CONGRUENCE OF LINE SEGMENTS, ANGLES, AND TRIANGLES

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1 HPTR HPTR TL O ONTNTS 4-1 Postulates of Lines, Line Sements, and nles 4-2 Usin Postulates and efinitions in Proofs 4-3 Provin Theorems bout nles 4-4 onruent Polyons and orrespondin Parts 4-5 Provin Trianles onruent Usin Side, nle, Side 4-6 Provin Trianles onruent Usin nle, Side, nle 4-7 Provin Trianles onruent Usin Side, Side, Side hapter Summary Vocabulary Review xercises umulative Review ONGRUN O LIN SGMNTS, NGLS, N TRINGLS One of the common notions stated by uclid was the followin: Thins which coincide with one another are equal to one another. uclid used this common notion to prove the conruence of trianles. or example, uclid s Proposition 4 states, If two trianles have the two sides equal to two sides respectively, and have the anles contained by the equal straiht lines equal, they will also have the base equal to the base, the trianle will be equal to the trianle, and the remainin anles will be equal to the remainin anles respectively, namely those which the equal sides subtend. In other words, uclid showed that the equal sides and anle of the first trianle can be made to coincide with the sides and anle of the second trianle so that the two trianles will coincide. We will expand on uclid s approach in our development of conruent trianles presented in this chapter.

2 Postulates of Lines, Line Sements, and nles POSTULTS O LINS, LIN SGMNTS, N NGLS Melissa planted a new azalea bush in the fall and wants to protect it from the cold and snow this winter. She drove four parallel stakes into the round around the bush and covered the structure with burlap fabric. urin the first winter storm, this protective barrier was pushed out of shape. Her neihbor suested that she make a tripod of three stakes fastened toether at the top, formin three trianles. Melissa found that this arranement was able to stand up to the storms. Why was this chane an improvement? What eometric fiure occurs most frequently in weiht-bearin structures? In this chapter we will study the properties of trianles to discover why trianles keep their shape. Recall that a line,, is an infinite set of points that extends endlessly in both directions, but a line sement,, is a part of and has a finite lenth. We can choose some point of that is not a point of to form a line sement of any lenth. When we do this, we say that we are extendin the line sement. Postulate 4.1 Postulate 4.2 line sement can be extended to any lenth in either direction. When we choose point on so that is the midpoint of, we say that we have extended but is not the oriinal sement,. In this case, we have chosen so that and 2. We will also accept the followin postulates: Throuh two iven points, one and only one line can be drawn. Two points determine a line. Throuh iven points and, one and only one line can be drawn. Postulate 4.3 Two lines cannot intersect in more than one point. and any other point. intersect at and cannot intersect at

3 136 onruence of Line Sements, nles, and Trianles Postulate 4.4 One and only one circle can be drawn with any iven point as center and the lenth of any iven line sement as a radius. O r Only one circle can be drawn that has point O as its center and a radius equal in lenth to sement r. We make use of this postulate in constructions when we use a compass to locate points at a iven distance from a iven point. Postulate 4.5 t a iven point on a iven line, one and only one perpendicular can be drawn to the line. t point P on P, exactly one line, P, can be drawn perpendicular to P and no other line throuh P is perpendicular to P. P Postulate 4.6 rom a iven point not on a iven line, one and only one perpendicular can be drawn to the line. rom point P not on, exactly one line, P, can be drawn perpendicular to and no other line from P is perpendicular to. P Postulate 4.7 or any two distinct points, there is only one positive real number that is the lenth of the line sement joinin the two points. or the distinct points and, there is only one positive real number, represented by, which is the lenth of. Since is also called the distance from to, we refer to Postulate 4.7 as the distance postulate. Postulate 4.8 The shortest distance between two points is the lenth of the line sement joinin these two points. The fiure shows three paths that can be taken in oin from to.

4 Postulates of Lines, Line Sements, and nles 137 The lenth of (the path throuh, a point collinear with and ) is less than the lenth of the path throuh or the path throuh. The measure of the shortest path from to is the distance. Postulate 4.9 line sement has one and only one midpoint. has a midpoint, point M, and no other point is a midpoint of. M Postulate 4.10 n anle has one and only one bisector. nle has one bisector, bisects. h, and no other ray XMPL 1 Use the fiure to answer the followin questions: n nswers a. What is the intersection point of m and n? b. o points and determine line m line m, n, or l? m l XMPL 2 Solution Lines p and n are two distinct lines that intersect line m at. If line n is perpendicular to line m, can line p be perpendicular to line m? xplain. No. Only one perpendicular can be drawn to a line at a iven point on the line. Since line n is perpendicular to m and lines n and p are distinct, line p cannot be perpendicular to m. nswer n p m

5 138 onruence of Line Sements, nles, and Trianles XMPL 3 Solution h If h bisects and point is not a point on, can h be the bisector of? No. n anle has one and only bisector. Since point is h not a point on h, h is not the same ray as. Therefore, h cannot be the bisector of. nswer onditional Statements as They Relate to Proof To prove a statement in eometry, we start with what is known to be true and use definitions, postulates, and previously proven theorems to arrive at the truth of what is to be proved. s we have seen in the text so far, the information that is known to be true is often stated as iven and what is to be proved as prove. When the information needed for a proof is presented in a conditional statement, we use the information in the hypothesis to form a iven statement, and the information in the conclusion to form a prove statement. Numerical and lebraic pplications XMPL 4 Rewrite the conditional statement in the iven and prove format: If a ray bisects a straiht anle, it is perpendicular to the line determined by the straiht anle. Solution raw and label a diaram. Use the hypothesis, a ray bisects a straiht anle, as the iven. Name a straiht anle usin the three letters from the diaram and state in the iven that this anle is a straiht anle. Name the ray that bisects the anle, usin the vertex of the anle as the endpoint of the ray that is the bisector. State in the iven that the ray bisects the anle: Given: is a straiht anle and h bisects. Use the conclusion, if (the bisector) is perpendicular to the line determined by the straiht anle, to write the prove. We have called the bisector h, and the line determined by the straiht anle is. Prove: h '

