HTR 174 5 HTR TL O ONTNTS 5-1 Line Segments ssociated with Triangles 5-2 Using ongruent Triangles to rove Line Segments ongruent and ngles ongruent 5-3 Isosceles and quilateral Triangles 5-4 Using Two airs of ongruent Triangles 5-5 roving Overlapping Triangles ongruent 5-6 erpendicular isector of a Line Segment 5-7 asic onstructions hapter Summary Vocabulary Review xercises umulative Review ONGRUN S ON TRINGLS The SSS postulate tells us that a triangle with sides of given lengths can have only one size and shape. Therefore, the area of the triangle is determined. We know that the area of a triangle is one-half the product of the lengths of one side and the altitude to that side. ut can the area of a triangle be found using only the lengths of the sides? formula to do this was known by mathematicians of India about 3200.. In the Western world, Heron of lexandria, who lived around 75.., provided in his book Metrica a formula that we now call Heron s formula: If is the area of the triangle with sides of length a, b, and c, and the semiperimeter, s, is one-half the 1 perimeter, that is, s (a b c), then 2 5 "s(s 2 a)(s 2 b)(s 2 c) In Metrica, Heron also provided a method of finding the approximate value of the square root of a number. This method, often called the divide and average method, continued to be used until calculators made the pencil and paper computation of a square root unnecessary.
Line Segments ssociated with Triangles 175 5-1 LIN SGMNTS SSOIT WITH TRINGLS Natalie is planting a small tree. efore filling in the soil around the tree, she places stakes on opposite sides of the tree at equal distances from the base of the tree. Then she fastens cords from the same point on the trunk of the tree to the stakes. The cords are not of equal length. Natalie reasons that the tree is not perpendicular to the ground and straightens the tree until the cords are of equal lengths. Natalie used her knowledge of geometry to help her plant the tree. What was the reasoning that helped Natalie to plant the tree? Geometric shapes are all around us. requently we use our knowledge of geometry to make decisions in our daily life. In this chapter you will write formal and informal proofs that will enable you to form the habit of looking for logical relationships before making a decision. ltitude of a Triangle INITION n altitude of a triangle is a line segment drawn from any vertex of the triangle, perpendicular to and ending in the line that contains the opposite side. G R altitude H altitude T S altitude In, if is perpendicular to, then is the altitude from vertex to the opposite side. In G, if K GH is perpendicular to g, the line that contains the side, then GH is the altitude from vertex G to the opposite side. In an obtuse triangle such as G above, the altitude from each of the acute angles lies outside the triangle. J L In right TSR, if RS is perpendicular to TS, then RS is the altitude from vertex R to the opposite side TS and TS is the altitude from T to the opposite side RS. In a right triangle such as TSR above, the altitude from each vertex of an acute angle is a leg of the triangle. very triangle has three altitudes as shown in JKL. Median of a Triangle INITION median of a triangle is a line segment that joins any vertex of the triangle to the midpoint of the opposite side.
176 ongruence ased on Triangles median M In, if M is the midpoint of, then M is the median drawn from vertex to side. We may also draw a median from vertex to the midpoint of side, and a median from vertex to the midpoint of side. Thus, every triangle has three medians. ngle isector of a Triangle INITION n angle bisector of a triangle is a line segment that bisects any angle of the triangle and terminates in the side opposite that angle. R angle bisector Q In QR, if is a point on Q such that R QR, then R is the angle bisector from R in QR. We may also draw an angle bisector from the vertex to some point on QR, and an angle bisector from the vertex Q to some point on R. Thus, every triangle has three angle bisectors. In a scalene triangle, the altitude, the median, and the angle bisector drawn from any common vertex are three distinct line segments. In, from the common vertex, three line segments are drawn: 1. is the altitude from because '. 2. is the angle bisector from because. 3. is the median from because is the midpoint of. median angle bisector altitude In some special triangles, such as an isosceles triangle and an equilateral triangle, some of these segments coincide, that is, are the same line. We will consider these examples later. XML 1 Given: KM is the angle bisector from K in JKL, and LK > JK. rove: JKM LKM K L M J
Line Segments ssociated with Triangles 177 roof Statements Reasons 1. LK > JK 1. Given. 2. KM is the angle bisector from 2. Given. K in JKL. 3. KM bisects JKL. 3. efinition of an angle bisector of a triangle. 4. JKM LKM 4. efinition of the bisector of an angle. 5. KM > KM 5. Reflexive property of congruence. 6. JKM LKM 6. SS (steps 1, 4, 5). xercises Writing bout Mathematics 1. xplain why the three altitudes of a right triangle intersect at the vertex of the right angle. 2. Triangle is a triangle with an obtuse angle. Where do the lines containing the three altitudes of the triangle intersect? eveloping Skills 3. Use a pencil, ruler, and protractor, or use geometry software, to draw, an acute, scalene triangle with altitude, angle bisector, and median. a. Name two congruent angles that have their vertices at. b. Name two congruent line segments. c. Name two perpendicular line segments. d. Name two right angles. 4. Use a pencil, ruler, and protractor, or use geometry software, to draw several triangles. Include acute, obtuse, and right triangles. a. raw three altitudes for each triangle. b. Make a conjecture regarding the intersection of the lines containing the three altitudes. 5. Use a pencil, ruler, and protractor, or use geometry software, to draw several triangles. Include acute, obtuse, and right triangles. a. raw three angle bisectors for each triangle. b. Make a conjecture regarding the intersection of these three angle bisectors.
