nobleman, even one so bereft of fortune as Orlando, may win him disfavor in the court. Such deference on Charles’s part speaks to the severe hierarchy of power that structures court life. Charles also provides necessary plot explication. Through Charles’s report to Oliver, Shakespeare sketches the backdrop of his comedy: the usurpation of Duke Senior by Duke Frederick, Rosalind’s precarious situation, and the qualities of life in the Forest of Ardenne. Although set in France, the forest to which Duke Senior and his loyal lords flee is intentionally reminiscent of Sherwood Forest, the home of Robin Hood. It is, in Charles’s estimation, a remnant of “the golden world,” a time of ease and abundance from which the modern world has fallen (I.i.103). Thus, before we ever see Ardenne, which cannot be located on any map, we understand it as a place where Orlando will find the remedy he so desperately seeks.

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Act II, scenes i–iv
Summary: Act II, scene i
[O]ur life, exempt from public haunt,
Finds tongues in trees, books in the running brooks,
Sermons in stones, and good in everything.
(See Important Quotations Explained)
The banished Duke Senior expounds on the wonders of life in the forest. He tells his associates that he prefers forest dwelling to the “painted pomp” of courtly existence (II.i.3). He reminds them that their existence in Ardenne is free from danger and that their greatest worry here is nothing worse than the cold winter wind. The woods provide Duke Senior with everything he needs, from conversation to education to spiritual edification, for he “[f]inds tongues in trees, books in the running brooks, / Sermons in stones, and good in everything” (II.i.16–17). Lord Amiens agrees with him. The duke suggests that they hunt some venison, but he cannot help but mourn the fate of the deer, who, though natives of Ardenne, are violently slaughtered. One lord announces that the melancholy lord Jaques has seconded this observation, declaring Senior guiltier of usurpation than his loveless brother, Duke Frederick. Duke Senior, in good humor, asks one of his men to bring him to Jaques, because arguing with him is such fun.
Summary: Act II, scene ii

