This supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry.


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1 Mat 444/445 Geometry for Teacers Summer 2008 Supplement : Similar Triangles Tis supplement is meant to be read after Venema s Section 9.2. Trougout tis section, we assume all nine axioms of uclidean geometry. Similar Triangles Te idea of scaling geometric objects is ubiquitous in our experience. Wen you draw a map to scale, or enlarge a poto, or tell your computer to use a larger font size, you are creating a new geometric object tat as te same sape as te old one, but as all of its parts reduced or enlarged in size or scaled by te same ratio. In geometry, two figures tat ave te same sape but not necessarily te same size are said to be similar to eac oter. (We will give a more precise matematical definition below.) To analyze tis concept in te context of axiomatic uclidean geometry, let us start wit triangles, te simplest geometric figures. Tere are two separate tings tat we migt expect to be te case wen two triangles are similar (Fig. 1): First, all tree pairs of corresponding angles sould be congruent (wic means F Figure 1: Similar Triangles. te triangles ave te same sape ), and second, te lengts of pairs of corresponding sides sould all ave te same ratio (wic means tey ave proportional sizes ). In some igscool geometry texts, including tat of Jacobs, te definition of similar triangles includes bot of tese properties. However, it is a fact (wic we will prove below) tat eac of tese conditions implies te oter. For us, it will be easier to coose one of tem as our official definition of similar triangles. Tus we make te following definition: Two triangles are said to be similar if tere is a correspondence between teir vertices suc tat corresponding angles are congruent. Te notation F means tat is similar to F under te correspondence,, and F, or more specifically tat =, =, = F. Te first ting to notice is tat in uclidean geometry, it is only necessary to ceck tat two of te corresponding angles are congruent. Teorem.1 ( Similarity Teorem). If and F are triangles suc tat = and =, ten F. Proof. Under tese ypoteses, it follows immediately from te nglesum Teorem tat = F. Te next teorem sows tat similar triangles can be readily constructed in uclidean geometry, once a new size is cosen for one of te sides. It is an analogue for similar triangles of Venema s Teorem Teorem.2 (Similar Triangle onstruction Teorem). If is a triangle, is a segment, and H is a alfplane bounded by, ten tere is a unique point F H suc tat F. 1
2 H Q P F Figure 2: onstruction of a triangle similar to a given one. Proof. Te ngle onstruction Teorem ensures tat tere exists a unique ray P lying in H suc tat P = (Fig. 2). Similarly, tere exists a unique ray Q in H suc tat Q =. ecause te measures of and sum to less tan 180 by orollary.9, it follows tat te measures of P and Q also sum to less tan 180. y uclid s Postulate V, terefore, tere is a point F on tat same side of were te lines P and Q intersect. y te Similarity Teorem, F. To prove uniqueness, just note tat if F is any oter point satisfying te conclusion of te teorem, ten te fact tat F = implies tat F must lie on P by te uniqueness part of te ngle onstruction Teorem, and similarly F must lie on Q. Since tese two rays intersect only at F, it follows tat F = F. It is important to observe tat altoug it is always possible in neutral geometry to construct a triangle congruent to a given one, as Teorem sowed, tis construction of similar triangles only works in uclidean geometry because it requires uclid s Postulate V. In fact, we will see later tat in yperbolic geometry, it is impossible to construct noncongruent similar triangles! Our main order of business in tis section is to sow tat similar triangles ave proportional sides. Our proof of tis fact is modeled on tat of uclid (wic e carries out in ook VI), using te teory of area. Tus before we get to our teorem about similar