178 ongruence ased on Triangles 6. Use a pencil, ruler, and protractor, or use geometry software, to draw several triangles. Include acute, obtuse, and right triangles. a. raw three medians to each triangle. b. Make a conjecture regarding the intersection of these three medians. In 7 9, draw and label each triangle described. omplete each required proof in two-column format. 7. Given: In QR, R > QR, Q, 8. Given: In, G is both an angle and RS is a median. bisector and an altitude. rove: SR QSR rove: G G 9. Given: is a median of but is not congruent to. rove: is not an altitude of. (Hint: Use an indirect proof.) pplying Skills In 10 13, complete each required proof in paragraph format. 10. In a scalene triangle, LNM, show that an altitude, NO, cannot be an angle bisector. (Hint: Use an indirect proof.) 11. telephone pole is braced by two wires that are fastened to the pole at point and to the ground at points and. The base of the pole is at point, the midpoint of. If the pole is perpendicular to the ground, are the wires of equal length? Justify your answer. 1 12. The formula for the area of a triangle is 2 bh with b the length of one side of a triangle and h the length of the altitude to that side. In, is the altitude from vertex to side and M is the midpoint of. Show that the median separates into two triangles of equal area, M and M. 13. farmer has a triangular piece of land that he wants to separate into two sections of equal area. How can the land be divided? 5-2 USING ONGRUNT TRINGLS TO ROV LIN SGMNTS ONGRUNT N NGLS ONGRUNT The definition of congruent triangles tells us that when two triangles are congruent, each pair of corresponding sides are congruent and each pair of corresponding angles are congruent. We use three pairs of corresponding parts, SS, S, or SSS, to prove that two triangles are congruent. We can then conclude that each of the other three pairs of corresponding parts are also congruent. In this section we will prove triangles congruent in order to prove that two line segments or two angles are congruent.
XML 1 Using ongruent Triangles to rove Line Segments ongruent and ngles ongruent 179 Given:, is a right angle, is a right angle, >, >. rove: > roof The line segments that we want to prove congruent are corresponding sides of and. Therefore we will first prove that. Then use that corresponding parts of congruent triangles are congruent. Statements 1. is a right angle, is a 1. Given. right angle. Reasons 2. 2. If two angles are right angles, then they are congruent. (Theorem 4.1) 3. > 3. Given. 4. > 4. Given. 5. 1 > 1 5. ddition postulate. 6. 6. Given. 7. 1 5 7. artition postulate. 1 5 8. > 8. Substitution postulate (steps 5, 7). 9. 9. SS (steps 3, 2, 8). 10. > 10. orresponding parts of congruent triangles are congruent. xercises Writing bout Mathematics 1. Triangles and are congruent triangles. If and are complementary angles, are and also complementary angles? Justify your answer. 2. leg and the vertex angle of one isosceles triangle are congruent respectively to a leg and the vertex angle of another isosceles triangle. Is this sufficient information to conclude that the triangles must be congruent? Justify your answer.