Back at court, Duke Frederick is enraged to discover the disappearances of Celia, Rosalind, and Touchstone; he cannot believe that the three could leave court without anyone’s notice. One attending lord reports that Celia’s gentlewoman overheard Celia and Rosalind complimenting Orlando, and she speculates that wherever the women are, Orlando is likely with them. Frederick seizes on this information and commands that Oliver be recruited to find his brother.
Summary: Act II, scene iii
Orlando returns to his former home, where the servant Adam greets him. News of the young man’s victory over Charles precedes him, and Adam worries that Orlando’s strength and bravery will be the keys to his downfall. Adam begs Orlando not to enter Oliver’s house. Oliver, he reports, having learned of Orlando’s triumph, plans to burn the place where Orlando sleeps in hopes of destroying Orlando with it. “Abhor it,” Adam warns, “fear it, do not enter it” (II.iii.29). Orlando wonders about his fate, speculating that without a home, he may be destined to eke out a living as a common highway robber. Adam suggests that the two of them take to the road with his modest life’s savings. Touched by Adam’s constant service, Orlando agrees.
Summary: Act II, scene iv
Rosalind, Celia, and Touchstone arrive, safe but exhausted, in the Forest of Ardenne. The three sit down to rest, but before long they are interrupted by two shepherds: young Corin and old Silvius. The shepherds are so wrapped up in their conversation about Silvius’s hopeless love and devotion to the shepherdess Phoebe that they do not notice the three travelers. Corin, who claims to have loved a thousand times, tries to advise Silvius, but the young man, maintaining that his companion could not possibly understand the depth of his feelings, wanders off. Rosalind, Celia, and Touchstone approach Corin and ask where they might find a place to rest. When Corin admits that his master’s modest holdings are up for sale, Rosalind and Celia decide to buy the property.
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Act II, scenes v–vii
Summary: Act II, scene v
As Amiens strolls through the Forest of Ardenne with Jaques in tow, he sings a song inviting his listeners to lie with him “[u]nder the greenwood tree” (II.v.1), where there are no enemies but “winter and rough weather” (II.v.8). Jaques begs him to continue, but Amiens hesitates, claiming that the song will only make Jaques melancholy. The warning does not deter Jaques, who proudly claims that he can “suck melancholy out of a song as a weasel sucks eggs” (II.v.11–12). While the other lords in attendance prepare for Duke Senior’s meal, Amiens leads them in finishing the song. Jaques follows with a verse set to the same tune, which he himself wrote. In it, he chides those foolish enough to leave their wealth and leisure for life in the forest. Amiens leaves to summon the duke to dinner.
Summary: Act II, scene vi
Orlando and Adam enter the Forest of Ardenne. Adam is exhausted from travel and claims that he will soon die from hunger. Orlando assures his loyal servant that he will find him food. Before he sets off to hunt, Orlando fears leaving Adam lying in “the bleak air” and carries him off to shelter (II.vi.12).
Summary: Act II, scene vii
And so from hour to hour we ripe and ripe,
And then from hour to hour we rot and rot;
And thereby hangs a tale.
(See Important Quotations Explained)
Duke Senior returns to camp to find that Jaques has disappeared. When a lord reports that Jaques has last been seen in good spirits, the duke worries that happiness in one who is typically so miserable portends discord in the universe. Just after the duke commands the lord to find Jaques, Jaques appears. He is uncharacteristically merry and explains that while wandering through the forest, he met a fool. He repeats the fool’s witty observations about Lady Fortune and proclaims that he himself would like to be a fool. In this position, Jaques reasons, he would be able to speak his mind freely, thereby cleansing “the foul body of th’infected world” with the “medicine” of his criticism (II.vii.60–61). The duke laments the sin of “chiding sin” and reminds Jaques that he himself is guilty of many of the evils he would inevitably criticize in others (II.v.64). Their playful argument is interrupted when Orlando barges onto the scene, drawing his sword and demanding food. The duke asks whether Orlando’s rudeness is a function of distress or bad breeding and, once Orlando has regained his composure, invites him to partake of the banquet. Orlando goes off to fetch Adam. Duke Senior observes that he and his men are far from alone in their unhappiness: there is much strife in the world. Jaques replies that the world is a stage and “all the men and women merely players” (II.vii.139). All humans pass through the stages of infancy, childhood, and adulthood; they experience love and seek honor, but all eventually succumb to the debility of old age and “mere oblivion” (II.vii.164). Orlando returns with Adam and all begin to eat. The duke soon realizes that Orlando is the son of Sir Rowland, the duke’s old friend, and heartily welcomes the young man.
Analysis: Act II, scenes v–vii
Both Act II, scene v and Act II, scene vi deal primarily with the melancholy lord, Jaques, who offers a sullen perspective on the otherwise comedic events in Ardenne. He turns Amiens’s song about the pleasures of leisurely life into a means of berating the foresters, and he comes close to playing the part of the fool, in the sense that he turns a critical eye on a world in which he lives but does not fully inhabit. But unlike Feste in Twelfth Night or the fool in King Lear,Jaques does not demonstrate the insight or wisdom that would make his observations truly arresting or illuminating. His most impressive speech in the play begins with a familiar set piece in Elizabethan drama: “All the world’s a stage, / And all the men and women merely players” (II.vii.138–139). He goes on to describe the seven stages of a man’s life, from infancy to death, through his roles as lover and soldier, but Jaques’s observations may strike us as untrue or banal. His estimation that lovers sigh “like furnace, with a woeful ballad / Made to his mistress’ eyebrow” is humorous, and it certainly describes the kind of intemperate, undiscriminating affection that Silvius shows to Phoebe, or Phoebe to Ganymede (II.vii.147–148). But the criticism seems ill-suited to a play as aware and forgiving of love’s silliness as As You Like It.As a philosopher, Jaques falls short of accurately describing the complexity of Rosalind’s feelings for Orlando; his musings bear the narrow and pinched shortcomings of the habitually sullen.
Jaques’s sullenness blinds him to his own foolishness regarding life. Jaques goes on to describe man’s later years, the decline into second childhood and obliviousness, without teeth, eyesight, taste, or anything else. Countering Jaques’s unflattering picture of old age, Orlando carries Adam to the duke’s banquet table, the old man entering his final years with his loyalty, generosity of spirit, and appetite intact. Although the thought of serving as Duke Frederick’s fool appeals to him, Jaques ultimately lacks the wit, wisdom, and heart to perform the task. When he meets Touchstone in the forest, he sings the clown’s praises, quoting with glee Touchstone’s nihilistic musings on the passage of time: “And so from hour to hour we ripe and ripe, / And then from hour to hour we rot and rot” (II.vii.26–27). Jaques does not realize that Touchstone’s “deep--contemplative” speech is a bawdy mockery of his own brooding behavior (II.vii.31). Indeed, throughout the play, Jaques remains so mired in his own moodiness that he sees very little of the world he so desperately wants to criticize. Knowing that Jaques’s eyes are trained on men’s baser instincts, the duke doubts Jaques’s ability to serve as a proper and entertaining fool. Jaques, he feels, would be a boor, berating the courtiers for sins that Jaques himself has committed. This exchange points to an important difference between Jaques and the duke: the former is committed to being unhappy in the world and will suffer in it, while the latter is happy to make the best of the world he is given and will thrive, as the title of the play seems to promise.