6 Postulates of Lines, Line Sements, and nles 139 nswer Given: is a straiht anle and h bisects. Prove: h ' In eometry, we are interested in provin that statements are true. These true statements can then be used to help solve numerical and alebraic problems, as in xample 5. XMPL 5 Solution SQ h bisects RST, m RSQ 4x, and m QST 3x 20. ind the measures of RSQ and QST. The bisector of an anle separates the anle into two conruent anles. Therefore, RSQ QST. Then since conruent anles have equal measures, we may write an equation that states that m RSQ m QST. 4x 3x 20 x 20 m RSQ 4x m QST 3x 20 4(20) 3(20) R 4x S Q 3x 20 T nswer m RSQ m QST 80 xercises Writin bout Mathematics 1. If two distinct lines and intersect at a point, what must be true about points and? Use a postulate to justify your answer. 2. If LM 10, can LM be extended so that LM 15? xplain why or why not.

7 140 onruence of Line Sements, nles, and Trianles evelopin Skills In 3 12, in each case: a. Rewrite the conditional statement in the Given/Prove format. b. Write a formal proof. 3. If and, 4. If > and >, then. then >. 5. If m 1 m 2 90 and 6. If m m,m 1 m, and m m 2, then m 1 m 90. m 2 m, then m 1 m If >, and >, 8. If 2 and GH 2, then >. then GH If, 2, and 10. If RT RS, R 2 RT, and R 2 RS, 2, then. then R R. 11. If and, 12. If and, then. then. In 13 15, Given: RST, SQ h, and SP h. 13. If QSR PST and m QSP 96, find m QSR. 14. If m QSR 40 and SQ h ' SP h, find m PST. 15. If m PSQ is twice m QSR and SQ h ' SP h, find m PST. Q P R S T pplyin Skills In 16 19, use the iven conditional to a. draw a diaram with eometry software or pencil and paper, b. write a iven and a prove, c. write a proof. 16. If a trianle is equilateral, then the measures of the sides are equal. 17. If and are distinct points and two lines intersect at, then they do not intersect at. 18. If a line throuh a vertex of a trianle is perpendicular to the opposite side, then it separates the trianle into two riht trianles. 19. If two points on a circle are the endpoints of a line sement, then the lenth of the line sement is less than the lenth of the portion of the circle (the arc) with the same endpoints. 20. Points and G are both on line l and on line m. If and G are distinct points, do l and m name the same line? Justify your answer.

8 Usin Postulates and efinitions in Proofs USING POSTULTS N INITIONS IN PROOS theorem was defined in hapter 3 as a statement proved by deductive reasonin. We use the laws of loic to combine definitions and postulates to prove a theorem. carefully drawn fiure is helpful in decidin upon the steps to use in the proof. However, recall from hapter 1 that we cannot use statements that appear to be true in the fiure drawn for a particular problem. or example, we may not assume that two line sements are conruent or are perpendicular because they appear to be so in the fiure. On the other hand, unless otherwise stated, we will assume that lines that appear to be straiht lines in a fiure actually are straiht lines and that points that appear to be on a iven line actually are on that line in the order shown. XMPL 1 Given: with > Prove: > Proof Statements Reasons 1.,,, and are collinear 1. Given. with between and and between and Partition postulate > 3. Given. 4. > 4. Reflexive property ddition postulate. 6. > 6. Substitution postulate. XMPL 2 Given: M is the midpoint of. Prove: M 1 2 and M 1 2 M

9 142 onruence of Line Sements, nles, and Trianles M Proof Statements Reasons 1. M is the midpoint of. 1. Given. 2. M > M 2. efinition of midpoint. 3. M M 3. efinition of conruent sements. 4. M M 4. Partition postulate. 5. M M or 2M 5. Substitution postulate M 2 6. Halves of equal quantities are equal. 7. M M or 2M 7. Substitution postulate M 2 8. Halves of equal quantities are equal. Note: We used definitions and postulates to prove statements about lenth. We could not use information from the diaram that appears to be true. xercises Writin bout Mathematics 1. xplain the difference between the symbols and. ould both of these describe sements of the same line? 2. Two lines, and, intersect and m is 90. re the measures of,, and also 90? Justify your answer. evelopin Skills In 3 12, in each case: a. Rewrite the conditional statement in the Given/Prove format. b. Write a proof that demonstrates that the conclusion is valid. 3. If, bisects, and 4. If h bisects, h bisects bisects, then., and, then.

10 Usin Postulates and efinitions in Proofs If, then. 6. If is a sement and, then. 7. If is a sement, is the midpoint 8. If P and T are distinct points and P is of, and is the midpoint of, then the midpoint of RS, then T is not the. midpoint of RS. R P T S 9. If, then. 10. If, is the midpoint of, and is the midpoint of, then. 11. If and and bisect 12. If R bisects, 3 1, and each other at, then. 4 2, then 3 4. R 3 h pplyin Skills 13. The rays h and G h separate into three conruent anles,, G, and G. If m 7a 10 and m G 10a 2, find: a. m G b. m c. m d. Is acute, riht, or obtuse?

