Further Concepts in Geometry

Transcription

1 ppendix F Further oncepts in Geometry F. Exporing ongruence and Simiarity Identifying ongruent Figures Identifying Simiar Figures Reading and Using Definitions ongruent Trianges assifying Trianges Identifying ongruent Figures Two figures are congruent if they have the same shape and the same size. Each of the trianges in Figure F. is congruent to each of the other trianges. The trianges in Figure F. are not congruent to each other. ongruent Figure F. Not ongruent Figure F. Notice that two figures can be congruent without having the same orientation. If two figures are congruent, then either one can be moved (and turned or fipped if necessary) so that it coincides with the other figure. Exampe Dividing Regions into ongruent Parts Divide the region into two congruent parts. Soution There are many soutions to this probem. Some of the soutions are shown in Figure F.3. an you think of others? F

2 F ppendix F Further oncepts in Geometry Figure F.3 Identifying Simiar Figures Two figures are simiar if they have the same shape. (They may or may not have the same size.) Each of the quadriateras in Figure F.4 is simiar to the others. The quadriateras in Figure F.5 are not simiar to each other. Simiar Figure F.4 Exampe Determining Simiarity Not Simiar Figure F.5 Two of the figures are simiar. Which two are they? (a) (b) (c) Soution The first figure has five sides and the other two figures have four sides. ecause simiar figures must have the same shape, the first figure is not simiar to either of the others. ecause you are tod that two figures are simiar, it foows that the second and third figures are simiar. Exampe 3 Determining Simiarity You wrote an essay on Eucid, the Greek mathematician who is famous for writing a geometry book tited Eements of Geometry. You are making a copy of the essay using a photocopier that is set at 75% reduction. Is each image on the copied pages simiar to its origina? Soution Every image on a copied page is simiar to its origina. The copied pages are smaer, but that doesn t matter because simiar figures do not have to be the same size.

3 PPENDIX F Further oncepts in Geometry F3 Exampe 4 Drawing an Object to Scae 9 4 in. You are drawing a foor pan of a buiding. You choose a scae of 8 inch to foot. That is, 8 inch of the foor pan represents foot of the actua buiding. What dimensions shoud you draw for a room that is feet wide and 8 feet ong? Soution ecause each foot is represented as inch, the width of the room shoud be and the ength of the room shoud be The scae dimensions of the room are inches by inches. See Figure F.6. 4 Figure F.6 3 in. Reading and Using Definitions definition uses known words to describe a new word. If no words were known, then no new words coud be defined. So, some words such as point, ine, and pane must be commony understood without being defined. Some statements such as a point ies on a ine and point ies between points and are aso not defined. See Figure F.7. D is between and. D is not between and. Figure F.7 Segments and Rays onsider the ine that contains the points and. (In geometry, the word ine means a straight ine.) The ine segment (or simpy segment) consists of the endpoints and and a points on the ine that ie between and. The ray consists of the initia point and a points on the ine that ie on the same side of as ies. If is between and, then and are opposite rays. Points, segments, or rays that ie on the same ine are coinear. Lines are drawn with two arrowheads, ine segments are drawn with no arrowhead, and rays are drawn with a singe arrowhead. See Figure F.8. Line or Line segment or Ray (a) (b) (c) Ray (d) Figure F.8 Opposite rays and (e)

