NAME DATE PERIOD. Study Guide and Intervention

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1 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 5-1 M IO tudy Guide and Intervention isectors, Medians, and ltitudes erpendicular isectors and ngle isectors perpendicular bisector of a side of a triangle is a line, segment, or ray in the same plane as the triangle that is perpendicular to the side and passes through its midpoint. nother special segment, ray, or line is an angle bisector, which divides an angle into two congruent angles. wo properties of perpendicular bisectors are: (1) a point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the segment, and (2) the three perpendicular bisectors of the sides of a triangle meet at a point, called the circumcenter of the triangle, that is equidistant from the three vertices of the triangle. wo properties of angle bisectors are: (1) a point is on the angle bisector of an angle if and only if it is equidistant from the sides of the angle, and (2) the three angle bisectors of a triangle meet at a point, called the incenter of the triangle, that is equidistant from the three sides of the triangle. xample 1 is the perpendicular xample 2 M is the angle bisector bisector of. ind x. of M. ind x if m 1 5x 8 and m 2 8x 16. 5x 6 3x 8 is the perpendicular bisector of, so. 3x 8 5x x 7 x 1 2 M M is the angle bisector of M, so m 1 m 2. 5x 8 8x x 8 x ind the value of each variable y 3x 8x 6x 2 7x 9 6x 10y 4 (4x 30) is the perpendicular is equilateral. bisects. bisector of. 4. or what kinds of triangle(s) can the perpendicular bisector of a side also be an angle bisector of the angle opposite the side? 5. or what kind of triangle do the perpendicular bisectors intersect in a point outside the triangle? hapter 5 6 Glencoe Geometry

2 5-1 M IO tudy Guide and Intervention (continued) isectors, Medians, and ltitudes Medians and ltitudes median is a line segment that connects the vertex of a triangle to the midpoint of the opposite side. he three medians of a triangle intersect at the centroid of the triangle. entroid heorem he centroid of a triangle is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. centroid L xample oints,, and are the midpoints of, and, respectively. ind x, y, and z. U 2 3 U 2 3 U 2 3 6x 2 3 (6x 15) (24 3y 3) 6z 4 2 (6z 4 11) 3 9x 6x y 3 3 (6z 4) 6z x y 9z 6 6z 15 x y 3z 9 5 y z 3 L 2 3, L 2 3, L U 11 6z 4 3y 3 6x Lesson 5-1 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. ind the value of each variable x 7x x 3 9x 6 3y is a median. ;,, and are midpoints G 7x 4 H 5y H H HG M y z 8y 24 6z x V 6x 32 G V is the centroid of ; is the centroid of. 18; M 15; 24 9z 6 9x 2 J 3y 5 O z 2x K M L 7. or what kind of triangle are the medians and angle bisectors the same segments? 8. or what kind of triangle is the centroid outside the triangle? hapter 5 7 Glencoe Geometry

3 5-2 M IO tudy Guide and Intervention Inequalities and riangles ngle Inequalities roperties of inequalities, including the ransitive, ddition, ubtraction, Multiplication, and ivision roperties of Inequality, can be used with measures of angles and segments. here is also a omparison roperty of Inequality. or any real numbers a and b, either a b, a b, or a b. he xterior ngle heorem can be used to prove this inequality involving an exterior angle. xterior ngle Inequality heorem If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles. 1 m 1 m, m 1 m opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. xample List all angles of G whose measures are less than m 1. he measure of an exterior angle is greater than the measure of either remote interior angle. o m 3 m 1 and m 4 m 1. List all angles that satisfy the stated condition. 1. all angles whose measures are less than m 1 2. all angles whose measures are greater than m 3 3. all angles whose measures are less than m 1 4. all angles whose measures are greater than m 1 5. all angles whose measures are less than m 7 6. all angles whose measures are greater than m 2 7. all angles whose measures are greater than m 5 8. all angles whose measures are less than m 4 9. all angles whose measures are less than m all angles whose measures are greater than m 4 G H L M J K 1 2 U X W V Q O 9 10 Lesson 5-2 hapter 5 13 Glencoe Geometry

4 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 5-2 M IO tudy Guide and Intervention (continued) Inequalities and riangles ngle-ide elationships When the sides of triangles are not congruent, there is a relationship between the sides and angles of the triangles. If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. If, then m m. If m m, then. xample 1 List the angles in order xample 2 List the sides in order from least to greatest measure. from shortest to longest. 6 cm 7 cm 35 9 cm,, 20,, 125 List the angles or sides in order from least to greatest measure cm cm cm etermine the relationship between the measures of the given angles. 4., U 5., U U V UV, etermine the relationship between the lengths of the given sides. 7., , , hapter 5 14 Glencoe Geometry

