Answers (Anticipation Guide and Lesson 5-1)

Transcription

1 hapter Glencoe Geometry nticipation Guide elationships in riangles tep efore you begin hapter ead each statement. ecide whether you gree () or isagree () with the statement. Write or in the first column O if you are not sure whether you agree or disagree, write N (Not ure). EP,, or N tatement. ny point that is on the perpendicular bisector of a segment is equidistant from the endpoints of that segment.. he circumcenter of a triangle is equidistant from the midpoints of each side of the triangle. EP or. he altitudes of a triangle meet at the orthocenter. 4. hree altitudes can be drawn for any one triangle.. median of a triangle is any segment that contains the midpoint of a side of the triangle. 6. he measure of an exterior angle of a triangle is always greater than the measures of either of its corresponding remote interior angles. 7. he longest side in a triangle is opposite the smallest angle in that triangle. 8. o write an indirect proof that two lines are perpendicular, begin by assuming the two lines are not perpendicular. 9. he length of the longest side of a triangle is always greater than the sum of the lengths of the other two sides. 0. In two triangles, if two pairs of sides are congruent, then the measure of the included angles determines which triangle has the longer third side. tep fter you complete hapter eread each statement and complete the last column by entering an or a. id any of your opinions about the statements change from the first column? For those statements that you mark with a, use a piece of paper to write an example of why you disagree. hapter Glencoe Geometry hapter esources - tudy Guide and Intervention isectors of riangles Perpendicular isector perpendicular bisector is a line, segment, or ray that is perpendicular to the given segment and passes through its midpoint. ome theorems deal with perpendicular bisectors. Perpendicular isector heorem onverse of Perpendicular isector heorem ircumcenter heorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. he perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle. Example Find the measure of FM. Example is the perpendicular bisector of. Find x..8 FK is the perpendicular bisector of GM. FG = FM.8 = FM Find each measure.. XW. F x - 6 x + 8 = x + 8 = x = x 7 = x Point P is the circumcenter of EMK. List any segment(s) congruent to each segment below.. MY YE 4. KP MP, EP. MN NK 6. E hapter Glencoe Geometry K Lesson - nswers (nticipation Guide and Lesson -) nswers

2 hapter Glencoe Geometry - tudy Guide and Intervention (continued) ngle isectors nother special segment, ray, or line is an angle bisector, which divides an angle into two congruent angles. ngle isector heorem onverse of ngle isector heorem Incenter heorem isectors of riangles If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. If a point in the interior of an angle if equidistant from the sides of the angle, then it is on the bisector of the angle. he angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the sides of the triangle. Example M is the angle bisector of NMP. Find x if m = x + 8 and m = 8x - 6. N M P M is the angle bisector of NMP, so m = m. x + 8 = 8x = x 8 = x Find each measure.. E. Y MK 4. EWL hapter 6 Glencoe Geometry (7x + ) (x + ) x + x Point U is the incenter of GHY. Find each measure below.. MU 6. UGM 8 7. PHU 8. HU 8 - kills Practice Find each measure. isectors of riangles. FG. KL x x U 4. LYF x + 4 x IU 6. MYW 9 9 x + 7x 7 Point P is the circumcenter of. List any segment(s) congruent to each segment below P P, P Point is the incenter of PQ. Find each measure below. 0. U 40. U 0. QPK hapter 7 Glencoe Geometry 8 (4x - ) (x + ) (x + ) (4x - 9) 0 40 Lesson - nswers (Lesson -)

3 hapter 4 Glencoe Geometry - Enrichment Inscribed and ircumscribed ircles he three angle bisectors of a triangle intersect in a single point called the incenter. his point is the center of a circle that just touches the three sides of the triangle. Except for the three points where the circle touches the sides, the circle is inside the triangle. he circle is said to be inscribed in the triangle.. With a compass and a straightedge, construct the inscribed circle for PQ by following the steps below. tep onstruct the bisectors of and Q. Label the point where the bisectors meet,. tep onstruct a perpendicular segment from to Q. Use the letter to label the point where the perpendicular segment intersects Q. tep Use a compass to draw the circle with center at and radius. onstruct the inscribed circle in each triangle... he three perpendicular bisectors of the sides of a triangle also meet in a single point. his point is the center of the circumscribed circle, which passes through each vertex of the triangle. Except for the three points where the circle touches the triangle, the circle is outside the triangle. G 4. Follow the steps below to construct the circumscribed circle for FGH. tep onstruct the perpendicular bisectors of FG and FH. Use the letter to label the point where the perpendicular bisectors meet. tep raw the circle that has center and radius F. F H onstruct the circumscribed circle for each triangle.. 6. hapter 0 Glencoe Geometry P Q - tudy Guide and Intervention Medians median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side. he three medians of a triangle intersect at the centroid of the triangle. he centroid is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. Example In, U is the centroid and U = 6. Find UK and K. U = K 6 = K 4 = K U + UK = K 6 + UK = 4 UK = 8 Medians and ltitudes of riangles In, U = 6, U =, and F = 8. Find each measure.. U 8. EU 6. U UF 6 6. E 8 In E, U is the centroid, UK =, EM =, and U = 9. Find each measure. 7. U 4 8. MU 7 9. K 6 0. JU 4.. EU 4. J. hapter Glencoe Geometry 6 9 Lesson - nswers (Lesson - and Lesson -)