4 F4 ppendix F Further oncepts in Geometry O P S R The top ange can be denoted by or by. In the ower figure, the ange ROS shoud not be denoted by O because the figure contains three anges whose vertex is O. Figure F.9 It foows that and denote the same segment, but and do not denote the same ray. No ength is given to ines or rays because each is infinitey ong. The ength of the ine segment is denoted by. nges n ange consists of two different rays that have the same initia point. The rays are the sides of the ange. The ange that consists of the rays and is denoted by,, or. The point is the vertex of the ange. See Figure F.9. The measure of is denoted by m. nges are cassified as acute, right, obtuse, and straight. cute Right Obtuse Straight 0 < m < 90 m < m < 80 m 80 3 In geometry, uness specificay stated otherwise, anges are assumed to have a measure that is greater than 0 and ess than or equa to 80. Every nonstraight ange has an interior and an exterior. point D is in the interior of if it is between points that ie on each side of the ange. Two anges (such as ROS and SOP shown in Figure F.9) are adjacent if they share a common vertex and side, but have no common interior points. In Figure F.0, and 3 share a vertex, but not a common side, so and 3 are not adjacent. and are adjacent. and 3 are not adjacent. Figure F.0 Segment and nge ongruence Two segments are congruent, D, if they have the same ength. Two anges are congruent, P Q, if they have the same measure. D Q P D m P m Q Definitions can aways be interpreted forward and backward. For instance, the definition of congruent segments means () if two segments have the same measure, then they are congruent, and () if two segments are congruent, then they have the same measure. You earned that two figures are congruent if they have the same shape and size.

5 ongruent Trianges PPENDIX F Further oncepts in Geometry F5 If is congruent to PQR, then there is a correspondence between their anges and sides such that corresponding anges are congruent and corresponding sides are congruent. The notation PQR indicates the congruence and the correspondence, as shown in Figure F.. If two trianges are congruent, then you know that they share many properties. PQR orresponding anges are P Q R orresponding sides are PQ QR RP Q Figure F. R P Exampe 5 Naming ongruent Parts You and a friend have identica drafting trianges, as shown in Figure F.. Name a congruent parts. E S D T Figure F. F R Soution Given that DEF RST, the congruent anges and sides are as foows. nges: D R, E S, F T Sides: DE RS, EF ST, FD TR

6 F6 ppendix F Further oncepts in Geometry assifying Trianges triange can be cassified by reationships among its sides or among its anges, as shown in the foowing definitions. assification by Sides. n equiatera triange has three congruent sides.. n isoscees triange has at east two congruent sides. 3. scaene triange has no sides congruent. Equiatera Isoscees Scaene assification by nges. n acute triange has three acute anges. If these anges are a congruent, then the triange is aso equianguar.. right triange has exacty one right ange. 3. n obtuse triange has exacty one obtuse ange. cute Equianguar Right Obtuse In, each of the points,, and is a vertex of the triange. (The pura of vertex is vertices.) The side is the side opposite. Two sides that share a common vertex are adjacent sides (see Figure F.3). Opposite side of djacent sides of Figure F.3 Vertex

7 PPENDIX F Further oncepts in Geometry F7 The sides of right trianges and isoscees trianges are given specia names. In a right triange, the sides adjacent to the right ange are the egs of the triange. The side opposite the right ange is the hypotenuse of the triange (see Figure F.4). Legs Hypotenuse ase Legs Figure F.4 Figure F.5 n isoscees triange can have three congruent sides. If it has ony two, then the two congruent sides are the egs of the triange. The third side is the base of the triange (see Figure F.5). F. Exercises In Exercises 3, copy the region on a piece of dot paper. Then divide the region into two congruent parts. How many different ways can you do this?. In Exercises 6 and 7, copy the region on a piece of paper. Then divide the region into four congruent parts In Exercises 8, use the trianguar grid beow. In the grid, each sma triange has sides of one unit. 4. Two of the figures are congruent. Which are they? (a) (b) (c) 5. Two of the figures are simiar. Which are they? (a) (b) (c) 8. How many congruent trianges with one-unit sides are in the grid? 9. How many congruent trianges with two-unit sides are in the grid? 0. How many congruent trianges with three-unit sides are in the grid?. Does the grid contain trianges that are not simiar to each other?