180 ongruence ased on Triangles eveloping Skills In 3 8, the figures have been marked to indicate pairs of congruent angles and pairs of congruent segments. a. In each figure, name two triangles that are congruent. b. State the reason why the triangles are congruent. c. or each pair of triangles, name three additional pairs of parts that are congruent because they are corresponding parts of congruent triangles. 3. M 4. 5. R S 6. 7. X 8. S R Q Y 9. Given: > and is the midpoint 10. Given: > and of. rove: rove: > 11. Given: and bisect each 12. Given: KLM and NML are right other. angles and KL = NM. rove: rove: K N K N L M
Isosceles and quilateral Triangles 181 13. Triangle is congruent to triangle, 3x 7, 5x 9, and 4x. ind: a. x b. c. d. 14. Triangle QR is congruent to triangle LMN,m 7a,m L 4a 15, and and Q are complementary. ind: a. a b. m c. m Q d. m M pplying Skills In 15 and 16, complete each required proof in paragraph format. 15. a. rove that the median from the vertex angle of an isosceles triangle separates the triangle into two congruent triangles. b. rove that the two congruent triangles in a are right triangles. 16. a. rove that if each pair of opposite sides of a quadrilateral are congruent, then a diagonal of the quadrilateral separates it into two congruent triangles. b. rove that a pair of opposite angles of the quadrilateral in a are congruent. In 17 and 18, complete each required proof in two-column format. 17. a. oints and separate into three congruent segments. is a point not on g such that > and >. raw a diagram that shows these line segments and write the information in a given statement. b. rove that. c. rove that. 18. The line segment M is both the altitude and the median from to LN in LN. a. rove that LN is isosceles. b. rove that M is also the angle bisector from in LN. 5-3 ISOSLS N QUILTRL TRINGLS When working with triangles, we observed that when two sides of a triangle are congruent, the median, the altitude, and the bisector of the vertex angle separate the triangle into two congruent triangles. These congruent triangles can be used to prove that the base angles of an isosceles triangle are congruent. This observation can be proved as a theorem called the Isosceles Triangle Theorem.
182 ongruence ased on Triangles Theorem 5.1 If two sides of a triangle are congruent, the angles opposite these sides are congruent. Given with > rove roof In order to prove this theorem, we will use the median to the base to separate the triangle into two congruent triangles. Statements Reasons 1. raw, the midpoint of. 1. line segment has one and only one midpoint. 2. is the median from 2. efinition of a median of a vertex. triangle. 3. > 3. Reflexive property of congruence. 4. > 4. efinition of a midpoint. 5. > 5. Given. 6. 6. SSS (steps 3, 4, 5). 7. 7. orresponding parts of congruent triangles are congruent. corollary is a theorem that can easily be deduced from another theorem. We can prove two other statements that are corollaries of the isosceles triangle theorem because their proofs follow directly from the proof of the theorem. orollary 5.1a The median from the vertex angle of an isosceles triangle bisects the vertex angle. roof: rom the preceding proof that, we can also conclude that since they, too, are corresponding parts of congruent triangles. orollary 5.1b The median from the vertex angle of an isosceles triangle is perpendicular to the base.
Isosceles and quilateral Triangles 183 roof: gain, from, we can say that because they are corresponding parts of congruent triangles. If two lines intersect to form congruent adjacent angles, then they are perpendicular. Therefore, '. roperties of an quilateral Triangle The isosceles triangle theorem has shown that in an isosceles triangle with two congruent sides, the angles opposite these sides are congruent. We may prove another corollary to this theorem for any equilateral triangle, where three sides are congruent. orollary 5.1c very equilateral triangle is equiangular. roof: If is equilateral, then > >. y the isosceles triangle theorem, since >,, and since >,. Therefore,. XML 1 Given: not on, >, and >. rove: > roof and are corresponding sides of and, and we will prove these triangles congruent by SS. We are given two pairs of congruent corresponding sides and must prove that the included angles are congruent. Statements 1. > 1. Given. Reasons 2. 2. Isosceles triangle theorem (Or: If two sides of a triangle are congruent, the angles opposite these sides are congruent.). 3. 3. Given. ontinued
184 ongruence ased on Triangles (ontinued) Statements Reasons 4. and are 4. If two angles form a linear pair, supplementary. and they are supplementary. are supplementary. 5. 5. The supplements of congruent angles are congruent. 6. > 6. Given. 7. 7. SS (steps 1, 5, 6). 8. > 8. orresponding parts of congruent triangles are congruent. xercises Writing bout Mathematics 1. Joel said that the proof given in xample 1 could have been done by proving that. o you agree with Joel? Justify your answer. 2. bel said that he could prove that equiangular triangle is equilateral by drawing median and showing that. What is wrong with bel s reasoning? eveloping Skills 3. In, if >, m 3x 15 and m 7x 5, find m and m. 4. Triangle RST is an isosceles right triangle with RS ST and R and T complementary angles. What is the measure of each angle of the triangle? 5. In equilateral,m 3x y,m 2x 40, and m 2y. ind x, y,m, m, and m. 6. Given: not on and 7. Given: Quadrilateral with > > and > rove: rove:
Isosceles and quilateral Triangles 185 8. Given: > and > 9. Given: > and rove: rove: 10. Given: Isosceles with >, is the midpoint of, is the midpoint of and is the midpoint of. a. rove: b. rove: is isosceles. In 11 and 12, complete each given proof with a partner or in a small group. 11. Given: with, G, 12. Given: not on, >, ', and G ' G. and is not congruent to. rove: > rove: is not congruent to. G (Hint: Use an indirect proof.) pplying Skills 13. rove the isosceles triangle theorem by drawing the bisector of the vertex angle instead of the median. 14. rove that the line segments joining the midpoints of the sides of an equilateral triangle are congruent. 15. is a point not on G and. rove that G. 16. In QR,m R m Q. rove that Q R.