4 hapter Glencoe Geometry - tudy Guide and Intervention (continued) ltitudes n altitude of a triangle is a segment from a vertex to the line containing the opposite side meeting at a right angle. Every triangle has three altitudes which meet at a point called the orthocenter. y (7, 7) Example he vertices of are (, ), (7, 7) and (9, ). Find the coordinates of the orthocenter of. Find the point where two of the three altitudes intersect. (, ) (9, ) x Find the equation of the altitude from to. If has a slope of, then the altitude has a slope of. y - y = m(x x ) Point-slope form y - = (x ) m =, (x, y ) = (, ) y - = x y = x + istributive Property implify. to. If has a slope of, then the altitude has a slope of -. y - y = m(x - x ) Point-slope form y - = - (x - 9) m = -, (x, y ) = (9, ) y - = - x + 7 y = - x + olve the system of equations and find where the altitudes meet. y = x + y = - x + x + = - x + = x + ubtract x from each side. ubtract from each side. 4 = x ivide both sides by -. 7 = x y = x + = (7) + = 7 + = 6 he coordinates of the orthocenter of is (6, 7). Medians and ltitudes of riangles istributive Property implify. OOINE GEOMEY Find the coordinates of the orthocenter of each triangle.. J(, 0), H(6, 0), I(, 6). (, 0), (4, 7), U(8, ) (, ), hapter Glencoe Geometry - kills Practice Medians and ltitudes of riangles In PQ, NQ = 6, K =, and PK = 4. Find each length.. KM. KQ 4. LK 4. L. 4.. NK 6. PM 6 In, H is the centroid, EH = 6, H = 4, and M = 4. Find each length. 7. H 8. HM H 0. H 8.. E 8 OOINE GEOMEY Find the coordinates of the centroid of each triangle.. X(, ) Y(, ), Z(, 0) 4. (, ), (6, ), (0, 0) (, 0) ( 6, ) OOINE GEOMEY Find the coordinates of the orthocenter of each triangle.. L(8, 0), M(0, 8), N(4, 0) 6. ( 9, 9), E( 6, 6), F(0, 6) (0, ) (-9, -) hapter Glencoe Geometry 4 Lesson - nswers (Lesson -) nswers

5 hapter 6 Glencoe Geometry - Practice Medians and ltitudes of riangles In, P = 0, EP = 8, and F = 9. Find each length.. P. FP. P P 6. E 6 4 In MIV, Z is the centroid, MZ = 6, YI = 8, and NZ =. Find each measure. 7. Z 8. YZ 6 9. M 0. ZV 9 4. NV. IZ 6 OOINE GEOMEY Find the coordinates of the centroid of each triangle.. I(, ), J(6, ), K(, ) 4. H(0, ), U(4, ), P(, ) (4, ) (, ) OOINE GEOMEY Find the coordinates of the orthocenter of each triangle.. P(-, ), Q(, ), (, ) 6. (0, 0), (, ), U(, 6) (, -) (0, 9) E 8 0 F P 7. MOILE 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. hapter 4 Glencoe Geometry - Word Problem Practice. LNING Johanna balanced a triangle flat on her finger tip. What point of the triangle must Johanna be touching? centroid. EFLEION Part of the working space in Paulette s loft is partitioned in the shape of a nearly equilateral triangle with mirrors hanging on all three partitions. From which point could someone see the opposite corner behind his or her reflection in any of the three mirrors? orthocenter Medians and ltitudes of riangles. INE For what kind of triangle is there a point where the distance to each side is half the distance to each vertex? Explain. equilateral: incenter = centroid = circumcenter 4. MEIN Look at the right triangle below. What do you notice about the orthocenter and the vertices of the triangle? he orthocenter coincides with one of the vertices.. PLZ n architect is designing a triangular plaza. For aesthetic purposes, the architect pays special attention to the location of the centroid and the circumcenter O. a. Give an example of a triangular plaza where = O. If no such example exists, state that this is impossible. an equilateral triangle b. Give an example of a triangular plaza where is inside the plaza and O is outside the plaza. If no such example exists, state that this is impossible. an obtuse triangle c. Give an example of a triangular plaza where is outside the plaza and O is inside the plaza. If no such example exists, state that this is impossible. impossible hapter Glencoe Geometry Lesson - nswers (Lesson -)

6 hapter 7 Glencoe Geometry - Enrichment onstructing entroids and Orthocenters he three medians of a triangle intersect at a single point called the centroid. You can use a straightedge and compass to find the centroid of a triangle.. With a straightedge and compass, construct the centroid for U by following the steps below. tep Locate the midpoints of sides U and U. Label the midpoints and respectively. tep raw the segments and. Use the letter H to label their point of intersection, which is the centroid of U. onstruct the centroid of each triangle... he three altitudes of a triangle meet in a single point called the orthocenter of the triangle. 4. Follow the steps below to construct the orthocenter of E using a straightedge and compass. tep Extend segments and E past point long enough to meet perpendiculars from E and as shown. tep onstruct the perpendicular from point to the line E and label the point of intersection X. Likewise, label the point of intersection of line with the perpendicular from E as point Z. In this case both X and Z lie outside E. tep Label O the point where perpendiculars X and EZ intersect. his is the orthocenter of E. onstruct the orthocenter of each triangle.. 6. hapter 6 Glencoe Geometry - tudy Guide and Intervention ngle Inequalities Properties of inequalities, including the ransitive, ddition, and ubtraction Properties of Inequality, can be used with measures of angles and segments. here is also a omparison Property of Inequality. For any real numbers a and b, either a < b, a = b, or a > b. he Exterior ngle Inequality heorem can be used to prove this inequality involving an exterior angle. Exterior ngle Inequality heorem he measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Example List all angles of EFG whose measures are less than m. he measure of an exterior angle is greater than the measure of either remote interior angle. o m < m and m 4 < m. Inequalities in One riangle Use the Exterior ngle Inequality heorem to list all of the angles that satisfy the stated condition.. measures are less than m. measures are greater than m. measures are less than m 4. measures are greater than m. measures are less than m 7 6. measures are greater than m 7. measures are greater than m 8. measures are less than m 4 9. measures are less than m 0. measures are greater than m 4, 4,, 6 7,,, 6, UV 4, 7, UV, 4,, 7, NP, 8, OPN, OQ m > m, m > m hapter 7 Glencoe Geometry H G 4 L E 4 M J K N 8 7 U F X W V 8 Q 6 4 O P 9 0 Lesson - nswers (Lesson - and Lesson -) nswers

