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The midpoint of a line segment divides the segment into two congruent segments. 5. Given. 6. Perpendicular lines intersect to form right angles. 7. If two angles are right angles, they are congruent. 8. ASA (steps 2, 4, 7). 1. Marty said that if two triangles are congruent and one of the triangles is a right triangle,... |
IDE, SIDE Proving Triangles Congruent Using Side, Side, Side 165 Cut three straws to any lengths and put their ends together to form a triangle. Now cut a second set of straws to the same lengths and try to form a different triangle. Repeated experiments lead to the conclusion that it cannot be done. As shown in ABC an... |
property of congruence. In each case, is the given information sufficient to prove congruent triangles? 3. C A B D 4. C D F E B A 5. W Z R S T In 6–8, two sides are marked to indicate that they are congruent. Name the pair of corresponding sides that would have to be proved congruent in order to prove the triangles co... |
on the angle bisector and. Prove that the triangles that she drew are congruent. draws and BD CD CHAPTER SUMMARY Definitions to Know • Adjacent angles are two angles in the same plane that have a common vertex and a common side but do not have any interior points in common. • Complementary angles are two angles the su... |
to each other. (Transitive Property) 4.14 Two triangles are congruent if two sides and the included angle of one triangle are congruent, respectively, to two sides and the included angle of the other. (SAS) 4.15 Two triangles are congruent if two angles and the included side of one triangle are congruent, respectively... |
a 12. Find the measure of K and of Q. 4. If and ABC DBE tulate that justifies your answer. intersect at F, what is true about B and F? State a pos- 5. If g LM'MN and g KM'MN, what is true about g LM and g KM? State a postulate that justifies your answer. 6. Point R is not on LMN. Is LM MN less than, equal to, or greate... |
(4) 3(7) (3) 3 (4 7) 3 (7 4) (4) 3 (4 7) (3 4) 7 14365C04.pgs 7/12/07 3:05 PM Page 171 Cumulative Review 171 2. If the sum of the measures of two angles is 90, the angles are (1) supplementary. (2) complementary. (3) a linear pair. (4) adjacent angles. 3. If AB BC AC, which of the following may be false?. AC (1) B is t... |
180 (2) LMN and NMP are supplementary angles. h ML h ML (3) (4) and and h MP h MN are opposite rays. are opposite rays. 10. The solution set of the equation 3(x 2) 5x is (1) {x x 3} (2) {x x 3} (3) {x x 1} (4) {x x 1} 14365C04.pgs 7/12/07 3:05 PM Page 172 172 Congruence of Line Segments, Angles, and Triangles Part II ... |
Find the measure of each angle of the triangle. 16. Triangle DEF is equilateral and equiangular. The midpoint of is L. Line segments is N, and of DE are drawn., and MN ML FD NL EF, is M, of a. Name three congruent triangles. b. Prove that the triangles named in a are congruent. c. Prove that NLM is equilateral. d. Pro... |
SEGMENTS ASSOCIATED WITH TRIANGLES Line Segments Associated with Triangles 175 Natalie is planting a small tree. Before 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 ... |
Bisector of a Triangle DEFINITION An angle bisector of a triangle is a line segment that bisects any angle of the triangle and terminates in the side opposite that angle. In PQR, if D is a point on is the angle bisector from R in PQR. We may also draw an angle bisector from the, and an angle bisector from the vertex Q... |
sca- lene triangle with altitude CD, angle bisector CE, and median. CF a. Name two congruent angles that have their vertices at C. 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 tr... |
and h the length of the altitude to that side. In ABC, side and M is the midpoint of angles of equal area, AMC and BMC. AB AB 1 2bh with b the length of one side of a triangle is the altitude from vertex C to. Show that the median separates ABC into two tri- CD 13. A farmer has a triangular piece of land that he wants... |
). 10. Corresponding parts of congruent triangles are congruent. Exercises Writing About Mathematics 1. Triangles ABC and DEF are congruent triangles. If A and B are complementary angles, are D and E also complementary angles? Justify your answer. 2. A leg and the vertex angle of one isosceles triangle are congruent re... |