186 ongruence ased on Triangles 5-4 USING TWO IRS O ONGRUNT TRINGLS Often the line segments or angles that we want to prove congruent are not corresponding parts of triangles that can be proved congruent by using the given information. However, it may be possible to use the given information to prove a different pair of triangles congruent. Then the congruent corresponding parts of this second pair of triangles can be used to prove the required triangles congruent. The following is an example of this method of proof. XML 1 Given:, >, and > rove: > roof Since and are corresponding parts of and, we can prove these two line segments congruent if we can prove and congruent. rom the given, we cannot prove immediately that and congruent. However, we can prove that. Using corresponding parts of these larger congruent triangles, we can then prove that the smaller triangles are congruent. Statements Reasons 1. > 1. Given. 2. > 2. Given. 3. > 3. Reflexive property of congruence. 4. 4. SSS (steps 1, 2, 3). 5. 5. orresponding parts of congruent triangles are congruent. 6. > 6. Reflexive property of congruence. 7. 7. SS (steps 1, 5, 6). 8. > 8. orresponding parts of congruent triangles are congruent.
Using Two airs of ongruent Triangles 187 xercises Writing bout Mathematics 1. an xample 1 be proved by proving that? Justify your answer. 2. Greg said that if it can be proved that two triangles are congruent, then it can be proved that the medians to corresponding sides of these triangles are congruent. o you agree with Greg? xplain why or why not. eveloping Skills 3. Given:, M is the 4. Given:, G midpoint of, and N is bisects, and H the midpoint of. bisects. rove: M N rove: G > H M G N H 5. Given: and bisect each other, 6. Given: M M and G intersects at G and at. > rove: is the midpoint of G. rove: > G M
188 ongruence ased on Triangles 7. Given: > and h bisects. 8. Given: R RQ and S SQ rove: h bisects. rove: RT ' Q R T S Q pplying Skills 9. In quadrilateral, =, =, and M is the midpoint of. line segment through M intersects at and at. rove that M bisects M at M. 10. omplete the following exercise with a partner or in a small group: Line l intersects at M, and and S are any two points on l. rove that if = and S = S, then M is the midpoint of and l is perpendicular to. a. Let half the group treat the case in which and S are on the same side of. b. Let half the group treat the case in which and S are on opposite sides of. c. ompare and contrast the methods used to prove the cases. 5-5 ROVING OVRLING TRINGLS ONGRUNT If we know that > and >, can we prove that? These two triangles overlap and share a common side. To make it easier to visualize the overlapping triangles that we want to prove congruent, it may be helpful to outline each with a different color as shown in the figure. Or the triangles can be redrawn as separate triangles. The segment is a side of each of the triangles and. Therefore, to the given information, > and >, we can add > and prove that by SSS.
roving Overlapping Triangles ongruent 189 XML 1 roof Given: and are medians to the legs of isosceles. rove: > Separate the triangles to see more clearly the triangles to be proved congruent. We know that the legs of an isosceles triangle are congruent. Therefore, >. We also know that the median is a line segment from a vertex to the midpoint of the opposite side. Therefore, and are midpoints of the congruent legs. The midpoint divides the line segment into two congruent segments, that is, in half, and halves of congruent segments are congruent: >. Now we have two pair of congruent sides: > and >. The included angle between each of these pairs of congruent sides is, and is congruent to itself by the reflexive property of congruence. Therefore, by SS and > because corresponding parts of congruent triangles are congruent. XML 2 Using the results of xample 1, find the length of x 15. if 5x 9 and Solution 5x 9 x 15 4x 24 x 6 5x 9 x 15 5(6) 9 6 15 30 9 21 21 nswer 21 xercises Writing bout Mathematics 1. In xample 1, the medians to the legs of isosceles were proved to be congruent by proving. ould the proof have been done by proving? Justify your answer.
190 ongruence ased on Triangles 2. In orollary 5.1b, we proved that the median to the base of an isosceles triangle is also the altitude to the base. If the median to a leg of an isosceles triangle is also the altitude to the leg of the triangle, what other type of triangle must this triangle be? eveloping Skills 3. Given:, >, >, 4. Given: SR > SQT, R > QT and and are right angles. rove: SRQ ST and R T rove: and > S Q R T 5. Given: >, ', and ' 6. Given:,,, > rove: and > rove: > and 7. Given: TM > TN, M is the midpoint of TR 8. Given: > and > and N is the midpoint of TS. rove: rove: RN > SM T M N R S pplying Skills In 9 11, complete each required proof in paragraph format. 9. rove that the angle bisectors of the base angles of an isosceles triangle are congruent. 10. rove that the triangle whose vertices are the midpoints of the sides of an isosceles triangle is an isosceles triangle. 11. rove that the median to any side of a scalene triangle is not the altitude to that side.