7 hapter 7 Glencoe Geometry - Enrichment onstructing entroids and Orthocenters he three medians of a triangle intersect at a single point called the centroid. You can use a straightedge and compass to find the centroid of a triangle.. With a straightedge and compass, construct the centroid for U by following the steps below. tep Locate the midpoints of sides U and U. Label the midpoints and respectively. tep raw the segments and. Use the letter H to label their point of intersection, which is the centroid of U. onstruct the centroid of each triangle... he three altitudes of a triangle meet in a single point called the orthocenter of the triangle. 4. Follow the steps below to construct the orthocenter of E using a straightedge and compass. tep Extend segments and E past point long enough to meet perpendiculars from E and as shown. tep onstruct the perpendicular from point to the line E and label the point of intersection X. Likewise, label the point of intersection of line with the perpendicular from E as point Z. In this case both X and Z lie outside E. tep Label O the point where perpendiculars X and EZ intersect. his is the orthocenter of E. onstruct the orthocenter of each triangle.. 6. hapter 6 Glencoe Geometry - tudy Guide and Intervention ngle Inequalities Properties of inequalities, including the ransitive, ddition, and ubtraction Properties of Inequality, can be used with measures of angles and segments. here is also a omparison Property of Inequality. For any real numbers a and b, either a < b, a = b, or a > b. he Exterior ngle Inequality heorem can be used to prove this inequality involving an exterior angle. Exterior ngle Inequality heorem he measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Example List all angles of EFG whose measures are less than m. he measure of an exterior angle is greater than the measure of either remote interior angle. o m < m and m 4 < m. Inequalities in One riangle Use the Exterior ngle Inequality heorem to list all of the angles that satisfy the stated condition.. measures are less than m. measures are greater than m. measures are less than m 4. measures are greater than m. measures are less than m 7 6. measures are greater than m 7. measures are greater than m 8. measures are less than m 4 9. measures are less than m 0. measures are greater than m 4, 4,, 6 7,,, 6, UV 4, 7, UV, 4,, 7, NP, 8, OPN, OQ m > m, m > m hapter 7 Glencoe Geometry H G 4 L E 4 M J K N 8 7 U F X W V 8 Q 6 4 O P 9 0 Lesson - nswers (Lesson - and Lesson -) nswers

8 hapter 8 Glencoe Geometry - tudy Guide and Intervention (continued) 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. Example List the angles in order Example List the sides in order from smallest to largest measure. from shortest to longest. 6 cm 7 cm 9 cm 0,,,, List the angles and sides in order from smallest to largest cm cm Inequalities in One riangle.7 cm hapter 8 Glencoe Geometry If >, then m > m. If m > m, then >.,,,,,,,,,,,,, U,,,,, Q, P,, U,, U,, P, Q, QP E,,, X, Z, Y,,,,, E, E YZ, XY, XZ,, - Use the Exterior ngle Inequality heorem to list all of the angles that satisfy the stated condition.. measures less than m,, 4,, 7, 8. measures less than m 9, 4, 6, 7. measures greater than m, 4. measures greater than m 8,, List the angles and sides of each triangle in order from smallest to largest Q,,, P, PQ, Q K, M, L, ML, KL, KM F, H, G, HG, FG, FH X, Y, Z, YZ, XZ, XY kills Practice Inequalities in One riangle 4,,,,,, U,, U,, U hapter 9 Glencoe Geometry Lesson - nswers (Lesson -)