11 144 onruence of Line Sements, nles, and Trianles 14. Sement is a bisector of and is perpendicular to. 2x 30 and x 10. a. raw a diaram that shows and. b. ind and. c. ind the distance from to. Justify your answer. 15. Two line sements, RS and LM, bisect each other and are perpendicular to each other at N, and RN LN. RS 3x 9, and LM 5x 17. a. raw a diaram that shows RS and LM. b. Write the iven usin the information in the first sentence. c. Prove that RS LM. d. ind RS and LM. e. ind the distance from L to RS. Justify your answer. 4-3 PROVING THORMS OUT NGLS In this section we will use definitions and postulates to prove some simple theorems about anles. Once a theorem is proved, we can use it as a reason in later proofs. Like the postulates, we will number the theorems for easy reference. Theorem 4.1 If two anles are riht anles, then they are conruent. Given Prove and are riht anles. Proof Statements Reasons 1. and are riht anles. 1. Given. 2. m 90 and m efinition of riht anle. 3. m m 3. Transitive property of equality efinition of conruent anles. We can write this proof in pararaph form as follows: Proof: riht anle is an anle whose deree measure is 90. Therefore, m is 90 and m is 90. Since m and m are both equal to the same quantity, they are equal to each other. Since and have equal measures, they are conruent.

12 Provin Theorems bout nles 145 Theorem 4.2 If two anles are straiht anles, then they are conruent. Given and are straiht anles. Prove The proof of this theorem, which is similar to the proof of Theorem 4.1, is left to the student. (See exercise 17.) efinitions Involvin Pairs of nles INITION djacent anles are two anles in the same plane that have a common vertex and a common side but do not have any interior points in common. W Z P Y X nle and are adjacent anles because they have as their common vertex, h as their common side, and no interior points in common. However, XWY and XWZ are not adjacent anles. lthouh XWY and XWZ have W as their common vertex and WX h as their common side, they have interior points in common. or example, point P is in the interior of both XWY and XWZ. INITION omplementary anles are two anles, the sum of whose deree measures is 90. W d c Z N b a M S L X Y T R ach anle is called the complement of the other. If m c 40 and m d 50, then c and d are complementary anles. If m a 35 and m b 55, then a and b are complementary anles. omplementary anles may be adjacent, as in the case of c and d, or they may be nonadjacent, as in the case of a and b. Note that if the two complementary anles are adjacent, their sum is a riht anle: c d WZY, a riht anle. Since m c m d 90, we say that c is the complement of d, and that d is the complement of c. When the deree measure of an anle is k, the deree measure of the complement of the anle is (90 k) because k (90 k) 90. INITION Supplementary anles are two anles, the sum of whose deree measures is 180.

13 146 onruence of Line Sements, nles, and Trianles XMPL 1 When two anles are supplementary, each anle is called the supplement of the other. If m c 40 and m d 140, then c and d are supplementary anles. If m a 35 and d T m b 145, then a and b are supplementary Q anles. c S R Supplementary anles may be adjacent, as in the case of c and d, or they may be nonadjacent, as in the case of a and b. Note that if the two supplementary anles are adjacent, their b a sum is a straiht anle. Here c d TQR, a straiht anle. Since m c m d 180, we say that c is the supplement of d and that d is the supplement of c. When the deree measure of an anle is k, the deree measure of the supplement of the anle is (180 k) because k (180 k) 180. ind the measure of an anle if its measure is 24 derees more than the measure of its complement. Solution nswer Let x measure of complement of anle. Then x 24 measure of anle. The sum of the deree measures of an anle and its complement is 90. x x x x 66 x 33 x The measure of the anle is 57 derees. Theorems Involvin Pairs of nles Theorem 4.3 If two anles are complements of the same anle, then they are conruent. Given Prove 1 is the complement of 2 and 3 is the complement of

14 Provin Theorems bout nles 147 Proof Statements Reasons 1. 1 is the complement of Given. 2. m 1 m omplementary anles are two anles the sum of whose deree measures is is the complement of Given. 4. m 3 m efinition of complementary anles. 5. m 1 m 2 m 3 m 2 5. Transitive property of equality (steps 2 and 4). 6. m 2 m 2 6. Reflexive property of equality. 7. m 1 m 3 7. Subtraction postulate onruent anles are anles that have the same measure. Note: In a proof, there are two acceptable ways to indicate a definition as a reason. In reason 2 of the proof above, the definition of complementary anles is stated in its complete form. It is also acceptable to indicate this reason by the phrase efinition of complementary anles, as in reason 4. We can also ive an alebraic proof for the theorem just proved. Proof: In the fiure, m x. oth and are complements to. Thus, m 90 x and m 90 x, and we conclude and have the same measure. Since anles that have the same measure are conruent,. (90 x) x (90 x) Theorem 4.4 If two anles are conruent, then their complements are conruent. Given H is the complement of. GH is the complement of H. G H Prove GH This theorem can be proved in a manner similar to Theorem 4.3 but with the use of the substitution postulate. We can also use an alebraic proof.

15 148 onruence of Line Sements, nles, and Trianles G Proof H onruent anles have the same measure. If H, we can represent the measure of each anle by the same variable: m m H x. Since is the complement of, and GH is the complement of H, then m 90 x and m GH 90 x. Therefore, m m GH and GH. Theorem 4.5 If two anles are supplements of the same anle, then they are conruent. Given Prove is the supplement of, and is the supplement of. Theorem 4.6 If two anles are conruent, then their supplements are conruent. Given Prove H, is the supplement of, and GH is the supplement of H. GH G H The proofs of Theorems 4.5 and 4.6 are similar to the proofs of Theorems 4.3 and 4.4 and will be left to the student. (See exercises 18 and 19.) More efinitions and Theorems Involvin Pairs of nles INITION linear pair of anles are two adjacent anles whose sum is a straiht anle. In the fiure, is a straiht anle and is not on. Therefore, +. Note that and are adjacent anles whose common side is h and whose remainin sides are opposite rays that toether form a straiht line,. Theorem 4.7 If two anles form a linear pair, then they are supplementary. Given Prove and form a linear pair. and are supplementary.