erpendicular isector of a Line Segment 191 5-6 RNIULR ISTOR O LIN SGMNT erpendicular lines were defined as lines that intersect to form right angles. We also proved that if two lines intersect to form congruent adjacent angles, then they are perpendicular. (Theorem 4.8) The bisector of a line segment was defined as any line or subset of a line that intersects a line segment at its midpoint. N Q R M M M M In the diagrams, M g, NM g, QM, and MR h are all bisectors of since they each intersect at its midpoint, M. Only NM g is both perpendicular to and the bisector of. NM g is the perpendicular bisector of. INITION The perpendicular bisector of a line segment is any line or subset of a line that is perpendicular to the line segment at its midpoint. In Section 3 of this chapter, we proved as a corollary to the isosceles triangle theorem that the median from the vertex angle of an isosceles triangle is perpendicular to the base. In the diagram below, since M is the median to the base of isosceles, M '. Therefore, M g is the perpendicular bisector of. (1) M is the midpoint of : M = M. M is equidistant, or is at an equal distance, from the endpoints of. (2) is the base of isosceles : =. is equidistant from the endpoints of. These two points, M and, determine the perpendicular bisector of. This suggests the following theorem. M
192 ongruence ased on Triangles Theorem 5.2 If two points are each equidistant from the endpoints of a line segment, then the points determine the perpendicular bisector of the line segment. Given and points and T such that = and T = T. rove T g is the perpendicular bisector of. T Strategy Let T g intersect at M. rove T T by SSS. Then, using the congruent corresponding angles, prove M M by SS. onsequently, M > M, so T g is a bisector. lso, M M. Since two lines that intersect to form congruent adjacent angles are perpendicular, ' T g. Therefore, T g is the perpendicular bisector of. M T The details of the proof of Theorem 5.2 will be left to the student. (See exercise 7.) Theorem 5.3a If a point is equidistant from the endpoints of a line segment, then it is on the perpendicular bisector of the line segment. Given oint such that =. rove lies on the perpendicular bisector of. roof hoose any other point that is equidistant from the endpoints of, for example, M, the midpoint of. Then M g is the perpendicular bisector of by Theorem 5.2. (If two points are each equidistant from the endpoints of a line segment, then the points determine the perpendicular bisector of the line segment.) lies on M g. M The converse of this theorem is also true.
erpendicular isector of a Line Segment 193 Theorem 5.3b If a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the line segment. Given oint on the perpendicular bisector of. rove = roof Let M be the midpoint of. Then M > M and M > M. erpendicular lines intersect to form right angles, so M M. y SS, M M. Since corresponding parts of congruent triangles are congruent, > and. M Theorems 5.3a and 5.3b can be written as a biconditional. Theorem 5.3 point is on the perpendicular bisector of a line segment if and only if it is equidistant from the endpoints of the line segment. Methods of roving Lines or Line Segments erpendicular To prove that two intersecting lines or line segments are perpendicular, prove that one of the following statements is true: 1. The two lines form right angles at their point of intersection. 2. The two lines form congruent adjacent angles at their point of intersection. 3. ach of two points on one line is equidistant from the endpoints of a segment of the other. Intersection of the erpendicular isectors of the Sides of a Triangle When we draw the three perpendicular bisectors of the sides of a triangle, it appears that the three lines are concurrent, that is, they intersect in one point. Theorems 5.3a and 5.3b allow us to prove the following perpendicular bisector concurrence theorem. N M L
194 ongruence ased on Triangles Theorem 5.4 The perpendicular bisectors of the sides of a triangle are concurrent. Given g MQ, the perpendicular bisector of N N S Q M Q M R R NR g, the perpendicular bisector of LS, the perpendicular bisector of g rove MQ g, NR g, and LS intersect in. roof L g g (1) We can assume from the diagram that MQ and NR g intersect. Let us call the point of intersection. (2) y theorem 5.3b, since is a point on MQ, the perpendicular bisector of, is equidistant from and. (3) Similarly, since is a point on NR g, the perpendicular bisector of, is equidistant from and. (4) In other words, is equidistant from,, and. y theorem 5.3a, since is equidistant from the endpoints of, is on the perpendicular bisector of. g (5) Therefore, MQ g, NR g, and LS, the three perpendicular bisectors of, intersect in a point,. The point where the three perpendicular bisectors of the sides of a triangle intersect is called the circumcenter. g N S Q M R L XML 1 rove that if a point lies on the perpendicular bisector of a line segment, then the point and the endpoints of the line segment are the vertices of an isosceles triangle. Given: lies on the perpendicular bisector of RS. rove: RS is isosceles. R S
erpendicular isector of a Line Segment 195 roof Statements Reasons 1. lies on the perpendicular 1. Given. bisector of RS. 2. R = S 2. If a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the line segment. (Theorem 5.3b) 3. R > S 3. Segments that have the same measure are congruent. 4. RS is isosceles. 4. n isosceles triangle is a triangle that has two congruent sides. xercises Writing bout Mathematics 1. Justify the three methods of proving that two lines are perpendicular given in this section. 2. ompare and contrast xample 1 with orollary 5.1b, The median from the vertex angle of an isosceles triangle is perpendicular to the base. eveloping Skills 3. If RS is the perpendicular bisector of 4. If R S and QR QS, prove S, prove that RS RS. that Q ' RS and RT ST. R S R Q T S 5. olygon is equilateral 6. Given and with ( = = = ). and, prove that rove that and bisect each is the perpendicular bisector of. other and are perpendicular to each other.