Study Questions
1. By putting on male clothes and adopting a masculine swagger, Rosalind easily passes as a man throughout the better part of the play. What does her behavior suggest about gender? Does the play imply that notions of gender are fixed or fluid? Explain.
Answer for Study Question 1 >>
Rosalind’s behavior suggests that she knows better than anyone else that her society makes different demands of men and women. For instance, she knows that, when dressed as Ganymede, she is forbidden from crying over a perceived slight from Orlando. Likewise, something as simple as a “doublet and hose”—her male disguise—stops her from celebrating the discovery that Orlando has authored love poems in her honor (III.ii.200–201). Indeed, as the clothes make the man, they also make the woman act like one. To Elizabethans, the fundamental divide between the sexes may have been as much a matter of external expressions of behavior and clothing as of anatomy. On one hand, this conception made gender a much more fluid notion than it is to many modern audiences. Codes of behavior were more a matter of mimicry than a function of chromosomal makeup, which Rosalind shows as she plays a swaggering young man imitating a woman.
On the other hand, this fluidity caused a great deal of anxiety among Elizabethans, who, in the end, wanted very much to believe that the categories that organized their world were stable. Thus, they insisted that certain behaviors and customs were established by one’s sex. Women might pretend to be men for a brief and entertaining moment, but they must, in the end, behave like women. Rosalind eases the anxieties surrounding her very deft performance by reverting, time and again, to the behaviors expected of her as a woman: to the Elizabethan mind, she would be a much more troubling character if she did not faint at the sight of Orlando’s blood. Although gender proves to be completely undefined in the Forest of Ardenne, everyone is returned to his or her supposedly proper place by the final act. Indeed, nowhere is the anxiety over gender--swapping quelled more than in the Epilogue, where the actor playing Rosalind, who is herself so talented at role-playing, unveils himself as an actor, thereby promising that with his bow comes an end to subversion and a return to the established social order.
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Character List
Rosalind - The daughter of Duke Senior. Rosalind, considered one of Shakespeare’s most delightful heroines, is independent minded, strong-willed, good-hearted, and terribly clever. Rather than slink off into defeated exile, Rosalind resourcefully uses her trip to the Forest of Ardenne as an opportunity to take control of her own destiny. When she disguises herself as Ganymede—a handsome young man—and offers herself as a tutor in the ways of love to her beloved Orlando, Rosalind’s talents and charms are on full display. Only Rosalind, for instance, is both aware of the foolishness of romantic love and delighted to be in love. She teaches those around her to think, feel, and love better than they have previously, and she ensures that the courtiers returning from Ardenne are far gentler than those who fled to it.
Read an in-depth analysis of Rosalind.
Orlando - The youngest son of Sir Rowland de Bois and younger brother of Oliver. Orlando is an attractive young man who, under his brother’s neglectful care, has languished without a gentleman’s education or training. Regardless, he considers himself to have great potential, and his victorious battle with Charles proves him right. Orlando cares for the aging Adam in the Forest of Ardenne and later risks his life to save Oliver from a hungry lioness, proving himself a proper gentleman. He is a fitting hero for the play and, though he proves no match for her wit or poetry, the most obvious romantic match for Rosalind.
Read an in-depth analysis of Orlando.
Duke Senior - The father of Rosalind and the rightful ruler of the dukedom in which the play is set. Having been banished by his usurping brother, Frederick, Duke Senior now lives in exile in the Forest of Ardenne with a number of loyal men, including Lord Amiens and Jaques. We have the sense that Senior did not put up much of a fight to keep his dukedom, for he seems to make the most of whatever life gives him. Content in the forest, where he claims to learn as much from stones and brooks as he would in a church or library, Duke Senior proves himself to be a kind and fair-minded ruler.