isosceles triangle separates the triangle into two congruent triangles. b. Prove that the two congruent triangles in a are right triangles. 16. a. Prove 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. Prove that a ... |
gruence. 4. Definition of a midpoint. 5. Given. 6. SSS (steps 3, 4, 5). 7. Corresponding parts of congruent triangles are congruent. A 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 fol... |
triangle are congruent, the angles opposite these sides are congruent.). 3. Given. Continued 14365C05.pgs 7/10/07 8:41 AM Page 184 184 Congruence Based on Triangles (Continued) E Statements 4. ABE and EBC are supplementary. DCE and ECB are supplementary. 5. ABE DCE A B C D AB > CD 6. 7. ABE DCE 8. AE > DE Reasons 4. I... |
isosceles. AC, C D E A F B In 11 and 12, complete each given proof with a partner or in a small group. 11. Given: ABC with AB AC, BG EC,, and 12. Given: E not on BE'DE CG'GF and EB. AB > CD,, ABCD is not congruent to. EC Prove: BD > CF Prove: AE is not congruent to DE. A D F B E G C Applying Skills E A B C D (Hint: Us... |
E DAE 8. CE > DE 1. Given. 2. Given. 3. Reflexive property of congruence. 4. SSS (steps 1, 2, 3). 5. Corresponding parts of congruent triangles are congruent. 6. Reflexive property of congruence. 7. SAS (steps 1, 5, 6). 8. Corresponding parts of congruent triangles are congruent. 14365C05.pgs 7/10/07 8:41 AM Page 187 U... |
PB and a. Let half the group treat the case in which P and S are on the same side of AB. b. Let half the group treat the case in which P and S are on opposite sides of AB. c. Compare and contrast the methods used to prove the cases. 5-5 PROVING OVERLAPPING TRIANGLES CONGRUENT AD > BC DB > CA, can we prove If we know t... |
21 CD x 15 6 15 21 Answer 21 Exercises Writing About Mathematics 1. In Example 1, the medians to the legs of isosceles ABC were proved to be congruent by proving ABE ACD. Could the proof have been done by proving DBC ECB? Justify your answer. 14365C05.pgs 7/10/07 8:41 AM Page 190 190 Congruence Based on Triangles 2. I... |
a line segment was defined as any line or subset of a line that intersects a line segment at its midpoint In the diagrams, g, PM g, NM QM, and h MR are all bisectors of AB since they each intersect AB at its midpoint, M. Only is both perpendicular to AB and g NM g NM is the perpendicular bisector of. AB the bisector o... |
points of a line segment, then it is on the perpendicular bisector of the line segment. Given Point P such that PA = PB. Prove P lies on the perpendicular bisector of. AB Proof Choose any other point that is equidistant from AB. Then AB by Theorem 5.2. (If two points, for example, M, the midg is the perpendicular PM th... |
urrence theorem. C N P L A M B 14365C05.pgs 7/10/07 8:41 AM Page 194 194 Congruence Based on Triangles Theorem 5.4 The perpendicular bisectors of the sides of a triangle are concurrent. Given g MQ, the perpendicular bisector of AB C C Proof EXAMPLE 1 g NR, the perpendicular bisector of AC g LS, the perpendicular bisect... |
ent sides. Exercises Writing About Mathematics 1. Justify the three methods of proving that two lines are perpendicular given in this section. 2. Compare and contrast Example 1 with Corollary 5.1b, “The median from the vertex angle of an isosceles triangle is perpendicular to the base.” Developing Skills 3. If RS ASB i... |
is on B. 3. Using the same compass radius, place the point on C and, with the pencil, draw an arc that h CX intersects of intersection D.. Label this point 14365C05.pgs 8/2/07 5:43 PM Page 197 Conclusion CD > AB Proof Since AB and CD are radii of congruent circles, they are congruent. Basic Constructions 197 Construct... |
icular bisector of AB. Finally, M is the point on AB where the perpendicular bisector intersects AB, so AM = BM. By definition, M is the midpoint of AB. M D B 4. Use a straight- edge to draw g CD intersecting AB at M. C A M D B 14365C05.pgs 8/2/07 5:43 PM Page 199 Basic Constructions 199 Construction 4 Bisect a Given A... |
idistant to C and D. 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 PE CD. Since CD is the perpendicular bisector of g'AB is a subset of line g AB g PE,. E P C D A B 14365C05.pgs 8/2/07 5:43 ... |