9 hapter 9 Glencoe Geometry - Practice Use the figure at the right to determine which angle has the greatest measure..,, 4. 4, 8, 9.,, , 8, 0 7 Use the Exterior ngle Inequality heorem to list all angles that satisfy the stated condition.. measures are less than m 6. measures are less than m 7. measures are greater than m 7 8. measures are greater than m Use the figure at the right to determine the relationship between the measures of the given angles. 9. m QW, m WQ 0. m W, m W. m, m. m WQ, m QW m > m m WQ < m QW Use the figure at the right to determine the relationship between the lengths of the given sides.. H, GH 4. E, G H > GH E < G. EG, FG Inequalities in One riangle, 4,, 7, 8, 7, 8,,, 9 6, 9 6. E, EG 7. PO he figure shows the position of three trees on one part of a Frisbee course. t which tree position is the angle between the trees the greatest? W E F 48 0 H 40 ft 7. ft ft hapter 0 Glencoe Geometry 4 0 m QW < m WQ m W < m W EG < FG E > EG Q G Word Problem Practice Inequalities in One riangle. INE arl and ose live on the same straight road. From their balconies they can see a flagpole in the distance. he angle that each person s line of sight to the flagpole makes with the road is the same. How do their distances from the flagpole compare? hey are equal.. OUE INGLE on notices that the side opposite the right angle in a right triangle is always the longest of the three sides. Is this also true of the side opposite the obtuse angle in an obtuse triangle? Explain. Yes. ince an obtuse triangle only has obtuse angle and acute angles, the side opposite the obtuse angle is the longest side.. ING Jake built a triangular structure with three black sticks. He tied one end of a string to vertex M and the other end to a point on the stick opposite M, pulling the string taut. Prove that the length of the string cannot exceed the longer of the two sides of the structure. M string ample answer: he string divides the triangle in two; one of these triangles is right or obtuse because one side of the string must make a right or obtuse angle with the stick. In this triangle, the side opposite the right or obtuse angle is longer than the string and that side is also a side of the triangle. 4. QUE Matthew has three different squares. He arranges the squares to form a triangle as shown. ased on the information, list the squares in order from the one with the smallest perimeter to the one with the largest perimeter.,, hapter Glencoe Geometry IIE tella is going to exas to visit a friend. 64 s she was looking at a map to see where 9 bilene she might want to go, she noticed the cities ustin, allas, and bilene formed a triangle. he wanted to determine how the distances between the cities were related, so she used a protractor to measure two angles. a. ased on the information in the figure, which of the two cities are nearest to each other? allas and bilene b. ased on the information in the figure, which of the two cities are farthest apart from each other? bilene and ustin allas ustin Lesson - nswers (Lesson -) nswers

10 hapter 0 Glencoe Geometry - Enrichment onstruction Problem 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. Existing oad P Q n E m k losed egion (Lake) Y X V Follow 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 P and Q on line m that lie well above the closed region. onstruct the perpendicular bisector, n, of PQ. 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 E. 4. Mark point X on line k so that X is below line n and so that EX is congruent to.. Mark points and V on line k and on opposite sides of X, so that X and XV are congruent. onstruct the perpendicular bisector, l, of V. all the point where the line l hits the boundary of the closed region point Y. XY corresponds to the new road. hapter Glencoe Geometry l - abri Junior can be used to investigate the relationships between angles and sides of a triangle. tep Use abri Junior. to draw and label a triangle. elect F riangle to draw a triangle. Move the cursor to where you want the first vertex. Press ENE. epeat this procedure to determine the next two vertices of the triangle. elect F lph-num to label each vertex. Move the cursor to a vertex, press ENE, enter, and press ENE again. epeat this procedure to label vertex and vertex. tep raw an exterior angle of. elect F Line to draw a line through. elect F Point, Point on to draw a point on so that is between and the new point. elect F lph-num to label the point. tep Find the measures of the three interior angles and the exterior angle,. elect F Measure, ngle. o find the measure of, select points,, and (with the vertex as the second point selected). epeat to find the remaining angle measures. tep 4 Find the measure of each side of. elect F Measure,. & Length. o find the length of, select point and then select point. epeat this procedure to find the lengths of and. Graphing alculator ctivity abri Junior: Inequalities in One riangle nalyze your drawing.. What is the relationship between m and m? m and m? ample answer: m > m ; m > m. Make a conjecture about the relationship between the measures of an exterior angle ( ) and its two remote interior angles ( and ). he measure of an exterior angle is equal to the sum of the measure of the two remote interior angles.. hange the dimensions of the triangle by moving point. (Press LE so the pointer becomes a black arrow. Move the pointer close to point until the arrow becomes transparent and point is blinking. Press LPH to change the arrow to a hand. hen move the point.) Is your conjecture still true? yes 4. Which side of the triangle is the longest? the shortest? ee students work.. Which angle measure (not including the exterior angle) is the greatest? the least? ee students work. 6. Make a conjecture about where the longest side is in relationship to the greatest angle and where the shortest side is in relationship to the least angle. he longest side is opposite the greatest angle. he shortest side is opposite the least angle. hapter Glencoe Geometry Lesson - nswers (Lesson -)