8 F8 PPENDIX F Further oncepts in Geometry. If two figures are congruent, are they simiar? Expain. 3. If two figures are simiar, are they congruent? Expain. 4. an a triange be simiar to a square? Expain. 5. re any two squares simiar? Expain. In Exercises 6 9, match the description with its correct notation. (a) PQ (b) PQ (c) PQ (d) PQ 6. The ine through P and Q 7. The ray from P through Q 8. The segment between P and Q 9. The ength of the segment between P and Q 0. The point R is between points S and T. Which of the foowing are true? (a) R, S, and T are coinear. (b) SR is the same as ST. (c) ST is the same as TS. (d) ST is the same as TS. In Exercises 3, use the figure beow. X Z Y W. The figure shows three anges whose vertex is X. Write two names for each ange. Which two anges are adjacent?. Is Y in the interior or exterior of WXZ? 3. Which is the best estimate for m WXY? (a) 5 (b) 30 (c) 45 In Exercises 4 9, match the triange with its name. (a) Equiatera (b) Scaene (c) Obtuse (d) Equianguar (e) Isoscees (f ) Right 4. Side engths: cm, 3 cm, 4 cm 5. nge measures: 60, 60, Side engths: 3 cm, cm, 3 cm 7. nge measures: 30, 60, Side engths: 4 cm, 4 cm, 4 cm 9. nge measures: 0, 45, 5 In Exercises 30 3, use the figure, in which LMP ONQ. L M N P Q O 30. Name three pairs of congruent anges. 3. Name three pairs of congruent sides. 3. If LMP is isoscees, expain why ONQ must be isoscees. In Exercises 33 36, use the definition of congruence to compete the statement. 33. If TUV, then m. 34. If PQR XYZ, then P. 35. If LMN TUV, then LN. 36. If DEF NOP, then DE. 37. opy and compete the tabe. Write Yes if it is possibe to sketch a triange with both characteristics. Write No if it is not possibe. Iustrate your resuts with sketches. (The first is done for you.) Scaene Isoscees Equiatera cute Yes?? Obtuse??? Right??? cute and Scaene

9 PPENDIX F Further oncepts in Geometry F9 In Exercises 38 4, is isoscees with. Sove for x. Then decide whether the triange is equiatera. (The figures are not necessariy drawn to scae.) x + x x + 4. x + 6 4x 4x 6 oordinate Geometry In Exercises 44 and 45, use the foowing figure. y R T U P Q x 44. Find a ocation of S such that PQR PQS. 45. Find two ocations of V such that PQR TUV. x 4x x x Logica Reasoning rrange 6 toothpicks as shown beow. What is the east number of toothpicks you must remove to create four congruent trianges? (Each toothpick must be the side of at east one triange.) Sketch your resut. 4. Landscape Design You are designing a patio. Your pans use a scae of 8 inch to foot. The patio is 4 feet by 36 feet. What are its dimensions on the pans? 43. rchitecture The Pentagon, near Washington D.., covers a region that is about 00 feet by 00 feet. bout how arge woud a 8-inch to -foot scae drawing of the Pentagon be? Woud such a scae be reasonabe? 47. Logica Reasoning Show how you coud arrange six toothpicks to form four congruent trianges. Each triange has one toothpick for each side, and you cannot bend, break, or overap the toothpicks. (Hint: The figure can be three-dimensiona.) F. nges Identifying Specia Pairs of nges nges Formed by a Transversa nges of a Triange Identifying Specia Pairs of nges You have been introduced to severa definitions concerning anges. For instance, you know that two anges are adjacent if they share a common vertex and side but have no common interior points. Here are some other definitions for pairs of anges. See Figure F.6. and are compementary anges. Figure F.6

10 F0 ppendix F Further oncepts in Geometry Definitions for Pairs of nges Two anges are vertica anges if their sides form two pairs of opposite rays. Two adjacent anges are a inear pair if their noncommon sides are opposite rays. Two anges are compementary if the sum of their measures is 90. Each ange is the compement of the other. Two anges are suppementary if the sum of their measures is 80. Each ange is the suppement of the other. Exampe Identifying Specia Pairs of nges Figure F.7 4 Use the terms defined above to describe reationships between the abeed anges in Figure F.7. Soution (a) 3 and 5 are vertica anges. So are 4 and 6. (b) There are four sets of inear pairs: 3 and 4, 4 and 5, 5 and 6, and 3 and 6. The anges in each of these pairs are aso suppementary anges. In Exampe (b), note that the inear pairs are aso suppementary. This resut is stated in the foowing postuate. Linear Pair Postuate If two anges form a inear pair, then they are suppementary i.e., the sum of their measures is 80. The reationship between vertica anges is stated in the foowing theorem. Vertica nges Theorem If two anges are vertica anges, then they are congruent. nges Formed by a Transversa transversa is a ine that intersects two or more copanar ines at different points. The anges that are formed when the transversa intersects the ines have the foowing names.