to a given line through a point not on the given line) are alike and how they are different. Developing Skills 3. Given: AB Construct: a. A line segment congruent to. AB b. A line segment whose measure is 2AB. A B c. The perpendicular bisector of d. A line segment whose measure is. AB 11. 2AB 4. Given: A Construct: a.... |
C05.pgs 8/2/07 5:43 PM Page 204 204 Congruence Based on Triangles CHAPTER SUMMARY Definitions to Know • An 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. • A median of a triangle is a line segment that joins any ve... |
05.pgs 8/2/07 5:44 PM Page 205 Review Exercises 205 3. In ABC, CD is both the median and the altitude. If AB 5x 3, AC 2x 8, and BC 3x 5, what is the perimeter of ABC? 4. Angle PQS and angle SQR are a linear pair of angles. If mPQS 5a 15 and mSQR 8a 35, find mPQS and mSQR. 5. Let D be the point at which the three perpen... |
proof? b. Did the reassembled proof match your partner’s original proof? c. Did you find any components missing or have any components left- over? Why? 14365C05.pgs 7/10/07 8:42 AM Page 206 206 Congruence Based on Triangles CUMULATIVE REVIEW Chapters 1–5 Part I Answer all questions in this part. Each correct answer wi... |
(1) SSS (2) SAS (3) ASA (4) SSA 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... |
diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 15. Given: Point P is not on ABCD and P PB PC. Prove: ABP DCP A B C D 16. Prove that if AC and BD are perpendicular bisectors of each other, quadrilateral ABCD is equilateral (AB BC ... |
a vertical line, the y-axis, which are perpendicular and intersect at a point called the origin. Every point on a plane can be described by two numbers, called the coordinates of the point, usually written as an ordered pair. The first number in the pair, called the x-coordinate or the abscissa, is the distance from t... |
a point in the coordinate plane uniquely determines the coordinates of the point. Procedure To find the coordinates of a point: 1. From the point, move along a vertical line to the x-axis.The number on the x-axis is the x-coordinate of the point. 2. From the point, move along a horizontal line to the y-axis.The number... |
Points on the same vertical line have the same x-coordinate. Therefore, we can find BC and DA by subtracting their y-coordinates. BC DA 2 (2) 2 2 4 14365C06.pgs 7/12/07 2:57 PM Page 213 The Coordinates of a Point in a Plane 213 EXAMPLE 1 Graph the following points: A(4, 1), B(1, 5), C(2,1). Then draw ABC and find its ... |
(5, 3), E(5, 3), N(2, 0) 9. B(3, 2), A(2, 2), R(2, 2), N(3, 2) 11. R(4, 2), A(0, 2), M(0, 7) 8. F(5, 1), A(5, 5), R(0, 5), M(2, 1) 10. P(3, 0), O(0, 0), N(2, 2), D(1, 2) 12. M(1, 1), I(3, 1), L(3, 3), K(1, 3) 13. Graph points A(1, 1), B(5, 1), and C(5, 4). What must be the coordinates of point D if ABCD is a quadrilate... |
of reflection, point A corresponds to point A (in symbols, A → A) and point B corresponds to point B (B → B). Point C is a fixed point because it is a point on the line of reflection. In other words, C corresponds to itself (C → C). Under a reflection in line k, then, ABC corresponds to ABC (ABC → ABC). Each of the po... |
this theorem by using the definition of a reflection and SAS to show that BDC BDC and that using the fact that if two angles are congruent, their complements are congruent (Theorem 4.4), we can show that ACB ACB. From this last statement, we can conclude that AB AB. BC > BrC. Then, k 14365C06.pgs 7/12/07 2:57 PM Page ... |
aries: B C D A M k M 14365C06.pgs 7/12/07 2:57 PM Page 217 Corollary 6.1a Under a line reflection, angle measure is preserved. Line Reflections 217 Proof: We found that ABC ABC. Therefore, ABC ABC because the angles are corresponding parts of congruent triangles. Corollary 6.1b Under a line reflection, collinearity is ... |
DN ND. DN g CM 4. Draw CrDr. Line Symmetry We know that the altitude to the base of an isosceles triangle is also the median and the perpendicular bisector of the base. If we imagine that isosceles triangle ABC, shown at the left, is folded along the perpendicular bisector of the base so that A falls on C, the line al... |