196 ongruence ased on Triangles pplying Skills 7. rove Theorem 5.2. 8. rove that if the bisector of an angle of a triangle is perpendicular to the opposite side of the triangle, the triangle is isosceles. 9. line through one vertex of a triangle intersects the opposite side of the triangle in 1 3 adjacent angles whose measures are represented by 2 a 27 and 2 a 15. Is the line perpendicular to the side of the triangle? Justify your answer. 5-7 SI ONSTRUTIONS geometric construction is a drawing of a geometric figure done using only a pencil, a compass, and a straightedge, or their equivalents. straightedge is used to draw a line segment but is not used to measure distance or to determine equal distances. compass is used to draw circles or arcs of circles to locate points at a fixed distance from given point. The six constructions presented in this section are the basic procedures used for all other constructions. The following postulate allows us to perform these basic constructions: ostulate 5.1 Radii of congruent circles are congruent. onstruction 1 onstruct a Line Segment ongruent to a Given Line Segment. Given onstruct, a line segment congruent to. X X X 1. With a straightedge, draw a ray, X h. 2. Open the compass so that the point is on and the point of the pencil is on. 3. Using the same compass radius, place the point on and, with the pencil, draw an arc that intersects X h. Label this point of intersection.
asic onstructions 197 onclusion > roof Since and are radii of congruent circles, they are congruent. onstruction 2 onstruct an ngle ongruent to a Given ngle. Given onstruct 1. raw a ray with endpoint. 2. With as center, draw an arc that intersects each ray of. Label the points of intersection and. Using the same radius, draw an arc with as the center that intersects the ray from at. 3. With as the center, draw an arc with radius equal to that intersects the arc drawn in step 3. Label the intersection. 4. raw h. onclusion roof We used congruent radii to draw >, >, and >. Therefore, by SSS and because they are corresponding parts of congruent triangles.
198 ongruence ased on Triangles onstruction 3 onstruct the erpendicular isector of a Given Line Segment and the Midpoint of a Given Line Segment. Given onstruct g ' at M, the midpoint of. M 1. Open the compass to a radius that is greater than one-half of. 2. lace the point of the compass at and draw an arc above and an arc below. 3. Using the same radius, place the point of the compass at and draw an arc above and an arc below intersecting the arcs drawn in step 2. Label the intersections and. 4. Use a straightedge to draw g intersecting at M. onclusion g ' at M, the midpoint of. roof Since they are congruent radii, > and >. Therefore, and are both equidistant from and. If two points are each equidistant from the endpoints of a line segment, then the points determine the perpendicular bisector of the line segment (Theorem 5.2). Thus, g is the perpendicular bisector of. inally, M is the point on where the perpendicular bisector intersects, so M = M. y definition, M is the midpoint of. M
asic onstructions 199 onstruction 4 isect a Given ngle. Given onstruct h, the bisector of 1. With as center and any convenient radius, draw an arc that intersects h at and h at. 2. With as center, draw an arc in the interior of. 3. Using the same radius, and with as center, draw an arc that intersects the arc drawn in step 2. Label this intersection. 4. raw h. onclusion h bisects ;. roof We used congruent radii to draw > and >. y the reflexive property of congruence, > so by SSS,. Therefore, because they are corresponding parts of congruent triangles. Then h bisects because an angle bisector separates an angle into two congruent angles. onstruction 5 will be similar to onstruction 4. In onstruction 4, any given angle is bisected. In onstruction 5, is a straight angle that is bisected by the construction. Therefore, and are right angles and g g.
