5- NM IO ractice isectors, Medians, and ltitudes LG In, is the angle bisector of,,, and are medians, and is the centroid.. ind x if 4x and 0.. ind y if y and 8.. ind z if 5z 0 and 4. 4. If m x and m m x 0, is an altitude? xplain. LG In, is an altitude and X is a median. 5. ind if X x 7 and X x. 6. ind if x 6 and m 8x 6. X LG In, G I is a perpendicular bisector. 7. ind x if H 6 and H 6x 5. G 8. ind y if G.y and G y 5. I H 9. ind z if m GH z. OOIN GOMY he vertices of U are (0, ), (4, 7), and U(8, ). ind the coordinates of the points of concurrency of U. 0. orthocenter. centroid. circumcenter. MOIL Nabuko wants to construct a mobile out of flat triangles so that the surfaces of the triangles hang parallel to the floor when the mobile is suspended. How can Nabuko be certain that she hangs the triangles to achieve this effect? Glencoe/McGraw-Hill 48 Glencoe Geometry
5- NM IO kills ractice Inequalities and riangles etermine which angle has the greatest measure..,, 4. 4, 5, 7 5 4 6 7 8.,, 6 4. 5, 6, 8 Use the xterior ngle Inequality heorem to list all angles that satisfy the stated condition. 4 7 5. all angles whose measures are less than m 5 6 8 9 6. all angles whose measures are less than m 9 7. all angles whose measures are greater than m 5 8. all angles whose measures are greater than m 8 Lesson 5- etermine the relationship between the measures of the given angles. 9. m, m 0. m, m 4 5 4 9. m, m. m, m etermine the relationship between the lengths of the given sides.. L M, L 4. M, M N L 8 M 8 57 79 N 44 59 5. M N, N 6. M, L Glencoe/McGraw-Hill 5 Glencoe Geometry
5- NM IO nrichment onstruction roblem he diagram below shows segment adjacent to a closed region. he problem requires that you construct another segment XY to the right of the closed region such that points,, X, and Y are collinear. You are not allowed to touch or cross the closed region with your compass or straightedge. n m k xisting oad losed egion (Lake) X V ollow these instructions to construct a segment XY so that it is collinear with segment.. onstruct the perpendicular bisector of. Label the midpoint as point, and the line as m.. Mark two points and on line m that lie well above the closed region. onstruct the perpendicular bisector n of. Label the intersection of lines m and n as point.. Mark points and on line n that lie well to the right of the closed region. onstruct the perpendicular bisector k of. Label the intersection of lines n and k as point. 4. Mark point X on line k so that X is below line n and so that X is congruent to. 5. Mark points and V on line k and on opposite sides of X, so that X and X V are congruent. onstruct the perpendicular bisector of V. all the point where the line hits the boundary of the closed region point Y. X Y corresponds to the new road. Glencoe/McGraw-Hill 56 Glencoe Geometry
5- NM IO kills ractice Indirect roof Write the assumption you would make to start an indirect proof of each statement.. m m.. Line a is perpendicular to line b. 4. 5 is supplementary to 6. OO Write an indirect proof. 5. Given: x 8 rove: x Lesson 5-6. Given:. rove: Glencoe/McGraw-Hill 59 Glencoe Geometry
5-4 NM IO tudy Guide and Intervention he riangle Inequality he riangle Inequality If you take three straws of lengths 8 inches, 5 inches, and inch and try to make a triangle with them, you will find that it is not possible. his illustrates the riangle Inequality heorem. riangle Inequality heorem he sum of the lengths of any two sides of a triangle is greater than the length of the third side. b c a xample he measures of two sides of a triangle are 5 and 8. ind a range for the length of the third side. y the riangle Inequality, all three of the following inequalities must be true. 5 x 8 8 x 5 5 8 x x x x herefore x must be between and. xercises etermine whether the given measures can be the lengths of the sides of a triangle. Write yes or no.., 4, 6. 6, 9, 5. 8, 8, 8 4., 4, 5 5. 4, 8, 6 6..5,.5, ind the range for the measure of the third side given the measures of two sides. 7. and 6 8. and 8 9..5 and 5.5 0. 8 and 8. uppose you have three different positive numbers arranged in order from least to greatest. What single comparison will let you see if the numbers can be the lengths of the sides of a triangle? Lesson 5-4 Glencoe/McGraw-Hill 6 Glencoe Geometry