16 Provin Theorems bout nles 149 Proof In the fiure, and form a linear pair. They share a common side, h, and their remainin sides, h and h, are opposite rays. The sum of a linear pair of anles is a straiht anle, and the deree measure of a straiht anle is 180. Therefore, m m 180. Then, and are supplementary because supplementary anles are two anles the sum of whose deree measure is 180. Theorem 4.8 If two lines intersect to form conruent adjacent anles, then they are perpendicular. Given and with Prove ' Proof The union of the opposite rays, and h, is the straiht anle,. The measure of straiht anle is 180. y the partition postulate, is the sum of and. Thus, m m m 180 Since, they have equal measures. Therefore, 1 m = m 2 (180) 90 The anles, and, are riht anles. Therefore, because perpendicular lines intersect to form riht anles. h ' INITION Vertical anles are two anles in which the sides of one anle are opposite rays to the sides of the second anle. h In the fiure, and are a pair of vertical anles because and h are opposite rays and h and h are opposite rays. lso, and h are a pair of vertical anles because h and are opposite rays and h and h are opposite rays. In each pair of vertical anles, the opposite rays, which are the sides of the anles, form straiht lines, and. When two straiht lines intersect, two pairs of vertical anles are formed.

17 150 onruence of Line Sements, nles, and Trianles Theorem 4.9 If two lines intersect, then the vertical anles are conruent. Given and intersect at. Prove Proof Statements Reasons 1. and intersect at. 1. Given. h 2. h and are opposite rays. 2. efinition of opposite rays. h and h are opposite rays. 3. and are 3. efinition of vertical anles. vertical anles. 4. and are a 4. efinition of a linear pair. linear pair. and are a linear pair. 5. and are 5. If two anles form a linear pair, supplementary. they are supplementary. and are (Theorem 4.7) supplementary If two anles are supplements of the same anle, they are conruent. (Theorem 4.5) In the proof above, reasons 5 and 6 demonstrate how previously proved theorems can be used as reasons in deducin statements in a proof. In this text, we have assined numbers to theorems that we will use frequently in provin exercises as well as in provin other theorems. You do not need to remember the numbers of the theorems but you should memorize the statements of the theorems in order to use them as reasons when writin a proof. You may find it useful to keep a list of definitions, postulates, and theorems in a special section in your notebook or on index cards for easy reference and as a study aid. In this chapter, we have seen the steps to be taken in presentin a proof in eometry usin deductive reasonin: 1. s an aid, draw a fiure that pictures the data of the theorem or the problem. Use letters to label points in the fiure. 2. State the iven, which is the hypothesis of the theorem, in terms of the fiure. 3. State the prove, which is the conclusion of the theorem, in terms of the fiure.

18 Provin Theorems bout nles Present the proof, which is a series of loical aruments used in the demonstration. ach step in the proof should consist of a statement about the fiure. ach statement should be justified by the iven, a definition, a postulate, or a previously proved theorem. The proof may be presented in a two-column format or in pararaph form. Proofs that involve the measures of anles or of line sements can often be presented as an alebraic proof. XMPL 2 If and intersect at and h bisects, prove that. Solution Given: and intersect at and h bisects. Prove: Proof Statements Reasons 1. h bisects. 1. Given efinition of a bisector of an anle. 3. and intersect at. 3. Given. 4. and are vertical 4. efinition of vertical anles. anles If two lines intersect, then the vertical anles are conruent Transitive property of conruence (steps 2 and 5). lternative Proof n alebraic proof can be iven: Let m 2x. It is iven that h bisects. The bisector of an anle separates the anle into two conruent anles: and. onruent anles have equal measures. Therefore, m m x. It is also iven that and intersect at. If two lines intersect, the vertical anles are conruent and therefore have equal measures: m m x. Then since m = x and m x, m = m and.

19 152 onruence of Line Sements, nles, and Trianles xercises Writin bout Mathematics 1. Josh said that Theorem 4.9 could also have been proved by showin that. o you aree with Josh? xplain. 2. The statement of Theorem 4.7 is If two anles form a linear pair then they are supplementary. Is the converse of this theorem true? Justify your answer. evelopin Skills In 3 11, in each case write a proof, usin the hypothesis as the iven and the conclusion as the statement to be proved. 3. If m m 90,, and, then and are complements. h h 4. If G and intersect at and 5. If is a riht anle and ',, then G. then. G 6. If and are riht anles, 7. If is a riht anle and and, then is a riht anle, then..

20 Provin Theorems bout nles 153 h h 8. If intersects at H and at G, 9. If and intersect at, and and m HG m GH, then, then. HG G. H G 10. If and intersect at, and 11. If is a riht anle, and is, then '. complementary to, then. In 12 15, and intersect at. 12. If m 70, find m,m, and m. 13. If m 2x 20 and m 3x 30, find m,m,m, and m. 14. If m 5x 25 and m 7x 65, find m,m,m, and m. 15. If m y,m 3x, and m 2x y, find m,m,m, and m.

21 154 onruence of Line Sements, nles, and Trianles 16. RS intersects LM at P,m RPL x y,m LPS 3x 2y,m MPS 3x 2y. a. Solve for x and y. b. ind m RPL,m LPS, and m MPS. 17. Prove Theorem 4.2, If two anles are straiht anles, then they are conruent. 18. Prove Theorem 4.5, If two anles are supplements of the same anle, then they are conruent. 19. Prove Theorem 4.6, If two anles are conruent, then their supplements are conruent. pplyin Skills 20. Two anles form a linear pair. The measure of the smaller anle is one-half the measure of the larer anle. ind the deree measure of the larer anle. 21. The measure of the supplement of an anle is 60 derees more than twice the measure of the anle. ind the deree measure of the anle. 22. The difference between the deree measures of two supplementary anles is 80. ind the deree measure of the larer anle. 23. Two anles are complementary. The measure of the larer anle is 5 times the measure of the smaller anle. ind the deree measure of the larer anle. 24. Two anles are complementary. The deree measure of the smaller anle is 50 less than the deree measure of the larer. ind the deree measure of the larer anle. 25. The measure of the complement of an anle exceeds the measure of the anle by 24 derees. ind the deree measure of the anle. 4-4 ONGRUNT POLYGONS N ORRSPONING PRTS old a rectanular sheet of paper in half by placin the opposite edes toether. If you tear the paper alon the fold, you will have two rectanles that fit exactly on one another. We call these rectanles conruent polyons. onruent polyons are polyons that have the same size and shape. ach anle of one polyon is conruent to an anle of the other and each ede of one polyon is conruent to an ede of the other. In the diaram at the top of pae 155, polyon is conruent to polyon GH. Note that the conruent polyons are named in such a way that each vertex of corresponds to exactly one vertex of GH and each vertex of GH corresponds to exactly one vertex of. This relationship is called a one-to-one correspondence. The order in which the vertices are named shows this one-to-one correspondence of points.