Jaques - A faithful lord who accompanies Duke Senior into exile in the Forest of Ardenne. Jaques is an example of a stock figure in Elizabethan comedy, the man possessed of a hopelessly melancholy disposition. Much like a referee in a football game, he stands on the sidelines, watching and judging the actions of the other characters without ever fully participating. Given his inability to participate in life, it is fitting that Jaques alone refuses to follow Duke Senior and the other courtiers back to court, and instead resolves to assume a solitary and contemplative life in a monastery.
Read an in-depth analysis of Jaques.
Celia - The daughter of Duke Frederick and Rosalind’s dearest friend. Celia’s devotion to Rosalind is unmatched, as evidenced by her decision to follow her cousin into exile. To make the trip, Celia assumes the disguise of a simple shepherdess and calls herself Aliena. As elucidated by her extreme love of Rosalind and her immediate devotion to Oliver, whom she marries at the end of the play, Celia possesses a loving heart, but is prone to deep, almost excessive emotions.
Duke Frederick - The brother of Duke Senior and usurper of his throne. Duke Frederick’s cruel nature and volatile temper are displayed when he banishes his niece, Rosalind, from court without reason. That Celia, his own daughter, cannot mitigate his unfounded anger demonstrates the intensity of the duke’s hatefulness. Frederick mounts an army against his exiled brother but aborts his vengeful mission after he meets an old religious man on the road to the Forest of Ardenne. He immediately changes his ways, dedicating himself to a monastic life and returning the crown to his brother, thus testifying to the ease and elegance with which humans can sometimes change for the better.
Touchstone - A clown in Duke Frederick’s court who accompanies Rosalind and Celia in their flight to Ardenne. Although Touchstone’s job, as fool, is to criticize the behavior and point out the folly of those around him, Touchstone fails to do so with even a fraction of Rosalind’s grace. Next to his mistress, the clown seems hopelessly vulgar and narrow-minded. Almost every line he speaks echoes with bawdy innuendo.
Oliver - The oldest son of Sir Rowland de Bois and sole inheritor of the de Bois estate. Oliver is a loveless young man who begrudges his brother, Orlando, a gentleman’s education. He admits to hating Orlando without cause or reason and goes to great lengths to ensure his brother’s downfall. When Duke Frederick employs Oliver to find his missing brother, Oliver finds himself living in despair in the Forest of Ardenne, where Orlando saves his life. This display of undeserved generosity prompts Oliver to change himself into a better, more loving person. His transformation is evidenced by his love for the disguised Celia, whom he takes to be a simple shepherdess.
Silvius - A young, suffering shepherd, who is desperately in love with the disdainful Phoebe. Conforming to the model of Petrarchan love, Silvius prostrates himself before a woman who refuses to return his affections. In the end, however, he wins the object of his desire.
Phoebe - A young shepherdess, who disdains the affections of Silvius. She falls in love with Ganymede, who is really Rosalind in disguise, but Rosalind tricks Phoebe into marrying Silvius.
Lord Amiens - A faithful lord who accompanies Duke Senior into exile in the Forest of Ardenne. Lord Amiens is rather jolly and loves to sing.
Charles - A professional wrestler in Duke Frederick’s court. Charles demonstrates both his caring nature and his political savvy when he asks Oliver to intercede in his upcoming fight with Orlando: he does not want to injure the young man and thereby lose favor among the nobles who support him. Charles’s concern for Orlando proves unwarranted when Orlando beats him senseless.
Adam - The elderly former servant of Sir Rowland de Bois. Having witnessed Orlando’s hardships, Adam offers not only to accompany his young master into exile but to fund their journey with the whole of his modest life’s savings. He is a model of loyalty and devoted service.
Sir Rowland de Bois - The father of Oliver and Orlando, friend of Duke Senior, and enemy of Duke Frederick. Upon Sir Rowland’s death, the vast majority of his estate was handed over to Oliver according to the custom of primogeniture.
Corin - A shepherd. Corin attempts to counsel his friend Silvius in the ways of love, but Silvius refuses to listen.
Audrey - A simpleminded goatherd who agrees to marry Touchstone.
William - A young country boy who is in love with Audrey.