11 hapter Glencoe Geometry - he Geometer s ketchpad can be used to investigate the relationships between angles and sides of a triangle. tep tep tep Geometer s ketchpad ctivity Inequalities in One riangle Use he Geometer s ketchpad to draw a triangle and one exterior angle. onstruct a ray by selecting the ay tool from the toolbar. First, click where you want the first point. hen click a second point to draw the ray. Next, select the egment tool from the toolbar. Use the endpoint of the ray as the first point for the segment and click on a second point to construct the segment. onstruct another segment joining the second point of the previous segment to a point on the ray. m 69.9 m.9 m 4.78 m cm.7 cm.49 cm isplay the labels for each point. Use the election rrow tool to select all four points. isplay the labels by selecting how Label from the isplay menu. Find the measures of each angle. o find the measure of, use the election rrow tool to select points,, and (with the vertex as the second point selected). hen, under the Measure menu, select ngle. Use this method to find the remaining angle measures, including the exterior angle,. Find the measures of each side of the triangle. o find the measure of side, select and then. Next, under the Measure menu, select istance. Use this method to find the length of the other two sides. nalyze your drawing.. What is the relationship between m and m? m and m? ample answer: m > m ; m > m. Make a conjecture about the relationship between the measures of an exterior angle ( ) and its two remote interior angles ( and ). he measure of an exterior angle is equal to the sum of the measure of the two remote interior angles.. hange the dimensions of the triangle by selecting point with the pointer tool and moving it. Is your conjecture still true? yes 4. Which side of the triangle is the longest? the shortest? ee students work.. Which angle measure (not including the exterior angle) is the greatest? the least? ee students work. 6. Make a conjecture about where the longest side is in relationship to the greatest angle and where the shortest side is in relationship to the least angle. he longest side is opposite the greatest angle. he shortest side is opposite the least angle. hapter 4 Glencoe Geometry -4 tudy Guide and Intervention Indirect Proof Indirect lgebraic Proof One way to prove that a statement is true is to temporarily assume that what you are trying to prove is false. y showing this assumption to be logically impossible, you prove your assumption false and the original conclusion true. his is known as an indirect proof. teps for Writing an Indirect Proof. ssume that the conclusion is false by assuming the oppposite is true.. how that this assumption leads to a contradiction of the hypothesis or some other fact.. Point out that the assumption must be false, and therefore, the conclusion must be true. Example Given: x + > 8 Prove: x > tep ssume that x is not greater than. hat is, x = or x <. tep Make a table for several possibilities for x = or x <. When x = or x <, then x + is not greater than 8. tep his contradicts the given information that x + > 8. he assumption that x is not greater than must be false, which means that the statement x > must be true. tate the assumption you would make to start an indirect proof of each statement.. If x > 4, then x > 7. x 7. For all real numbers, if a + b > c, then a > c - b. a c - b omplete the indirect proof. Given: n is an integer and n is even. Prove: n is even.. ssume that n is not even. hat is, assume n is odd. 4. hen n can be expressed as a + by the meaning of odd number.. n = (a + ) ubstitution 6. = (a + )(a + ) Multiply. 7. = 4a + 4a + implify. 8. = (a + a) + istributive Property 9. (a + a)+ is an odd number. his contradicts the given that n is even, so the assumption must be false. 0. herefore, n is even. x x hapter Glencoe Geometry Lesson -4 nswers (Lesson - and Lesson -4) nswers

12 hapter Glencoe Geometry - he Geometer s ketchpad can be used to investigate the relationships between angles and sides of a triangle. tep tep tep Geometer s ketchpad ctivity Inequalities in One riangle Use he Geometer s ketchpad to draw a triangle and one exterior angle. onstruct a ray by selecting the ay tool from the toolbar. First, click where you want the first point. hen click a second point to draw the ray. Next, select the egment tool from the toolbar. Use the endpoint of the ray as the first point for the segment and click on a second point to construct the segment. onstruct another segment joining the second point of the previous segment to a point on the ray. m 69.9 m.9 m 4.78 m cm.7 cm.49 cm isplay the labels for each point. Use the election rrow tool to select all four points. isplay the labels by selecting how Label from the isplay menu. Find the measures of each angle. o find the measure of, use the election rrow tool to select points,, and (with the vertex as the second point selected). hen, under the Measure menu, select ngle. Use this method to find the remaining angle measures, including the exterior angle,. Find the measures of each side of the triangle. o find the measure of side, select and then. Next, under the Measure menu, select istance. Use this method to find the length of the other two sides. nalyze your drawing.. What is the relationship between m and m? m and m? ample answer: m > m ; m > m. Make a conjecture about the relationship between the measures of an exterior angle ( ) and its two remote interior angles ( and ). he measure of an exterior angle is equal to the sum of the measure of the two remote interior angles.. hange the dimensions of the triangle by selecting point with the pointer tool and moving it. Is your conjecture still true? yes 4. Which side of the triangle is the longest? the shortest? ee students work.. Which angle measure (not including the exterior angle) is the greatest? the least? ee students work. 6. Make a conjecture about where the longest side is in relationship to the greatest angle and where the shortest side is in relationship to the least angle. he longest side is opposite the greatest angle. he shortest side is opposite the least angle. hapter 4 Glencoe Geometry -4 tudy Guide and Intervention Indirect Proof Indirect lgebraic Proof One way to prove that a statement is true is to temporarily assume that what you are trying to prove is false. y showing this assumption to be logically impossible, you prove your assumption false and the original conclusion true. his is known as an indirect proof. teps for Writing an Indirect Proof. ssume that the conclusion is false by assuming the oppposite is true.. how that this assumption leads to a contradiction of the hypothesis or some other fact.. Point out that the assumption must be false, and therefore, the conclusion must be true. Example Given: x + > 8 Prove: x > tep ssume that x is not greater than. hat is, x = or x <. tep Make a table for several possibilities for x = or x <. When x = or x <, then x + is not greater than 8. tep his contradicts the given information that x + > 8. he assumption that x is not greater than must be false, which means that the statement x > must be true. tate the assumption you would make to start an indirect proof of each statement.. If x > 4, then x > 7. x 7. For all real numbers, if a + b > c, then a > c - b. a c - b omplete the indirect proof. Given: n is an integer and n is even. Prove: n is even.. ssume that n is not even. hat is, assume n is odd. 4. hen n can be expressed as a + by the meaning of odd number.. n = (a + ) ubstitution 6. = (a + )(a + ) Multiply. 7. = 4a + 4a + implify. 8. = (a + a) + istributive Property 9. (a + a)+ is an odd number. his contradicts the given that n is even, so the assumption must be false. 0. herefore, n is even. x x hapter Glencoe Geometry Lesson -4 nswers (Lesson - and Lesson -4) nswers