22 onruent Polyons and orrespondin Parts 155 GH indicates that: corresponds to ; corresponds to. corresponds to ; corresponds to. corresponds to G; G corresponds to. corresponds to H; H corresponds to. onruent polyons should always be named so as to indicate the correspondences between the vertices of the polyons. H G orrespondin Parts of onruent Polyons In conruent polyons and GH shown above, vertex corresponds to vertex. nles and are called correspondin anles, and. In this example, there are four pairs of such correspondin anles: G H In conruent polyons, correspondin anles are conruent. In conruent polyons and GH, since corresponds to and corresponds to, and are correspondin sides, and. In this example, there are four pairs of such correspondin sides: > G > GH > H In conruent polyons, correspondin sides are conruent. The pairs of conruent anles and the pairs of conruent sides are called the correspondin parts of conruent polyons. We can now present the formal definition for conruent polyons. INITION Two polyons are conruent if and only if there is a one-to-one correspondence between their vertices such that correspondin anles are conruent and correspondin sides are conruent. This definition can be stated more simply as follows: orrespondin parts of conruent polyons are conruent. onruent Trianles The smallest number of sides that a polyon can have is three. trianle is a polyon with exactly three sides. In the fiure, and are conruent trianles.

23 156 onruence of Line Sements, nles, and Trianles The correspondence establishes six facts about these trianles: three facts about correspondin sides and three onruences > qualities facts about correspondin anles. In the > table at the riht, these six facts are stated as equalities. Since each conruence > statement is equivalent to an equal- m m ity statement, we will use whichever m m notation serves our purpose better in a particular situation. m m or example, in one proof, we may prefer to write > and in another proof to write =. In the same way, we miht write or we miht write m = m. rom the definition, we may now say: orrespondin parts of conruent trianles are equal in measure. In two conruent trianles, pairs of correspondin sides are always opposite pairs of correspondin anles. In the precedin fiure,. The order in which we write the names of the vertices of the trianles indicates the one-to-one correspondence. 1. and are correspondin conruent anles. 2. is opposite, and is opposite. 3. and are correspondin conruent sides. quivalence Relation of onruence In Section 3-5 we saw that the relation is conruent to is an equivalence relation for the set of line sements and the set of anles. Therefore, is conruent to must be an equivalence relation for the set of trianles or the set of polyons with a iven number of sides. 1. Reflexive property:. 2. Symmetric property: If, then. 3. Transitive property: If and RST, then RST. Therefore, we state these properties of conruence as three postulates: Postulate 4.11 ny eometric fiure is conruent to itself. (Reflexive Property) Postulate 4.12 conruence may be expressed in either order. (Symmetric Property)

24 onruent Polyons and orrespondin Parts 157 Postulate 4.13 Two eometric fiures conruent to the same eometric fiure are conruent to each other. (Transitive Property) xercises Writin bout Mathematics 1. If, then >. Is the converse of this statement true? Justify your answer. 2. Jesse said that since RST and STR name the same trianle, it is correct to say RST STR. o you aree with Jesse? Justify your answer. evelopin Skills In 3 5, in each case name three pairs of correspondin anles and three pairs of correspondin sides in the iven conruent trianles. Use the symbol to indicate that the anles named and also the sides named in your answers are conruent In 6 10, LMNP is a square and LSN and PSM bisect each other. Name the property that justifies each statement. 6. LSP LSP 7. If LSP NSM, then NSM LSP. P N 8. If LSP NSM and NSM NSP, then LSP NSP. S 9. If LS SN, then SN LS. 10. If PLM PNM, then PNM PLM. L M

25 158 onruence of Line Sements, nles, and Trianles 4-5 PROVING TRINGLS ONGRUNT USING SI, NGL, SI The definition of conruent polyons states that two polyons are conruent if and only if each pair of correspondin sides and each pair of correspondin anles are conruent. However, it is possible to prove two trianles conruent by provin that fewer than three pairs of sides and three pairs of anles are conruent. Hands-On ctivity In this activity, we will use a protractor and ruler, or eometry software. Use the procedure below to draw a trianle iven the measures of two sides and of the included anle. STP 1. Use the protractor or eometry software to draw an anle with the iven measure. STP 2. raw two sements alon the rays of the anle with the iven lenths. The two sements should share the vertex of the anle as a common endpoint. STP 3. Join the endpoints of the two sements to form a trianle. STP 4. Repeat steps 1 throuh 3 to draw a second trianle usin the same anle measure and sement lenths. a. ollow the steps to draw two different trianles with each of the iven side-anle-side measures. (1) 3 in., 90, 4 in. (3) 5 cm, 115,8 cm (2) 5 in., 40, 5 in. (4) 10 cm, 30,8 cm b. or each pair of trianles, measure the side and anles that were not iven. o they have equal measures? c. re the trianles of each pair conruent? oes it appear that when two sides and the included anle of one trianle are conruent to the correspondin sides and anle of another, that the trianles are conruent? This activity leads to the followin statement of side-anle-side or SS trianle conruence, whose truth will be assumed without proof: Postulate 4.14 Two trianles are conruent if two sides and the included anle of one trianle are conruent, respectively, to two sides and the included anle of the other. (SS) In and, >, and >. It follows that. The postulate used here is abbreviated SS.