5-4 NM IO tudy Guide and Intervention (continued) he riangle Inequality istance etween a oint and a Line he perpendicular segment from a point to a line is the shortest segment from the point to the line. is the shortest segment from to. he perpendicular segment from a point to a plane is the shortest segment from the point to the plane. is the shortest segment from to plane N. N xample Given: oint is equidistant from the sides of an angle. rove: roof:. raw and to. ist. is measured the sides of. along a.. and are right angles.. ef. of lines. and are right triangles.. ef. of rt. 4. 4. t. angles are. 5. is equidistant from the sides of. 5. Given 6. 6. ef. of equidistant 7. 7. eflexive roperty 8. 8. HL 9. 9. xercises omplete the proof. Given: ; U rove: U roof:. ; U..... 4. and are a linear pair; 4. ef. of and U are a linear pair. 5. and are supplementary; 5. and U are supplementary. 6. 6. ngles suppl. to angles are. 7. U 7. 8. 8. U Glencoe/McGraw-Hill 64 Glencoe Geometry
5-5 NM IO kills ractice Inequalities Involving wo riangles Write an inequality relating the given pair of angles or segment measures.. m X, m X., 6 8 X 9 Write an inequality relating the given pair of angles or segment measures.. m, m U 4., U 0 95 85 7 7 5. In the figure,,,, and are congruent and. How does m compare with m? xplain your thinking. 6. Write a two-column proof. Given: rove: m m Lesson 5-5 Glencoe/McGraw-Hill 7 Glencoe Geometry
nswers (Lesson 5-) 5- NM IO kills ractice isectors, Medians, and ltitudes Glencoe/McGraw-Hill 47 Glencoe Geometry nswers Lesson 5- LG or xercises 4, use the given information to find each value.. ind x if G is a median of.. ind x and if U is a median of. 5x 7 G x x 9 x 8; 0 x 4 U 5x 0. ind x and if is an angle bisector. 4. ind x and IJ if H K is an altitude of HIJ. I G x 8 x 6 4x (x ) K H x.5; x 9; IJ 57 x 9 J LG or xercises 5 7, use the following information. In LMN,,, and are the midpoints of L M, M N, and L N, respectively. 5. ind x. 4 6. ind y. 0.8 7. ind z. 0.7 M x z.6 y.8 L N LG Lines a, b, and c are perpendicular bisectors of and meet at. 8. ind x. 9. ind y. 6 0. ind z. 5y 6 8 7z 4 8x 6 4 b a c OOIN GOMY he vertices of HIJ are G(, 0), H(6, 0), and I(, 6). ind the coordinates of the points of concurrency of HIJ.. orthocenter. centroid. circumcenter (, ) 0, 7, 5 NM IO 5- ractice (verage) isectors, Medians, and ltitudes LG In, is the angle bisector of,,, and are medians, and is the centroid.. ind x if 4x and 0. 4.5. ind y if y and 8. 6. ind z if 5z 0 and 4.. 4. If m x and m m x 0, is an altitude? xplain. Yes; since x 40 and is an angle bisector, it follows that m 70 and m 0. o m 90, and. LG In, is an altitude and X is a median. 5. ind if X x 7 and X x. 6. ind if x 6 and m 8x 6. 6 X LG In, G I is a perpendicular bisector. 7. ind x if H 6 and H 6x 5..5 G 8. ind y if G.y and G y 5. 5 I H 9. ind z if m GH z. 7.5 OOIN GOMY he vertices of U are (0, ), (4, 7), and U(8, ). ind the coordinates of the points of concurrency of U. 0. orthocenter. centroid. circumcenter 5 4, 4, 5 4 7, 8 4 or (5.75,.75). MOIL Nabuko wants to construct a mobile out of flat triangles so that the surfaces of the triangles hang parallel to the floor when the mobile is suspended. How can Nabuko be certain that she hangs the triangles to achieve this effect? he needs to hang each triangle from its center of gravity or centroid, which is the point at which the three medians of the triangle intersect. Glencoe/McGraw-Hill 48 Glencoe Geometry Glencoe/McGraw-Hill Glencoe Geometry