triangles, let us establis two simple facts about areas of triangles. Te first of tese is uclid s Proposition I.37, wile te second is is first proposition in ook VI. For us, tey are bot simple consequences of te area formula for triangles. Lemma.3. Suppose and are two distinct triangles tat ave a common side, suc tat (Fig. 3). Ten α( ) = α( ). Figure 3: Lemma.3. Proof. Let and be te feet of te perpendiculars from and, respectively, to. ecause and are bot perpendicular to, tey are parallel to eac oter by orollary It ten follows tat is a parallelogram (actually a rectangle, but all we need to know is tat it is a parallelogram), and 2
3 so = by Teorem.19. Set = =, so tat is te eigt of bot triangles and. It ten follows from te area formula for triangles tat α( ) = 1 2 () = α( ). Lemma.4. Suppose is a triangle, and is a point suc tat (Fig. 4). Ten α( ) α( ) =. Figure 4: Lemma.4. Proof. Note tat bot and ave te same eigt, wic is just te distance from to. Terefore, te formula for te area of a triangle sows tat as claimed. 1 α( ) α( ) = 2 =, 1 2 Our main tool for analyzing proportionality in similar triangles will be te following teorem, wic sows tat a line parallel to one side of a triangle cuts off proportional segments from te oter two sides. Teorem.5 (Te SideSplitter Teorem). Suppose is a triangle, and l is a line parallel to tat intersects at an interior point (Fig. 5). Ten l also intersects at an interior point, and =. l Figure 5: Te SideSplitter Teorem. 3
4 Figure 6: Proof of te SideSplitter Teorem. Proof. ecause l is parallel to, it does not contain or ; and because it intersects at an interior point, it does not contain eiter. Terefore, Pasc s Teorem guarantees tat l also intersects te interior of one oter side of. Since it cannot intersect, it must intersect te interior of. Let denote te intersection point. raw, and consider (Fig. 6). From Lemma.4, we conclude tat α( ) α( ) =. (.1) Similarly, drawing and considering, we obtain α( ) α( ) =. (.2) It follows from Teorem and additivity of area tat α( ) = α( ) + α( ), α( ) = α( ) + α( ). ut it follows from Lemma.3 tat α( ) = α( ), and terefore tat α( ) = α( ). (.3) ombining (.1), (.2), and (.3), we obtain as desired. = α( ) α( ) = α( ) α( ) = Now we come to te main teorem of tis section, wic says tat similar triangles ave proportional corresponding sides. It is often called simply te Similar Triangles Teorem. Teorem.6 (Fundamental Teorem on Similar Triangles). If F, ten = F = F. Proof. Suppose F. If =, ten is congruent to F by SS, and te teorem is true because all te ratios in (.4) are equal to 1. So let us suppose tat. One of tem is larger, say >. We will prove te first equality in (.4); te proof of te oter equality is exactly te same. oose a point P in te interior of suc tat P =, and let l be te line troug P and parallel to F (Fig. 7). It follows from te SideSplitter Teorem tat l intersects F at an interior point Q, and (.4) 4
5 P Q l F Figure 7: Proof of te Similar Triangles Teorem. tat P = Q F. (.5) y te converse to te orresponding ngles Teorem, PQ =, wic by ypotesis is congruent in turn to. It also follows from te ypotesis tat =. Since P = by construction, we ave PQ = by SS. Substituting P = and Q = into (.5), we obtain te first equation in (.4). Tis is one of te most important teorems in uclidean geometry. Nearly every geometry book as some version of it, but you will find it treated differently in different books. For example, in a textbook tat includes proportionality of te sides as part of te definition of similar triangles, tis teorem would be reprased to say tat if two triangles ave congruent corresponding angles, ten tey are similar. Some textbooks, for some reason, do not prove tis teorem at all, but instead take it as an additional postulate (often called te Similarity Postulate or some suc ting). We ave followed te lead of uclid (as does Jacobs) in using te teory of area to prove tis teorem. In fact, it is possible to give a proof tat does