11 PPENDIX F Further oncepts in Geometry F Figure F.8 t m nges Formed by a Transversa In Figure F.8, the transversa t intersects the ines and m. Two anges are corresponding anges if they occupy corresponding positions, such as and 5. Two anges are aternate interior anges if they ie between and m on opposite sides of t, such as and 8. Two anges are aternate exterior anges if they ie outside and m on opposite sides of t, such as and 7. Two anges are consecutive interior anges if they ie between and m on the same side of t, such as and 5. Exampe Naming Pairs of nges n 4 3 Figure F m 0 9 In Figure F.9, how is 9 reated to the other anges? Soution You can consider that 9 is formed by the transversa as it intersects m and n, or you can consider 9 to be formed by the transversa m as it intersects and n. onsidering one or the other of these, you have the foowing. (a) 9 and 0 are a inear pair. So are 9 and. (b) 9 and are vertica anges. (c) 9 and 7 are aternate exterior anges. So are 9 and 3. (d) 9 and 5 are corresponding anges. So are 9 and. To hep buid understanding regarding anges formed by a transversa, consider reationships between two ines. Parae ines are copanar ines that do not intersect. (Reca that two nonvertica ines are parae if and ony if they have the same sope.) Intersecting ines are copanar and have exacty one point in common. If intersecting ines meet at right anges, they are perpendicuar; otherwise, they are obique. See Figure F.0. Parae Figure F.0 Perpendicuar Obique Many of the anges formed by a transversa that intersects parae ines are congruent. The foowing postuate and theorems ist usefu resuts. orresponding nges Postuate If two parae ines are cut by a transversa, then the pairs of corresponding anges are congruent.

12 F ppendix F Further oncepts in Geometry Note that the hypothesis of this postuate states that the ines must be parae, as shown in Figure F.. If the ines are not parae, then the corresponding anges are not congruent, as shown in Figure F.. t m, Figure F. m nge Theorems ternate Interior nges Theorem If two parae ines are cut by a transversa, then the pairs of aternate interior anges are congruent. onsecutive Interior nges Theorem If two parae ines are cut by a transversa, then the pairs of consecutive interior anges are suppementary. ternate Exterior nges Theorem If two parae ines are cut by a transversa, then the pairs of aternate exterior anges are congruent. Perpendicuar Transversa Theorem If a transversa is perpendicuar to one of two parae ines, then it is perpendicuar to the second. t m, Figure F. m Exampe 3 Using Properties of Parae Lines In Figure F.3, ines r and s are parae ines cut by a transversa,. Find the measure of each abeed ange r s Figure F.3 Soution and the given ange are aternate exterior anges and are congruent. So m and the given ange are vertica anges. ecause vertica anges are congruent, they have the same measure. So, m Simiary, 4 and m m There are severa sets of inear pairs, incuding and ; 3 and 4; 5 and 6; 5 and 7. The anges in each of these pairs are aso suppementary anges; the sum of the measures of each pair of anges is 80. ecause one ange of each pair measures 75, the suppements each measure 05. So, 3, 6, and 7 each measure 05. nges of a Triange The word triange means three anges. When the sides of a triange are extended, however, other anges are formed. The origina three anges of the triange are the interior anges. The anges that are adjacent to the interior anges are the exterior anges of the triange. Each vertex has a pair of exterior anges, as shown in Figure F.4 on the foowing page.