13 hapter Glencoe Geometry -4 tudy Guide and Intervention (continued) Indirect Proof 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. Example Given: m = 00 Prove: is not a right angle. tep ssume that is a right angle. tep how that this leads to a contradiction. If is a right angle, then m = 90 and m + m = = 90. hus the sum of the measures of the angles of is greater than 80. tep he conclusion that the sum of the measures of the angles of is greater than 80 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. Indirect Proof tate the assumption you would make to start an indirect proof of each statement.. If m = 90, then m = 4. m 4. If V is not congruent to VE, then VE is not isosceles. VE is isosceles. omplete the indirect proof. Given: and G is not congruent to FG. Prove: E is not congruent to FE.. ssume that E FE. ssume the conclusion is false. 4. EG EG eflexive Property. EG EFG 6. G FG P 7. his contradicts the given information, so the assumption must be false. 8. herefore, E is not congruent to FE. hapter 6 Glencoe Geometry E G F -4 kills Practice Indirect Proof tate the assumption you would make to start an indirect proof of each statement.. m < m m m. EF EF. Line a is perpendicular to line b. Line a is not perpendicular to line b. 4. is supplementary to 6. is not supplementary to 6. Write an indirect proof of each statement.. Given: x + 8 Prove: x 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: F Prove: E EF E F tep : ssume E = EF. tep : If E = EF, then E EF by the definition of congruent segments. ut if E EF, then F by the Isosceles riangle heorem. his contradicts the given information that F. tep : ince the assumption that E = EF leads to a contradiction, it must be false. herefore, it must be true that E EF. hapter 7 Glencoe Geometry Lesson -4 nswers (Lesson -4)

14 hapter Glencoe Geometry -4 Practice Indirect Proof tate the assumption you would make to start an indirect proof of each statement.. bisects. does not bisect.. = Write an indirect proof of each statement.. Given: -4x + < -0 Prove: x > 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 Prove: a b 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.. PHYI ound travels through air at about 44 meters per second when the temperature is 0. If Enrique lives kilometers from the fire station and it takes seconds for the sound of the fire station siren to reach him, how can you prove indirectly that it is not 0 when Enrique hears the siren? ssume that it is 0 when Enrique hears the siren, then show that at this temperature it will take more than 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 Enrique hears the siren. hapter 8 Glencoe Geometry a b -4 Word Problem Practice Indirect Proof. NOE hirty-five students went on a canoeing expedition. hey rented 7 canoes for the trip. Use an indirect proof to show that at least one canoe had more than two students in it. ample answer: uppose all canoes had students, then the total would be less than or equal to 7 = 4, a contradiction.. E he area of the United tates is about 6,000,000 square miles. he area of Hawaii is about,000 square miles. Use an indirect proof to show that at least one of the fifty states has an area greater than 0,000 square miles. ample answer: uppose no state has area > 0,000 mi. hen the total area could not exceed 0, ,000 =,89,000, a contradiction.. ONEUIVE NUME avid was trying to find a common factor other than between various pairs of consecutive integers. Write an indirect proof to show avid that two consecutive integers do not share a common factor other than. ample answer: ssume x and y are integers with a common factor greater than. For consecutive integers one is even and the other is odd, so x = a and y = a +, for an integer a. Let n be the common factor greater that. herefore x n = a n is an integer and y n = a+ n is also an integer. ut a+ = a n n + n and is not an integer unless n n =, a contradiction. 4. WO he words accomplishment, counterexample, and extemporaneous all have 4 letters. Use an indirect proof to show that any word with 4 letters must use a repeated letter or have two letters that are consecutive in the alphabet. uppose the letters are distinct and nonconsecutive. hen the alphabet must have at least 4 + or 7 letters, a contradiction.. LIE INGLE lattice point is a point whose coordinates are both integers. lattice triangle is a triangle whose vertices are lattice points. It is a fact that a lattice triangle has an area of at least 0. square units. hapter 9 Glencoe Geometry y O x a. uppose has a lattice point in its interior. how that the lattice triangle can be partitioned into three smaller lattice triangles. ample answer in diagram above. b. Prove indirectly that a lattice triangle with area 0. square units contains no lattice point. (eing on the boundary does not count as inside.) ample answer: From Exercise a, the lattice triangle contains smaller lattice triangles, each of which has area at least 0. square units. he original would then have area at least. square units, a contradiction. Lesson -4 nswers (Lesson -4) nswers