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Similarity
Similarity is much like congruence, except in order for polygons to be similar, they only need to have the same shape. As we did with congruence, we will study similarity in triangles to simplify things. Formally speaking, two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional. For example, if triangles ABC and DEF are similar, then angle pairs AB and DE, BC and EF, and CA and FE are all equal. Also, AB/DE=BC/EF=CA/FD. If these three ratios are equal, then the corresponding sides are said to be proportional.

Figure %: Triangles ABC, DEF, and HIJ are similar

An easy to way to create similar triangles is by drawing a line through a triangle such that it intersects with two sides and is parallel to the third side. This new line will form a new triangle that is smaller that the original, but similar to it.

Figure %: Similar triangles
The line l , parallel to AC, creates the triangle DEB, which is similar to triangle ACB. Another thing about the above diagram: because the two triangles ACB and DEB are similar, DB/AB = EB/CB. We also know that AD/DB = CE/EB. This last relationship is just another application of similarity in triangles. In the next lesson we'll see how to prove triangles are simi lar.
Geometry: Congruence

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Problems
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Problems
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Proving Similarity of Triangles
There are three easy ways to prove similarity. These techniques are much like those employed to prove congruence--they are methods to show that all corresponding angles are congruent and all corresponding sides are proportional without actually needing to know the measure of all six parts of each triangle.
AA (Angle-Angle)
If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. We know this because if two angle pairs are the same, then the third pair must also be equal. When the three angle pairs are all equal, the three pairs of sides must also be in proportion. Picture three angles of a triangle floating around. If they are the vertices of a triangle, they don't determine the size of the triangle by themselves, because they can move farther away or closer to each other. But when they move, the triangle they create always retains its shape. Thus, they always form similar triangles. The diagram below makes this much more clear.

Figure %: Three pairs of congruent angles determine similar triangles
In the above figure, angles A, B, and C are vertices of a triangle. If one angle moves, the other two must move in accordance to create a triangle. So with any movement, the three angles move in concert to create a new triangle with the same shape. Hence, any triangles with three pairs of congruent angles will be similar. Also, note that if the three vertices are exactly the same distance from each other, then the triangle will be congruent. In other words, congruent triangles are a subset of similar triangles.
SSS (Side-Side-Side)
Another way to prove triangles are similar is by SSS, side-side-side. If the measures of corresponding sides are known, then their proportionality can be calculated. If all three pairs are in proportion, then the triangles are similar.

Figure %: If all three pairs of sides of corresponding triangles are in proportion, the triangles are similar
SAS (Side-Angle-Side)
If two pairs of corresponding sides are in proportion, and the included angle of each pair is equal, then the two triangles they form are similar. Any time two sides of a triangle and their included angle are fixed, then all three vertices of that triangle are fixed. With all three vertices fixed and two of the pairs of sides proportional, the third pair of sides must also be proportional.

Figure %: Two pairs of proportional sides and a pair of equal included angles determines similar triangles
Conclusion
These are the main techniques for proving congruence and similarity. With these tools, we can now do two things. * Given limited information about two geometric figures, we may be able to prove their congruence or similarity. * Given that figures are congruent or similar, we can deduce information about their corresponding parts that we didn't previously know.
The link between the corresponding parts of a triangle and the whole triangle