nswers (Lesson 5-) Lesson 5-5- NM IO kills ractice Inequalities and riangles etermine which angle has the greatest measure..,, 4. 4, 5, 7 4 5 4 6 7 8.,, 6 4. 5, 6, 8 6 8 Use the xterior ngle Inequality heorem to list all angles that satisfy the stated condition. 4 7 5. all angles whose measures are less than m,, 4, 5, 7, 8 5 6 8 9 6. all angles whose measures are less than m 9, 4, 6, 7 7. all angles whose measures are greater than m 5, 8. all angles whose measures are greater than m 8,, 5 etermine the relationship between the measures of the given angles. 9. m, m 0. m, m m m m m 9 5 4 4. m, m. m, m m m m m etermine the relationship between the lengths of the given sides.. L M, L 4. M, M N LM L M MN L 8 M 8 57 79 59 44 N 5. M N, N 6. M, L MN N M L Glencoe/McGraw-Hill 5 Glencoe Geometry NM IO 5- ractice (verage) Inequalities and riangles etermine which angle has the greatest measure..,, 4. 4, 8, 9 4.,, 7 4. 7, 8, 0 7 0 8 9 7 4 6 5 0 Use the xterior ngle Inequality heorem to list all angles that satisfy the stated condition. 5. all angles whose measures are less than m, 4, 5, 7, 8 4 5 6 7 8 9 6. all angles whose measures are less than m 5, 7, 8 7. all angles whose measures are greater than m 7,, 5, 9 8. all angles whose measures are greater than m 6, 9 etermine the relationship between the measures of the given angles. 9. m W, m W 0. m W, m W m W W m W W 47 45 4 W 5 44 4. m, m. m W, m W m m W W etermine the relationship between the lengths of the given sides. 48. H, G H 4., G H GH G 5. G, G 6., G G G G H 0 G 7 7. O he figure shows the position of three trees on one part of a risbee course. t which tree position is the angle between the trees the greatest? 40 ft 7.5 ft 5 ft Glencoe/McGraw-Hill 54 Glencoe Geometry Glencoe/McGraw-Hill 6 Glencoe Geometry
nswers (Lesson 5-) 5- NM IO kills ractice Indirect roof Glencoe/McGraw-Hill 59 Glencoe Geometry nswers Lesson 5- Write the assumption you would make to start an indirect proof of each statement.. m m m m.. Line a is perpendicular to line b. Line a is not perpendicular to line b. 4. 5 is supplementary to 6. 5 is not supplementary to 6. OO Write an indirect proof. 5. Given: x 8 rove: x roof: tep : ssume x. tep : If x, then x 4. ut if x 4, it follows that x 8. his contradicts the given fact that x 8. tep : ince the assumption of x leads to a contradiction, it must be false. herefore, x must be true. 6. Given:. rove: roof: tep : ssume. tep : If,then by the definition of congruent segments. ut if, then by the Isosceles riangle heorem. his contradicts the given information that. tep : ince the assumption that leads to a contradiction, it must be false. herefore, it must be true that. NM IO 5- ractice (verage) Indirect roof Write the assumption you would make to start an indirect proof of each statement.. bisects. does not bisect.. OO Write an indirect proof.. Given: 4x 0 rove: x roof: tep : ssume x. tep : If x, then 4x. ut 4x implies that 4x 0, which contradicts the given inequality. tep : ince the assumption that x leads to a contradiction, it must be true that x. 4. Given: m m 80 rove: a b roof: tep : ssume a b. tep : If a b, then the consecutive interior angles and are supplementary. hus m m 80. his contradicts the given statement that m m 80. tep : ince the assumption leads to a contradiction, the statement a b must be false. herefore, a b must be true. a b 5. HYI ound travels through air at about 44 meters per second when the temperature is 0. If nrique lives kilometers from the fire station and it takes 5 seconds for the sound of the fire station siren to reach him, how can you prove indirectly that it is not 0 when nrique hears the siren? ssume that it is 0 when nrique hears the siren, then show that at this temperature it will take more than 5 seconds for the sound of the siren to reach him. ince the assumption is false, you will have proved that it is not 0 when nrique hears the siren. Glencoe/McGraw-Hill 60 Glencoe Geometry Glencoe/McGraw-Hill 9 Glencoe Geometry