not use areas at all (and terefore does not require eiter te Neutral rea xiom or te uclidean rea xiom). Venema gives suc a proof in Sections , wic you migt wis to read if you re curious. Tat proof is considerably more involved tan te one we ave given ere; because of tat, and because te areabased proof is perfectly rigorous and is muc more likely to be found in igscool texts, we ave cosen to stick wit uclid s approac. Tere is little to be gained by avoiding te use of area in te treatment of similar triangles. Tere are many useful consequences of te Similar Triangles Teorem. Te first one is really just a simple reprasing of te teorem. orollary.7. If F, ten tere is a positive number r suc tat = r, = r F, = r F. Proof. Just define r to be te ratio /, and use (.4). Teorem.8 (SS Similarity Teorem). If and F are triangles suc tat = and / = /F, ten F. Proof. xercise.1. Teorem.9 (SSS Similarity Teorem). If and F are triangles suc tat / = /F = /F, ten F. Proof. xercise.2. Note tat te preceding teorem is actually te converse to te Similar Triangles Teorem. Taken togeter, tese two teorems say tat two triangles ave equal corresponding angles if and only if tey ave proportional corresponding sides. 5
6 Teorem.10 (rea Scaling Teorem). If two triangles are similar, ten te ratio of teir areas is te square of te ratio of any two corresponding sides; tat is, if F and = r, ten α( ) = r 2 α( F). Proof. xercise.3. Finally, using te teory of similar triangles, we can give yet anoter proof of te Pytagorean Teorem. (Since it is one of te most significant teorems in all of matematics, it never urts to see anoter proof!) s usual, we will label te vertices of our rigt triangle,, and, wit te rigt angle at ; and we will denote te lengt of te leg opposite by a, te lengt of te leg opposite by b, and te lengt of te ypotenuse by c (Fig. 8). b a c Figure 8: Te Pytagorean Teorem: a 2 + b 2 = c 2. Teorem.11 (Te Pytagorean Teorem). Suppose is a rigt triangle wit rigt angle, and let a =, b =, and c =. Ten a 2 + b 2 = c 2. Proof. Let be te foot of te perpendicular from to (Fig. 9). Notice tat cannot coincide wit b a x y c Figure 9: Proof of te Pytagorean Teorem. Figure 10: cannot lie outside. or, for ten would ave a second rigt angle at or, contradicting orollary.9. lso, because and are acute, cannot be outside of for example, if, ten would be an exterior angle for tat is smaller tan te remote interior angle at (Fig. 10), contradicting te xterior ngle Teorem. Tus is an interior point of. Set x =, y =, and =. y te nglesum Teorem applied to te rigt triangles and, µ + µ = 90 and µ + µ = 90. It follows by algebra tat =, so by te Similarity Teorem, we conclude tat. Te Similar Triangles Teorem ten gives, among oter tings, a c = y a. 6
7 Te same argument also sows tat, so b c = x b. Simplifying bot equations and adding tem togeter, we obtain Since x + y = c, tis proves te teorem. a 2 = cy, b 2 = cx, a 2 + b 2 = c(x + y). Our last teorem in tis section is te last teorem in uclid s ook I. Teorem.12 (onverse to te Pytagorean Teorem). Suppose is a triangle, wit side lengts a =, b =, and c =. If a 2 + b 2 = c 2, ten is a rigt angle. Proof. xercise.4. xercises.1. Prove Teorem.8 (te SS Similarity Teorem). [Hint: If <, coose a point P in te interior of suc tat P =, and let l be te line troug P parallel to F. Prove tat l intersects te interior of F at a point Q, and use te ypotesis, te SideSplitter Teorem, and some algebra to sow tat = PQ.].2. Prove Teorem.9 (te SSS Similarity Teorem). [Hint: Te proof is very similar to tat of te SS Similarity Teorem.].3. Prove Teorem.10 (te rea Scaling Teorem). [Hint: rop a perpendicular from a vertex to te opposite side in eac triangle, and use te Similarity Teorem to sow tat tis forms two pairs of similar triangles.].4. Prove Teorem.12 (te converse to te Pytagorean Teorem). [Hint: onstruct a rigt triangle wose legs ave lengts a and b, and sow tat it is congruent to.] 7
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