200 ongruence ased on Triangles onstruction 5 onstruct a Line erpendicular to a Given Line Through a Given oint on the Line. Given oint on g. onstruct g ' g 1. With as center and any convenient radius, draw arcs that intersect h at and h at. 2. With and as centers and a radius greater than that used in step 1, draw arcs intersecting at. 3. raw g. onclusion g ' g roof Since points and were constructed using congruent radii, = and is equidistant to and. Similarly, since was constructed using congruent radii,, and is equidistant to and. If two points are each equidistant from the endpoints of a line segment, then the points determine the perpendic- ular bisector of the line segment (Theorem 5.2). Therefore, g is the perpendicular bisector of. Since is a subset of line g, g ' g.
asic onstructions 201 onstruction 6 onstruct a Line erpendicular to a Given Line Through a oint Not on the Given Line. Given oint not on g. onstruct g ' g 1. With as center and any convenient radius, draw an arc that intersects g in two points, and. 2. Open the compass to a radius greater than onehalf of. With and as centers, draw intersecting arcs. Label the point of intersection. 3. raw g intersecting g at. onclusion g ' g roof Statements Reasons 1. >, > 1. Radii of congruent circles are congruent. 2., 2. Segments that are congruent have the same measure. 3. g ' 3. If two points are each equidistant from the endpoints of a line segment, then the points determine the perpendicular bisector of the line segment. (Theorem 5.2) 4. g ' g 4. is a subset of line g.
202 ongruence ased on Triangles XML 1 onstruct the median to in. onstruction median of a triangle is a line segment that joins any vertex of the triangle to the midpoint of the opposite side. To construct the median to, we must first find the midpoint of. 1. onstruct the perpendicular bisector of to locate the midpoint. all the midpoint M. R 2. raw M. M onclusion M is the median to in. S xercises Writing bout Mathematics 1. xplain the difference between the altitude of a triangle and the perpendicular bisector of a side of a triangle. 2. xplain how onstruction 3 (onstruct the perpendicular bisector of a given segment) and onstruction 6 (onstruct a line perpendicular to a given line through a point not on the given line) are alike and how they are different. eveloping Skills 3. Given: onstruct: a. line segment congruent to. b. line segment whose measure is 2. c. The perpendicular bisector of. d. line segment whose measure is 1 1 2. 4. Given: onstruct: a. n angle congruent to. b. n angle whose measure is 2m. c. The bisector of. d. n angle whose measure is 2 1 2 m/.
asic onstructions 203 5. Given: Line segment. onstruct: a. line segment congruent to. b. triangle with sides congruent to,, and. c. n isosceles triangle with the base congruent to and with legs congruent to. d. n equilateral triangle with sides congruent to. 6. Given: with m 60. onstruct: a. n angle whose measure is 30. b. n angle whose measure is 15. c. n angle whose measure is 45. 7. Given: onstruct: a. The median from vertex. b. The altitude to. c. The altitude to. d. The angle bisector of the triangle from vertex. 8. a. raw. onstruct the three perpendicular bisectors of the sides of. Let be the point at which the three perpendicular bisectors intersect. b. Is it possible to draw a circle that passes through each of the vertices of the triangle? xplain your answer. Hands-On ctivity ompass and straightedge constructions can also be done on the computer by using only the point, line segment, line, and circle creation tools of your geometry software and no other software tools. Working with a partner, use either a compass and straightedge, or geometry software to complete the following constructions: a. square with side. b. n equilateral triangle with side. c. 45 angle. d. 30 angle. e. circle passing through points, and. (Hint: See the proof of Theorem 5.4 or use Theorem 5.3.)
204 ongruence ased on Triangles HTR SUMMRY efinitions to Know n altitude of a triangle is a line segment drawn from any vertex of the triangle, perpendicular to and ending in the line that contains the opposite side. median of a triangle is a line segment that joins any vertex of the triangle to the midpoint of the opposite side. n angle bisector of a triangle is a line segment that bisects any angle of the triangle and terminates in the side opposite that angle. The perpendicular bisector of a line segment is a line, a line segment, or a ray that is perpendicular to the line segment at its midpoint. ostulates Theorems and orollaries 5.1 Radii of congruent circles are congruent. 5.1 If two sides of a triangle are congruent, the angles opposite these sides are congruent. 5.1a The median from the vertex angle of an isosceles triangle bisects the vertex angle. 5.1b The median from the vertex angle of an isosceles triangle is perpendicular to the base. 5.1c very equilateral triangle is equiangular. 5.2 If two points are each equidistant from the endpoints of a line segment, then the points determine the perpendicular bisector of the line segment. 5.3 point is on the perpendicular bisector of a line segment if and only if it is equidistant from the endpoints of the line segment. 5.4 The perpendicular bisectors of the sides of a triangle are concurrent. VOULRY 5-1 ltitude of a triangle Median of a triangle ngle bisector of a triangle 5-3 Isosceles triangle theorem orollary 5-6 erpendicular bisector of a line segment quidistant oncurrent erpendicular bisector concurrence theorem ircumcenter 5-7 Geometric construction Straightedge ompass RVIW XRISS g 1. If LMN ' KM, m LMK x y and m KMN 2x y, find the value of x and of y. 2. The bisector of QR in QR is QS. If m QS x 20 and m SQR 5x, find m QR.