26 Provin Trianles onruent Usin Side, nle, Side 159 Note: When conruent sides and anles are listed, a correspondence is established. Since the vertices of conruent anles, and, are and, corresponds to. Since and corresponds to, then corresponds to, and since and corresponds to, then corresponds to.we can write. ut, when namin the trianle, if we chane the order of the vertices in one trianle we must chane the order in the other. or example,,, and name the same trianle. If, we may write or, but we may not write. XMPL 1 Given:, is the bisector of, and '. Prove: Prove the trianles conruent by SS. Proof Statements Reasons 1. bisects. 1. Given. 2. is the midpoint of. 2. The bisector of a line sement intersects the sement at its midpoint. S 3. > 4. ' 4. Given. 3. The midpoint of a line sement divides the sement into two conruent sements. 5. and are 5. Perpendicular lines intersect to riht anles. form riht anles. 6. S 7. > 6. If two anles are riht anles, then they are conruent. 7. Reflexive property of conruence SS (steps 3, 6, 7). Note that it is often helpful to mark sements and anles that are conruent with the same number of strokes or arcs. or example, in the diaram for this proof, and are marked with a sinle stroke, and are marked with the symbol for riht anles, and is marked with an to indicate a side common to both trianles.

27 160 onruence of Line Sements, nles, and Trianles xercises Writin bout Mathematics 1. ach of two telephone poles is perpendicular to the round and braced by a wire that extends from the top of the pole to a point on the level round 5 feet from the foot of the pole. The wires used to brace the poles are of unequal lenths. Is it possible for the telephone poles to be of equal heiht? xplain your answer. 2. Is the followin statement true? If two trianles are not conruent, then each pair of correspondin sides and each pair of correspondin anles are not conruent. Justify your answer. evelopin Skills In 3 8, pairs of line sements marked with the same number of strokes are conruent. Pairs of anles marked with the same number of arcs are conruent. line sement or an anle marked with an is conruent to itself by the reflexive property of conruence. In each case, is the iven information sufficient to prove conruent trianles usin SS? In 9 11, two sides or a side and an anle are marked to indicate that they are conruent. Name the pair of correspondin sides or correspondin anles that would have to be proved conruent in order to prove the trianles conruent by SS

28 Provin Trianles onruent Usin nle, Side, nle 161 pplyin Skills In 12 14: a. raw a diaram with eometry software or pencil and paper and write a iven statement usin the information in the first sentence. b. Use the information in the second sentence to write a prove statement. c. Write a proof. 12. and bisect each other. Prove that. 13. is a quadrilateral; = ; = ; and,,, and are riht anles. Prove that the diaonal separates the quadrilateral into two conruent trianles. 14. PQR and RQS are a linear pair of anles that are conruent and PQ QS. Prove that PQR RQS. 4-6 PROVING TRINGLS ONGRUNT USING NGL, SI, NGL In the last section we saw that it is possible to prove two trianles conruent by provin that fewer than three pairs of sides and three pairs of anles are conruent, that is, by provin that two sides and the included anle of one trianle are conruent to the correspondin parts of another. There are also other ways of provin two trianles conruent. Hands-On ctivity In this activity, we will use a protractor and ruler, or eometry software. Use the procedure below to draw a trianle iven the measures of two anles and of the included side. STP 1. Use the protractor or eometry software to draw an anle with the first iven anle measure. all the vertex of that anle. STP 2. raw a sement with the iven lenth alon one of the rays of. One of the endpoints of the sement should be. all the other endpoint.

29 162 onruence of Line Sements, nles, and Trianles STP 3. raw a second anle of the trianle usin the other iven anle measure. Let be the vertex of this second anle and let h be one of the sides of this anle. STP 4.Let be the intersection of the rays of and that are not on. STP 5. Repeat steps 1 throuh 4. Let the vertex of the first anle, the vertex of the second anle, and the intersection of the rays of and. a. ollow the steps to draw two different trianles with each of the iven anle-side-anle measures. (1) 65, 4 in., 35 (2) 60, 4 in., 60 (3) 120, 9 cm, 30 (4) 30, 9 cm, 30 b. or each pair of trianles, measure the anle and sides that were not iven. o they have equal measures? c. or each pair of trianles, you can conclude that the trianles are conruent. The trianles formed, and, can be placed on top of one another so that and coincide, and coincide, and and coincide. Therefore,. This activity leads to the followin statement of anle-side-anle or S trianle conruence, whose truth will be assumed without proof: Postulate 4.15 Two trianles are conruent if two anles and the included side of one trianle are conruent, respectively, to two anles and the included side of the other. (S) Thus, in and, if, >, and, it follows that.the postulate used here is abbreviated S.We will now use this postulate to prove two trianles conruent.

30 Provin Trianles onruent Usin nle, Side, nle 163 XMPL 1 Given: and intersect at, is the midpoint of, ', and '. Prove: Prove the trianles conruent by S. Proof Statements Reasons 1. and intersect at. 1. Given is the midpoint of. 3. Given. S 4. > 5. ' and ' 5. Given. 2. If two lines intersect, the vertical anles are conruent. 4. The midpoint of a line sement divides the sement into two conruent sements. 6. and are riht anles. 6. Perpendicular lines intersect to form riht anles If two anles are riht anles, they are conruent S (steps 2, 4, 7). xercises Writin bout Mathematics 1. Marty said that if two trianles are conruent and one of the trianles is a riht trianle, then the other trianle must be a riht trianle. o you aree with Marty? xplain why or why not. 2. In and, >, and / > /. a. ora said that if is conruent to, then it follows that. o you aree with ora? Justify your answer. b. Usin only the SS and S postulates, if is not conruent to, what sides and what anles cannot be conruent?