5 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 5-3 M IO tudy Guide and Intervention Indirect roof Indirect roof with lgebra One way to prove that a statement is true is to assume that its conclusion is false and then show that this assumption leads to a contradiction of the hypothesis, a definition, postulate, theorem, or other statement that is accepted as true. hat contradiction means that the conclusion cannot be false, so the conclusion must be true. his is known as indirect proof. teps for Writing an Indirect roof 1. ssume that the conclusion is false. 2. how that this assumption leads to a contradiction. 3. oint out that the assumption must be false, and therefore, the conclusion must be true. xample Given: 3x 5 8 rove: x 1 tep 1 ssume that x is not greater than 1. hat is, x 1 or x 1. tep 2 Make a table for several possibilities for x 1 or x 1. he contradiction is that when x 1 or x 1, then 3x 5 is not greater than 8. tep 3 his contradicts the given information that 3x 5 8. he assumption that x is not greater than 1 must be false, which means that the statement x 1 must be true. x 3x Write the assumption you would make to start an indirect proof of each statement. 1. If 2x 14, then x or all real numbers, if a b c, then a c b. omplete the proof. Given: n is an integer and n 2 is even. rove: n is even. 3. ssume that 4. hen n can be expressed as 2a 1 by 5. n 2 ubstitution 6. Multiply. 7. implify. 8. 2(2a 2 2a) (2a 2 2a) 1 is an odd number. his contradicts the given that n 2 is even, so the assumption must be 10. herefore, hapter 5 22 Glencoe Geometry

6 5-3 M IO tudy Guide and Intervention (continued) Indirect roof Indirect roof with Geometry o write an indirect proof in geometry, you assume that the conclusion is false. hen you show that the assumption leads to a contradiction. he contradiction shows that the conclusion cannot be false, so it must be true. xample Given: m 100 rove: is not a right angle. tep 1 ssume that is a right angle. tep 2 how that this leads to a contradiction. If is a right angle, then m 90 and m m hus the sum of the measures of the angles of is greater than 180. tep 3 he conclusion that the sum of the measures of the angles of is greater than 180 is a contradiction of a known property. he assumption that is a right angle must be false, which means that the statement is not a right angle must be true. Write the assumption you would make to start an indirect proof of each statement. 1. If m 90, then m 45. opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 2. If V is not congruent to V, then V is not isosceles. omplete the proof. Given: 1 2 and G is not congruent to G. rove: is not congruent to. 3. ssume that ssume the conclusion is false. 4. G G 5. G G his contradicts the given information, so the assumption must 1 2 G Lesson 5-3 be 8. herefore, hapter 5 23 Glencoe Geometry

7 5-4 M IO tudy Guide and Intervention he riangle Inequality he riangle Inequality If you take three straws of lengths 8 inches, 5 inches, and 1 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 x x 3 x 3 13 x herefore x must be between 3 and 13. etermine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. 1. 3, 4, , 9, 15 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 3. 8, 8, , 4, , 8, , 2.5, 3 ind the range for the measure of the third side given the measures of two sides and and and and 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 hapter 5 29 Glencoe Geometry

8 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 5-4 M 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. Q Q is the shortest segment from Q to plane. xample Given: oint is equidistant from the sides of an angle. rove: roof: 1. raw and to 1. ist. is measured the sides of. along a. 2. and are right angles. 2. ef. of lines 3. and are right triangles. 3. ef. of rt t. angles are. 5. is equidistant from the sides of. 5. Given ef. of equidistant eflexive roperty HL omplete the proof. Given: ; U rove: U roof: 1. ; U and are a linear pair; 4. ef. of and U are a linear pair. 5. and are supplementary; 5. and U are supplementary ngles suppl. to angles are. 7. U U hapter 5 30 Glencoe Geometry

9 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 5-5 M IO tudy Guide and Intervention Inequalities Involving wo riangles Inequality he following theorem involves the relationship between the sides of two triangles and an angle in each triangle. Inequality/Hinge heorem If two sides of a triangle are congruent to two sides of another triangle and the included angle in one triangle has a greater measure than the included angle in the other, then the third side of the first triangle is longer than the third side of the second triangle If,, and m m, then. xample Write an inequality relating the lengths of and. wo sides of are congruent to two sides of and m m. y the Inequality/Hinge heorem, Write an inequality relating the given pair of segment measures. 1. M M,, J G H K M G, HK M, Write an inequality to describe the possible values of x (4x 10) cm cm cm 24 cm 1.8 cm cm 1.8 cm (3x 2.1) cm hapter 5 36 Glencoe Geometry

10 5-5 M IO tudy Guide and Intervention (continued) Inequalities Involving wo riangles Inequality he converse of the Hinge heorem is also useful when two triangles have two pairs of congruent sides. Inequality If two sides of a triangle are congruent to two sides of another triangle and the third side in one triangle is longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the corresponding angle in the second triangle. M If M, M, and, then m M m. 38 xample Write an inequality relating the measures of and. wo sides of are congruent to two sides of, and. y the Inequality, m m Write an inequality relating the given pair of angle measures. opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 1. M m M, m m, m 3. X 4. X 50 Y Z 30 Y W Z m, m Z m XYW, m WYZ Write an inequality to describe the possible values of x ( 1 2 x 6) cm 60 cm 33 (3x 3) 30 cm 60 cm Lesson 5-5 hapter 5 37 Glencoe Geometry