Review xercises 205 3. In, is both the median and the altitude. If 5x 3, 2x 8, and 3x 5, what is the perimeter of? 4. ngle QS and angle SQR are a linear pair of angles. If m QS 5a 15 and m SQR 8a 35, find m QS and m SQR. 5. Let be the point at which the three perpendicular bisectors of the sides of equilateral intersect. rove that,, and are congruent isosceles triangles. 6. rove that if the median,, to side of is not the altitude to side, then is not congruent to. 7. is the base of isosceles and is also the base of isosceles. rove that g is the perpendicular bisector of. 8. In, is the median to and >. rove that m m m. (Hint: Use Theorem 5.1, If two sides of a triangle are congruent, the angles opposite these sides are congruent. ) 9. a. raw a line,. onstruct '. b. Use to construct such that m 45. c. What is the measure of? d. What is the measure of? 10. a. raw obtuse QR with the obtuse angle at vertex Q. b. onstruct the altitude from vertex. xploration g g s you have noticed, proofs may be completed using a variety of methods. In this activity, you will explore the reasoning of others. 1. omplete a two-column proof of the following: oints L, M, and N separate into four congruent segments. oint is not on and M is an altitude of LM. rove that >. 2. ut out each statement and each reason from your proof, omitting the step numbers. 3. Trade proofs with a partner. 4. ttempt to reassemble your partner s proof. 5. nswer the following questions: a. Were you able to reassemble the proof? L M N b. id the reassembled proof match your partner s original proof? c. id you find any components missing or have any components leftover? Why?
206 ongruence ased on Triangles UMULTIV RVIW hapters 1 5 art I nswer all questions in this part. ach correct answer will receive 2 credits. No partial credit will be allowed. g 1. The symbol represents (1) a line segment with between and. (2) a line with the midpoint of. (3) a line with between and. (4) a ray with endpoint. 2. triangle with no two sides congruent is (1) a right triangle. (3) an isosceles triangle. (2) an equilateral triangle. (4) a scalene triangle. 3. Opposite rays have (1) no points in common. (3) two points in common. (2) one point in common. (4) all points in common. 4. The equality a 1 1 a is an illustration of (1) the commutative property of addition. (2) the additive inverse property. (3) the multiplicative identity property. (4) the closure property of addition. 5. The solution set of the equation 1.5x 7 0.25x 8 is (1) 120 (2) 12 (3) 11 (4) 1.2 6. What is the inverse of the statement When spiders weave their webs by noon, fine weather is coming soon? (1) When spiders do not weave their webs by noon, fine weather is not coming soon. (2) When fine weather is coming soon, then spiders weave their webs by noon. (3) When fine weather is not coming soon, spiders do not weave their webs by noon. (4) When spiders weave their webs by noon, fine weather is not coming soon. 7. If and are a linear pair of angles, then they must be (1) congruent angles. (3) supplementary angles. (2) complementary angles. (4) vertical angles.
8. Which of the following is not an abbreviation for a postulate that is used to prove triangles congruent? (1) SSS (2) SS (3) S (4) SS 9. If the statement If two angles are right angles, then they are congruent is true, which of the following statements must also be true? (1) If two angles are not right angles, then they are not congruent. (2) If two angles are congruent, then they are right angles. (3) If two angles are not congruent, then they are not right angles. (4) Two angles are congruent only if they are right angles. g hapter Summary 207 10. If is a line, which of the following may be false? (1) is on. (3) is between and. (2) 1 5 (4) is the midpoint of. art II nswer all questions in this part. ach correct answer will receive 2 credits. learly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. or all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 11. ngle QS and angle SQR are a linear pair of angles. If m QS 3a 18 and m SQR = 7a 2, find the measure of each angle of the linear pair. 12. Give a reason for each step in the solution of the given equation. 5(4 x) 32 x 20 5x 32 x 20 5x x 32 x x 20 6x 32 0 20 6x 32 6x 12 x 2 art III nswer all questions in this part. ach correct answer will receive 4 credits. learly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. or all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
208 ongruence ased on Triangles 13. Given: >, is the 14. Given: > and is any supplement of, and point on h, the bisector >. of. rove: rove: > art IV nswer all questions in this part. ach correct answer will receive 6 credits. learly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. or all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 15. Given: oint is not on and. rove: 16. rove that if and are perpendicular bisectors of each other, quadrilateral is equilateral ( ).