31 164 onruence of Line Sements, nles, and Trianles evelopin Skills In 3 5, tell whether the trianles in each pair can be proved conruent by S, usin only the marked conruent parts in establishin the conruence. Give the reason for your answer In 6 8, in each case name the pair of correspondin sides or the pair of correspondin anles that would have to be proved conruent (in addition to those pairs marked conruent) in order to prove that the trianles are conruent by S pplyin Skills 9. Given:,, 10. Given: bisects and and is the midpoint of. h bisects. Prove: Prove: h 11. Given: ' and h bisects 12. Given: G and. bisects at. Prove: Prove: G

32 Provin Trianles onruent Usin Side, Side, Side PROVING TRINGLS ONGRUNT USING SI, SI, SI ut three straws to any lenths and put their ends toether to form a trianle. Now cut a second set of straws to the same lenths and try to form a different trianle. Repeated experiments lead to the conclusion that it cannot be done. s shown in and, when all three pairs of correspondin sides of a trianle are conruent, the trianles must be conruent. The truth of this statement of side-side-side or SSS trianle conruence is assumed without proof. Postulate 4.16 Two trianles are conruent if the three sides of one trianle are conruent, respectively, to the three sides of the other. (SSS) Thus, in and above, >, >, and >. It follows that. The postulate used here is abbreviated SSS. XMPL 1 Given: Isosceles JKL with JK > KL and M the midpoint of JL. K Prove: JKM LKM Proof Prove the trianles conruent by usin SSS. J M L Statements S 1. JK > KL 1. Given. 2. M is the midpoint of JL. 2. Given. S 3. JM > LM S 4. KM > KM Reasons 3. efinition of midpoint. 4. Reflexive property of conruence. 5. JKM LKM 5. SSS (steps 1, 3, 4).

33 166 onruence of Line Sements, nles, and Trianles xercises Writin bout Mathematics 1. Josh said that if two trianles are conruent and one of the trianles is isosceles, then the other must be isosceles. o you aree with Josh? xplain why or why not. 2. lvan said that if two trianles are not conruent, then at least one of the three sides of one trianle is not conruent to the correspondin side of the other trianle. o you aree with lvan? Justify your answer. evelopin Skills In 3 5, pairs of line sements marked with the same number of strokes are conruent. line sement marked with is conruent to itself by the reflexive property of conruence. In each case, is the iven information sufficient to prove conruent trianles? W Z R S T In 6 8, two sides are marked to indicate that they are conruent. Name the pair of correspondin sides that would have to be proved conruent in order to prove the trianles conruent by SSS In 9 14, pairs of line sements marked with the same number of strokes are conruent. Pairs of anles marked with the same number of arcs are conruent. line sement or an anle marked with is conruent to itself by the reflexive property of conruence. In each case, is the iven information sufficient to prove conruent trianles? If so, write the abbreviation for the postulate that proves the trianles conruent P S 11. R T Q

34 hapter Summary S R M P Q 15. If two sides and the anle opposite one of those sides in a trianle are conruent to the correspondin sides and anle of another trianle, are the two trianles conruent? Justify your answer or draw a counterexample provin that they are not. (Hint: ould one trianle be an acute trianle and the other an obtuse trianle?) pplyin Skills 16. Given: bisects, ', and '. Prove: 17. Given: is equilateral, is the midpoint of. Prove: 18. Given: Trianle PQR with S on PQ and RS ' PQ; PSR is not conruent to QSR. Prove: PS QS 19. Gina is drawin a pattern for a kite. She wants it to consist of two conruent trianles that share a common side. She draws an anle with its vertex at and marks two points, and, one on each of the rays of the anle. ach point, and, is 15 inches from the vertex of the anle. Then she draws the bisector of, marks a point on the anle bisector and draws and. Prove that the trianles that she drew are conruent. HPTR SUMMRY efinitions to Know djacent anles are two anles in the same plane that have a common vertex and a common side but do not have any interior points in common. omplementary anles are two anles the sum of whose deree measures is 90. Supplementary anles are two anles the sum of whose deree measures is 180. linear pair of anles are two adjacent anles whose sum is a straiht anle. Vertical anles are two anles in which the sides of one anle are opposite rays to the sides of the second anle.

35 168 onruence of Line Sements, nles, and Trianles Two polyons are conruent if and only if there is a one-to-one correspondence between their vertices such that correspondin anles are conruent and correspondin sides are conruent. orrespondin parts of conruent polyons are conruent. orrespondin parts of conruent polyons are equal in measure. Postulates Theorems 4.1 line sement can be extended to any lenth in either direction. 4.2 Throuh two iven points, one and only one line can be drawn. (Two points determine a line.) 4.3 Two lines cannot intersect in more than one point. 4.4 One and only one circle can be drawn with any iven point as center and the lenth of any iven line sement as a radius. 4.5 t a iven point on a iven line, one and only one perpendicular can be drawn to the line. 4.6 rom a iven point not on a iven line, one and only one perpendicular can be drawn to the line. 4.7 or any two distinct points, there is only one positive real number that is the lenth of the line sement joinin the two points. (istance Postulate) 4.8 The shortest distance between two points is the lenth of the line sement joinin these two points. 4.9 line sement has one and only one midpoint n anle has one and only one bisector ny eometric fiure is conruent to itself. (Reflexive Property) 4.12 conruence may be expressed in either order. (Symmetric Property) 4.13 Two eometric fiures conruent to the same eometric fiure are conruent to each other. (Transitive Property) 4.14 Two trianles are conruent if two sides and the included anle of one trianle are conruent, respectively, to two sides and the included anle of the other. (SS) 4.15 Two trianles are conruent if two anles and the included side of one trianle are conruent, respectively, to two anles and the included side of the other. (S) 4.16 Two trianles are conruent if the three sides of one trianle are conruent, respectively, to the three sides of the other. (SSS) 4.1 If two anles are riht anles, then they are conruent. 4.2 If two anles are straiht anles, then they are conruent. 4.3 If two anles are complements of the same anle, then they are conruent. 4.4 If two anles are conruent, then their complements are conruent. 4.5 If two anles are supplements of the same anle, then they are conruent. 4.6 If two anles are conruent, then their supplements are conruent. 4.7 If two anles form a linear pair, then they are supplementary. 4.8 If two lines intersect to form conruent adjacent anles, then they are perpendicular. 4.9 If two lines intersect, then the vertical anles are conruent.