nswers (Lesson 5-4) 5-4 NM IO tudy Guide and Intervention he riangle Inequality Glencoe/McGraw-Hill 6 Glencoe Geometry nswers Lesson 5-4 he riangle Inequality If you take three straws of lengths 8 inches, 5 inches, and inch and try to make a triangle with them, you will find that it is not possible. his illustrates the riangle Inequality heorem. riangle Inequality he sum of the lengths of any two sides of a heorem triangle is greater than the length of the third side. b c a xample he measures of two sides of a triangle are 5 and 8. ind a range for the length of the third side. y the riangle Inequality, all three of the following inequalities must be true. 5 x 8 8 x 5 5 8 x x x x herefore x must be between and. xercises etermine whether the given measures can be the lengths of the sides of a triangle. Write yes or no.., 4, 6 yes. 6, 9, 5 no. 8, 8, 8 yes 4., 4, 5 yes 5. 4, 8, 6 no 6..5,.5, yes ind the range for the measure of the third side given the measures of two sides. 7. and 6 8. and 8 5 n 7 6 n 0 9..5 and 5.5 0. 8 and 8 4 n 7 74 n 90. uppose you have three different positive numbers arranged in order from least to greatest. What single comparison will let you see if the numbers can be the lengths of the sides of a triangle? ind the sum of the two smaller numbers. If that sum is greater than the largest number, then the three numbers can be the lengths of the sides of a triangle. 5-4 NM IO tudy Guide and Intervention (continued) he riangle Inequality istance etween a oint and a Line he perpendicular segment from a point to a line is the shortest segment from the point to the line. is the shortest segment from to. he perpendicular segment from a point to a plane is the shortest segment from the point to the plane. is the shortest segment from to plane N. N xample Given: oint is equidistant from the sides of an angle. rove: roof:. raw and to. ist. is measured the sides of. along a.. and are right angles.. ef. of lines. and are right triangles.. ef. of rt. 4. 4. t. angles are. 5. is equidistant from the sides of. 5. Given 6. 6. ef. of equidistant 7. 7. eflexive roperty 8. 8. HL 9. 9. xercises omplete the proof. Given: ; U rove: U roof:. ; U. Given.... 4. and are a linear pair; 4. ef. of linear pair and U are a linear pair. 5. and are supplementary; 5. Linear pairs are suppl. and U are supplementary. 6. U 6. ngles suppl. to angles are. 7. U 7. 8. U 8. U Glencoe/McGraw-Hill 64 Glencoe Geometry Glencoe/McGraw-Hill Glencoe Geometry
nswers (Lesson 5-5) 5-5 NM IO kills ractice Inequalities Involving wo riangles Glencoe/McGraw-Hill 7 Glencoe Geometry nswers Lesson 5-5 Write an inequality relating the given pair of angles or segment measures.. m X, m X m X m X 6 8 X 9., Write an inequality relating the given pair of angles or segment measures.. m, m U 4., U 0 95 85 7 7 m m U 5. In the figure,,,, and are congruent and. How does m compare with m? xplain your thinking. m m ; rom the given information and the Inequality heorem, it follows that in and we have m m. ince m m m and m m m, it follows that m m m m. ubtract m from each side of the last inequality to get m m. 6. Write a two-column proof. Given: rove: m m roof: tatements easons.. Given.. Given.. eflexive roperty 4. m m 4. Inequality NM IO 5-5 ractice (verage) Inequalities Involving wo riangles Write an inequality relating the given pair of angles or segment measures.., K., (x ) (x ) 0 0 0 40 60 K M K. m, m 4. m, m 4 4 4 J K 0 m m m m 5. Write a two-column proof. Given: G is the midpoint of. m m rove: roof: tatements easons. G is the midpoint of.. Given. G G. efinition of midpoint. G G. eflexive roperty 4. m m 4. Given 5. 5. Inequality G 6. OOL ebecca used a spring clamp to hold together a chair leg she repaired with wood glue. When she opened the clamp, she noticed that the angle between the handles of the clamp decreased as the distance between the handles of the clamp decreased. t the same time, the distance between the gripping ends of the clamp increased. When she released the handles, the distance between the gripping end of the clamp decreased and the distance between the handles increased. Is the clamp an example of the or Inequality? Inequality Glencoe/McGraw-Hill 7 Glencoe Geometry Glencoe/McGraw-Hill 5 Glencoe Geometry