13 PPENDIX F Further oncepts in Geometry F3 Origina triange Extend sides Exterior ange Exterior ange Interior ange Figure F You coud cut a triange out of a piece of paper. Tear off the three anges and pace them adjacent to each other, as shown in Figure F.5. What do you observe? (You coud perform this investigation by measuring with a protractor or using a computer drawing program.) You shoud arrive at the concusion given in the foowing theorem. Triange Sum Theorem The sum of the measures of the interior anges of a triange is 80. Figure F.5 Exampe 4 Using the Triange Sum Theorem 8 5 In the triange in Figure F.6, find m, m, and m 3. Soution To find the measure of 3, use the Triange Sum Theorem, as foows. 3 4 m Knowing the measure of 3, you can use the Linear Pair Postuate to write m Using the Triange Sum Theorem, you have Figure F.6 m Figure F.7 The next theorem is one that you might have anticipated from the investigation earier. s shown in Figure F.7, if you had torn ony two of the anges from the paper triange, you coud put them together to cover exacty one of the exterior anges. Exterior nge Theorem Remote interior anges Exterior ange The measure of an exterior ange of a triange is equa to the sum of the measures of the two remote (nonadjacent) interior anges. (See Figure F.8.) Figure F.8

14 F4 PPENDIX F Further oncepts in Geometry F. Exercises In Exercises 6, sketch a pair of anges that fits the description. Labe the anges as and.. inear pair of anges. Suppementary anges for which is acute 3. cute vertica anges 4. djacent congruent compementary anges 5. Obtuse vertica anges 6. djacent congruent suppementary anges In Exercises 9 4, find the vaue of x x x 30 In Exercises 7, use the figure to determine reationships between the given anges. 75 x 5 90 F 90 D. 3. O E 3x + 6 3x + 0 5x O and OD 8. O and O 9. O and OE 0. O and EOD. O and OF. O and OE 4. x + 30 x 4x In Exercises 3 8, use the foowing information to decide whether the statement is true or fase. (Hint: Make a sketch.) Vertica anges: and ; Linear pairs: and 3, and If m 3 30, then m If m 50, then m and 3 are congruent. 6. m 3 m m 4 m m 3 80 m 5. In the figure, P, S, and T are coinear. If m P 40 and m QST 0, what is m Q? (Hint: m P m Q m PSQ 80 (a) 40 (b) 55 (c) 70 (d) 0 (e) 40 Q 40 0 P S T

15 PPENDIX F Further oncepts in Geometry F5 In Exercises 6 9, use the figure beow. 36. n t m (b 30) k a b In Exercises 37 39, use the foowing figure. 6. Name two corresponding anges. 7. Name two aternate interior anges. 8. Name two aternate exterior anges. 9. Name two consecutive interior anges. In Exercises 30 33,. Find the measures of and. Expain your reasoning In Exercises 34 36, m n and k. Determine the vaues of a and b m a (b 0) n a (60 b) 70 0 (4a 90) (b) m k n 37. Name the interior anges of the triange. 38. Name the exterior anges of the triange. 39. Two ange measures are given in the figure. Find the measure of the eight abeed anges. In Exercises 40 43, use the foowing figure, in which DEF E 40. What is the measure of D? 4. What is the measure of? 4. What is the measure of? 43. What is the measure of F? an a right triange have an obtuse ange? Expain. 45. Is it true that a triange that has two 60 anges must be equianguar? Expain. 46. If a right triange has two congruent anges, does it have two 45 anges? Expain. D F

16 F6 PPENDIX F Further oncepts in Geometry In Exercises 47 and 48, find the measure of each abeed ange. 47. In Exercises 53 56, find the measures of the interior anges. 53. (x + 30) x x (x + 60) (x + 5) (x 5) (6x + ) (3x + ) (5x ) In Exercises 49 and 50, draw and abe a right triange,, for which the right ange is. What is m? 49. m m 47 In Exercises 5 and 5, draw two noncongruent, isoscees trianges that have an exterior ange with the given measure. 56. (3x 0) (0 x) (x)