36 Review xercises 169 VOULRY 4-1 istance postulate 4-3 djacent anles omplementary anles omplement Supplementary anles Supplement Linear pair of anles Vertical anles 4-4 One-to-one correspondence orrespondin anles orrespondin sides onruent polyons 4-5 SS trianle conruence 4-6 S trianle conruence 4-7 SSS trianle conruence RVIW XRISS 1. The deree measure of an anle is 15 more than twice the measure of its complement. ind the measure of the anle and its complement. 2. Two anles, LMP and PMN, are a linear pair of anles. If the deree measure of LMP is 12 less than three times that of PMN, find the measure of each anle. 3. Trianle JKL is conruent to trianle PQR and m K 3a 18 and m Q 5a 12. ind the measure of K and of Q. 4. If and intersect at, what is true about and? State a postulate that justifies your answer. 5. If LM ' MN and KM ' MN, what is true about LM and KM? State a postulate that justifies your answer. 6. Point R is not on LMN. Is LM MN less than, equal to, or reater than LR RN? State a postulate that justifies your answer. 7. If h and h are bisectors of, does lie on? State a postulate that justifies your answer. 8. The midpoint of is M. If MN and PM are bisectors of, does P lie on MN? Justify your answer. 9. The midpoint of is M. If MN and PM are perpendicular to, does P lie on MN? Justify your answer.

37 170 onruence of Line Sements, nles, and Trianles 10. Given:m 50, m 45, 11. Given: GH bisects and = 10 cm, m 50, m m. m 45, and 10 cm. Prove: G H Prove: H G 12. Given: >, >, is not conruent to. Prove: is not conruent to. xploration 1. If three anles of one trianle are conruent to the correspondin anles of another trianle, the trianles may or may not be conruent. raw diarams to show that this is true. 2. STUVWXYZ is a cube. Write a pararaph proof that would convince someone that STX, UTX, and STU are all conruent to one another. W Z X Y V U S T UMULTIV RVIW HPTRS 1 4 Part I nswer all questions in this part. ach correct answer will receive 2 credits. No partial credit will be allowed. 1. Which of the followin is an illustration of the associative property of addition? (1) 3(4 7) 3(7 4) (3) 3 (4 7) 3 (7 4) (2) 3(4 7) 3(4) 3(7) (4) 3 (4 7) (3 4) 7

38 2. If the sum of the measures of two anles is 90, the anles are (1) supplementary. (3) a linear pair. (2) complementary. (4) adjacent anles. 3. If, which of the followin may be false? (1) is the midpoint of. (3) is between and. (2) is a point of. (4) h and h are opposite rays. 4. If b is a real number, then b has a multiplicative inverse only if (1) b 1 (2) b 0 (3) b 0 (4) b 0 5. The contrapositive of Two anles are conruent if they have the same measures is (1) Two anles are not conruent if they do not have the same measures. (2) If two anles have the same measures, then they are conruent. (3) If two anles are not conruent, then they do not have the same measures. (4) If two anles do not have the same measures, then they are not conruent. 6. The statement Today is Saturday and I am oin to the movies is true. Which of the followin statements is false? (1) Today is Saturday or I am not oin to the movies. (2) Today is not Saturday or I am not oin to the movies. (3) If today is not Saturday, then I am not oin to the movies. (4) If today is not Saturday, then I am oin to the movies. 7. If, then and must be (1) obtuse. (2) scalene. (3) isosceles. (4) equilateral. umulative Review If and intersect at, and are (1) conruent vertical anles. (3) conruent adjacent anles. (2) supplementary vertical anles. (4) supplementary adjacent anles. 9. LMN and NMP form a linear pair of anles. Which of the followin statements is false? (1) m LMN m NMP 180 (2) LMN and NMP are supplementary anles. h h (3) ML and MP are opposite rays. h (4) ML and MN h are opposite rays. 10. The solution set of the equation 3(x 2) 5x is (1) {x x 3} (3) {x x 1} (2) {x x 3} (4) {x x 1}

39 172 onruence of Line Sements, nles, and Trianles Part II nswer all questions in this part. ach correct answer will receive 2 credits. learly indicate the necessary steps, includin appropriate formula substitutions, diarams, raphs, charts, etc. or all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 11. Given: PQ bisects RS at M 12. Given: Quadrilateral G with and R S. G and G. Prove: RMQ SMP Prove: G P G R M S Q Part III nswer all questions in this part. ach correct answer will receive 4 credits. learly indicate the necessary steps, includin appropriate formula substitutions, diarams, raphs, charts, etc. or all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. The followin statements are true: If our team does not win, we will not celebrate. We will celebrate or we will practice. We do not practice. id our team win? Justify your answer. 14. The two anles of a linear pair of anles are conruent. If the measure of one anle is represented by 2x y and the measure of the other anle by x 4y, find the values of x and of y.

40 umulative Review 173 Part IV nswer all questions in this part. ach correct answer will receive 6 credits. learly indicate the necessary steps, includin appropriate formula substitutions, diarams, raphs, charts, etc. or all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 15. Josie is makin a pattern for quilt pieces. One pattern is a riht trianle with two acute anles that are complementary. The measure of one of the acute anles is to be 12 derees more than half the measure of the other acute anle. ind the measure of each anle of the trianle. 16. Trianle is equilateral and equianular. The midpoint of is M, of is N, and of is L. Line sements MN, ML, and NL are drawn. a. Name three conruent trianles. b. Prove that the trianles named in a are conruent. c. Prove that NLM is equilateral. d. Prove that NLM is equianular.