15 hapter 4 Glencoe Geometry -4 Enrichment More ounterexamples ome statements in mathematics can be proven false by counterexamples. onsider the following statement. For any numbers a and b, a - b = b - a. You can prove that this statement is false in general if you can find one example for which the statement is false. Let a = 7 and b =. ubstitute these values in the equation above In general, for any numbers a and b, the statement a - b = b - a is false. You can make the equivalent verbal statement: subtraction is not a commutative operation. In each of the following exercises a, b, and c are any numbers. Prove that the statement is false by counterexample. ample answers are given.. a - (b - c) (a - b) - c. a (b c) (a b) c 6 - (4 - ) (6-4) - 6 (4 ) (6 4) a b b a 4. a (b + c) (a b) + (a c) (4 + ) (6 4) +(6 ) a + (bc) (a + b)(a + c) 6. a + a a (4. ) (6 + 4) (6 + ) (0) (8) Write the verbal equivalents for,, and.. ubtraction is not an associative operation.. ivision is not an associative operation.. ivision is not a commutative operation. 8. For the istributive Property, a(b + c) = ab + ac, it is said that multiplication distributes over addition. 4 and prove that some operations do not distribute. Write a statement for each exercise that indicates this. 4. ivision does not distribute over addition.. ddition does not distribute over multiplication. hapter 0 Glencoe Geometry - tudy Guide and Intervention he riangle Inequality If you take three straws of lengths 8 inches, 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 riangle Inequality he sum of the lengths of any two sides of a triangle must be greater than the length of the third side. a a + b > c b + c > a a + c > b Example he measures of two sides of a triangle are and 8. Find a range for the length of the third side. y the riangle Inequality heorem, all three of the following inequalities must be true. + x > x > + 8 > x x > x > - > x herefore x must be between and. Is it possible to form a triangle with the given side lengths? If not, explain why not.., 4, 6 yes. 6, 9, no; =. 8, 8, 8 yes 4., 4, yes. 4, 8, 6 no; < 6 6..,., yes Find the range for the measure of the third side of a triangle given the measures of two sides. 7. cm and 6 cm 8. yd and 8 yd cm < n < 7 cm 6 yd < n < 0 yd 9.. ft and. ft 0. 8 m and 8 m 4 ft < n < 7 ft 74 m < n < 90 m. 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? Find 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. hapter Glencoe Geometry b c Lesson - nswers (Lesson -4 and Lesson -)

16 hapter 4 Glencoe Geometry -4 Enrichment More ounterexamples ome statements in mathematics can be proven false by counterexamples. onsider the following statement. For any numbers a and b, a - b = b - a. You can prove that this statement is false in general if you can find one example for which the statement is false. Let a = 7 and b =. ubstitute these values in the equation above In general, for any numbers a and b, the statement a - b = b - a is false. You can make the equivalent verbal statement: subtraction is not a commutative operation. In each of the following exercises a, b, and c are any numbers. Prove that the statement is false by counterexample. ample answers are given.. a - (b - c) (a - b) - c. a (b c) (a b) c 6 - (4 - ) (6-4) - 6 (4 ) (6 4) a b b a 4. a (b + c) (a b) + (a c) (4 + ) (6 4) +(6 ) a + (bc) (a + b)(a + c) 6. a + a a (4. ) (6 + 4) (6 + ) (0) (8) Write the verbal equivalents for,, and.. ubtraction is not an associative operation.. ivision is not an associative operation.. ivision is not a commutative operation. 8. For the istributive Property, a(b + c) = ab + ac, it is said that multiplication distributes over addition. 4 and prove that some operations do not distribute. Write a statement for each exercise that indicates this. 4. ivision does not distribute over addition.. ddition does not distribute over multiplication. hapter 0 Glencoe Geometry - tudy Guide and Intervention he riangle Inequality If you take three straws of lengths 8 inches, 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 riangle Inequality he sum of the lengths of any two sides of a triangle must be greater than the length of the third side. a a + b > c b + c > a a + c > b Example he measures of two sides of a triangle are and 8. Find a range for the length of the third side. y the riangle Inequality heorem, all three of the following inequalities must be true. + x > x > + 8 > x x > x > - > x herefore x must be between and. Is it possible to form a triangle with the given side lengths? If not, explain why not.., 4, 6 yes. 6, 9, no; =. 8, 8, 8 yes 4., 4, yes. 4, 8, 6 no; < 6 6..,., yes Find the range for the measure of the third side of a triangle given the measures of two sides. 7. cm and 6 cm 8. yd and 8 yd cm < n < 7 cm 6 yd < n < 0 yd 9.. ft and. ft 0. 8 m and 8 m 4 ft < n < 7 ft 74 m < n < 90 m. 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? Find 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. hapter Glencoe Geometry b c Lesson - nswers (Lesson -4 and Lesson -)

17 hapter Glencoe Geometry - tudy Guide and Intervention (continued) he riangle Inequality Proofs Using he riangle Inequality heorem You can use the riangle Inequality heorem as a reason in proofs. omplete the following proof. Given: E Prove: + E > E tatements. E. + > E + E >. > E > E 4. + E > E. + E > E 6. + E > + - ( + E) 7. + = + E = E 8. + E > - E easons. Given. riangle Inequality heorem. ubtraction 4. ddition. ommutative 6. istributive 7. egment ddition Postulate 8. ubstitution POOF Write a two column proof. Given: PL M K is the midpoint of P. Prove: PK + KM > PL tatements easons. PL M. Given. P. K is the midpoint of P. lternate Interior ngles heorem.. Given 4. PK = K 4. efinition of midpoint. PKL MK. Vertical ngles heorem 6. PKL KM PK + KL > PL 7. riangle Inequality heorem 8. KL = KM 8. P 9. PK + KM > PL 9. ubstitution hapter Glencoe Geometry - kills Practice he riangle Inequality Is it possible to form a triangle with the given side lengths? If not, explain why not.. ft, ft, 4 ft. m, 7 m, 9 m yes yes. 4 mm, 8 mm, mm 4. in., in., 6 in. yes no; cm, 0 cm, 0 cm 6. km, 7 km, 9 km no; yes 7. 4 yd, 7 yd, yd 8. 6 m, 7 m, m no; yes Find the range for the measure of the third side of a triangle given the measures of two sides. 9. ft, 9 ft 0. 7 in., 4 in. 4 ft < n < 4 ft. 8 m, m. 0 mm, mm m < n < m mm < n < mm. yd, yd 4. km, 7 km yd < n < 7 yd. 7 cm, 8 cm, 6. 8 ft, ft cm < n < 4 cm 4 ft < n < 40 ft 7. Proof omplete the proof. Given: and E Prove: E > E tatements. + > + E > E easons. riangle Inequality heorem E > + E. ddition Property of Equality. + E = E. eg. ddition Post E > E 4. ubstitution 7 in. < n < in. km < n < 4 km hapter Glencoe Geometry Lesson - nswers (Lesson -) nswers