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Terms
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Congruence
Much of the study of geometry that we've done so far has consisted of defining terms and describing charateristics of various figures and their special cases. All of this study lays a foundation for one of the most important applications of geometry: proving shapes and figures are congruent. We've already discussed the congruence of segments and angles, but in the real world the congruence of regions in a plane is even more relevant. Since we can't easily prove the congruence of any region in the plane, we'll focus on simpler regions like those bound by polygons. And, like always, the study of polygons results in the study of triangles.
For two polygons to be congruent, they must have exactly the same size and shape. This means that their interior angles and sides must all be congruent. Not only must these parts be congruent, but they must be situated in a one-to- one correspondence, meaning each side in one polygon specifically corresponds to another side in the other polygon, and each pair of parts is congruent. To prove such a situation would be a tough task. That's why studying the congruence of triangles is so important--it allows us to draw conclusions about the congruence of polygons, too. We'll see how the six parts of a triangle correspond to one another, and how they must be aligned to signify congruence. We'll also study some techniques--shortcuts, really--to prove the congruence of triangles. We'll only work on informal proofs, the study of formal proofs in geometry will have to wait until the SparkNotes in Geometry 3. Finally, we'll take a look at similarity between triangles. Similarity is a lot like congruence, except it only requires the same shape, not size. After this section, we can focus on refining our skills for proving congruence. For now, we'll have to learn exactly what it means.
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Proving Similarity of Triangles
There are three easy ways to prove similarity. These techniques are much like those employed to prove congruence--they are methods to show that all corresponding angles are congruent and all corresponding sides are proportional without actually needing to know the measure of all six parts of each triangle.
AA (Angle-Angle) If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. We know this because if two angle pairs are the same, then the third pair must also be equal. When the three angle pairs are all equal, the three pairs of sides must also be in proportion. Picture three angles of a triangle floating around. If they are the vertices of a triangle, they don't determine the size of the triangle by themselves, because they can move farther away or closer to each other. But when they move, the triangle they create always retains its shape. Thus, they always form similar triangles. The diagram below makes this much more clear.

Figure %: Three pairs of congruent angles determine similar triangles
In the above figure, angles A, B, and C are vertices of a triangle. If one angle moves, the other two must move in accordance to create a triangle. So with any movement, the three angles move in concert to create a new triangle with the same shape. Hence, any triangles with three pairs of congruent angles will be similar. Also, note that if the three vertices are exactly the same distance from each other, then the triangle will be congruent. In other words, congruent triangles are a subset of similar triangles.
SSS (Side-Side-Side)
Another way to prove triangles are similar is by SSS, side-side-side. If the measures of corresponding sides are known, then their proportionality can be calculated. If all three pairs are in proportion, then the triangles are similar.

Figure %: If all three pairs of sides of corresponding triangles are in proportion, the triangles are similar
SAS (Side-Angle-Side)
If two pairs of corresponding sides are in proportion, and the included angle of each pair is equal, then the two triangles they form are similar. Any time two sides of a triangle and their included angle are fixed, then all three vertices of that triangle are fixed. With all three vertices fixed and two of the pairs of sides proportional, the third pair of sides must also be proportional.

Figure %: Two pairs of proportional sides and a pair of equal included angles determines similar triangles
Conclusion
These are the main techniques for proving congruence and similarity. With these tools, we can now do two things. * Given limited information about two geometric figures, we may be able to prove their congruence or similarity. * Given that figures are congruent or similar, we can deduce information about their corresponding parts that we didn't previously know.
The link between the corresponding parts of a triangle and the whole triangle is a two-way street, and we can go in whichever direction we want.
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Problems
Problem : If triangle ABC is similar to triangle DEF, which side is proportional to side BC?
Solution for Problem 1 >>
Side EF.
Close

Problem : Triangles ABC and DEC are similar, and line l is parallel to segment AB. What is the length of CE?

Solution for Problem 2 >>
8
Close

Problem : Can two triangles be similar but not congruent? Can they be congruent but not similar?
Solution for Problem 3 >>
Every pair of congruent triangles are similar, but every pair of similar triangles isn't necessarily congruent.
Close

Problem : When a line is drawn through an equilateral triangle such that it is parallel to one side and intersects one time with each of the other two sides, how many similar pairs of triangles are created?
Solution for Problem 4 >>
Thirty-six. If the equilateral triangle is triangle ABC, and the line intersects with sides AC and BC, then a new triangle, DFC, is created. Let point D be the intersection point of the line and side AC and let point E be the intersection point of side BC. Then triangle DFC is similar to triangles ABC, BCA, CAB, ACB, BAC, and CBA. The five other triangles, FCD, CDF, DCF, FDC, and CFD are also all similar to these six triangles, making 36 pairs of similar triangles.
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