18 hapter 6 Glencoe Geometry - Practice he riangle Inequality Is it possible to form a triangle with the given side lengths? If not explain why not.. 9,, 8 yes. 8, 9, 7 no; = 7. 4, 4, 9 yes 4., 6, 0 no; + 6 < 0., 4, 6 yes 6..7,., 4. yes ,.4,. no; =. 8..,.9,. yes Find the range for the measure of the third side of a triangle given the measures of two sides ft and 9 ft 0. 7 km and 9 km ft < n < ft km < n < 6 km. in. and 7 in.. 8 ft and ft 4 in. < n < 40 in. ft < n < 4 ft. yd and 8 yd 4. cm and 9 cm yd < n < 6 yd 8 cm < n < 70 cm. 4 m and 6 m 6. 4 in. and 7 in. 6 m < n < 48 m 47 in. < n < 6 in. 7. Given: H is the centroid of EF Prove: EY + FY > E tatements easons. H is the centroid of EF. EY is a median.. Y is the midpoint of F 4. Y = FY. EY + Y > E 6. EY + FY > E. Given. efinition of centroid. efinition of median 4. efinition of midpoint. riangle Inequality heorem 6. ubstitution 8. GENING Ha Poong has 4 lengths of wood from which he plans to make a border for a triangular-shaped herb garden. he lengths of the wood borders are 8 inches, 0 inches, inches, and 8 inches. How many different triangular borders can Ha Poong make? hapter 4 Glencoe Geometry - Word Problem Practice he riangle Inequality. IK Jamila has sticks of lengths, 4, 6, 8, and 0 inches. Using three sticks at a time as the sides of triangles, how many triangles can she make? Use the figure at the upermarket right for and. ailroad. PH o get to the nearest super market, anya must walk over a railroad track. here are two places where anya s home she can cross the track (points and ). Which path is longer? Explain. y the riangle Inequality heorem, the distance from anya s home to point and on to the supermarket is greater than the straight distance from anya s home to the upermarket.. PH While out walking one day anya finds a third place to cross the railroad tracks. how that the path through point is longer than the path through point. ample answer: Let be the upermarket and be anya s home. ecause is 90, m < 90, so m > 90, making >. imilarly, >. herefore + > IIE he distance between New York ity and oston is 87 miles and the distance between New York ity and Hartford is 97 miles. Hartford, oston, and New York ity form a triangle on a map. What must the distance between oston and Hartford be greater than? 90 mi. INGLE he figure shows an equilateral triangle and a point P outside the triangle. hapter Glencoe Geometry P a. raw the figure that is the result of turning the original figure 60 counterclockwise about. enote by P', the image of P under this turn. ee figure. b. Note that P' is congruent to P. It is also true that PP' is congruent to P. Why? ample answer: P is congruent to P' and m PP' is 60, o by, triangle PP' is equilateral. hus, PP' = P c. how that P, P, and P satisfy the triangle inequalities. ample answer: P'P is a triangle with side lengths equal to P, P, and P. P Lesson - nswers (Lesson -)

19 hapter 7 Glencoe Geometry - Enrichment onstructing riangles he measurements of the sides of a triangle are given. If a triangle having sides with these measurements is not possible, then write impossible. If a triangle is possible, draw it and measure each angle with a protractor.. = cm m = 0. PI = 8 cm m P = = cm m = 94 IN = cm m I = = 6 cm m = 6 PN = cm m N = impossible. ON = 0 cm m O = 4. W = 6 cm m = NE =. cm m N = WO = 7 cm m W = OE = 4.6 cm m E = O = cm m O = impossible W. =.l cm m = 6 6. = 4 cm m = 90 = 8 cm m = M = cm m = 7 = cm m = 7 M = cm m M = hapter 6 Glencoe Geometry M O -6 tudy Guide and Intervention Hinge heorem he following theorem and its converse involve the relationship between the sides of two triangles and an angle in each triangle. Hinge heorem onverse of the Hinge heorem Example of GF and FE. H If two sides of a triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. > If two sides of a triangle are congruent to two sides of another triangle, and the N third side in the first is longer than the 6 third side in the second, then the included angle in the first triangle is greater than M P the included angle in the second triangle. m M > m ompare the given measures.. M and P Inequalities in wo riangles ompare the measures 8 wo sides of HGF are congruent to two sides of HEF, and m GHF > m EHF. y the Hinge heorem, GF > FE. P M 9 N. and 8 M > P >. m and m Z 4. m XYW and m WYZ 4 G F E 48 0 X m < m Z m XYW < m WYZ Write an inequality for the range of values of x.. (4x - 0) (x - ) 0 60 x >. x < 4 Z 0 0 Y Example ompare the measures of and. hapter 7 Glencoe Geometry wo sides of are congruent to two sides of, and >. y the onverse of the Hinge heorem, m > m. W 6 8 X 4 Y Z Lesson - nswers (Lesson - and Lesson -6) nswers