of the figure. This line of reflection is a line of symmetry, or an axis of symmetry, and the figure has line symmetry. DEFINITION A figure has line symmetry when the figure is its own image under a line reflection. 14365C06.pgs 7/12/07 2:57 PM Page 220 220 Transformations and the Coordinate Plane It is possible for a... |
a right triangle with mN 90, prove that LMN is a right triangle. In 5–9, under a reflection in line k, the image of A is A, the image of B is B, the image of C is C, and the image of D is D. 5. Is ABC ABC? Justify your answer. 6. If AC'BC, is ArCr'BrCr? Justify your answer? 7. The midpoint of AB is M. If the image of ... |
under a reflection in the y-axis, the y-coordinate of the image is the same as the y-coordinate of the point; the x-coordinate of the image is the opposite of the x-coordinate of the point. Note that for a reflection in the y-axis, the image of (1, 2) is (1, 2) and the image of (1, 2) is (1, 2). O x 1 A reflection in ... |
is the absolute value of the difference of the x-coordinates of the endpoints. intersects the PPr PQ a 0 a and PQ a 0 = a Since PQ PQ, Q is the midpoint of Steps 1 and 2 prove that if P has the coordinates (a, b) and P has the coor, and therefore, dinates (a, b), the y-axis is the perpendicular bisector of the image o... |
Transformations and the Coordinate Plane EXAMPLE 1 On graph paper: a. Locate A(3, 1). b. Locate A, the image of A under a reflection in the y-axis, and write its coordinates. c. Locate A, the image of A under a reflection in the x-axis, and write its coordinates. Answers y 1 1O x Reflection in the Line y = x In the fi... |
the distance between them is the absolute value of their y-coordinates and if two points have the same y-coordinate, then the distance between them is the absolute value of their x-coordinates. PQ b a and PQ b a Therefore, Q is equidistant from P and P. y P(a, b) R(b, b) Q(a, a) P(b, a) x PR b a and PR = b a Therefore... |
(3, 5) 9. (1, 4) 10. (2, 3) 11. (2, 3) 12. (1, 0) In 13–17: a. On graph paper, locate each point and its image under ryx. b. Write the coordinates of the image point. 13. (3, 5) 14. (3, 5) 15. (4, 2) 16. (1, 5) 17. (2, 2) Applying Skills 18. Prove Theorem 6.3, “Under a reflection in the x-axis, the image of P(a, b) is... |
: ArBr is the image of AB under a reflection in line k. 14365C06.pgs 7/12/07 2:57 PM Page 227 For each figure, construct the reflection in the given line. Point Reflections in the Coordinate Plane 227 AB a. Segment with vertices A(4, 2) and B(2, 4), and line k through points C(2, 1) and D(0, 5). b. Angle EFG with verti... |
, CP PC, and P is the midpoint of.CCr A C P B BBr. DEFINITION A point reflection in P is a transformation of the plane such that: 1. If point A is not point P, then the image of A is A and P the midpoint of 2. The point P is its own image. AAr. 14365C06.pgs 7/12/07 2:57 PM Page 228 228 Transformations and the Coordinat... |
5 and its corollaries can be summarized in the following statement. Under a point reflection, distance, angle measure, collinearity, and midpoint are preserved. We use RP as a symbol for the image under a reflection in point P. For example, RP(A) B means “The image of A under a reflection in point P is B.” R(1, 2)(A) A... |
the x-axis and a vertical line through P. Then: OB 0 a a Let B be the point (a, 0). Then: OB 0 (a) a OB OB and OB > OBr POB POB because vertical angles are congruent. Therefore, POB POB by SAS. In particular, P PB PB 0 b b Since OBP is a right angle, OBP is a right angle and is a vertical line. Therefore, P has the sa... |
) (a, b) while rx-axis(a, b) (a, b) ry-axis(a, b) (a, b) Therefore, a reflection in the origin gives the same result as a reflection in the x-axis followed by a reflection in the y-axis. Answer Exercises Writing About Mathematics 1. Ada said if the image of AB under a reflection in point P is ArBr, then the image of Ar... |
same as the reflection in the x-axis? Justify your answer. 16. a. What is the image of A(4, 4) under a reflection in the point P(4, 2)? b. What is the image of C(1, 2) under a reflection in the point P(4, 2)? c. The point D(2, 0) lies on the segment AB. Does the image of D lie on the image of AB? Justify your answer. ... |
coordinate of the point: x → x 4 2. The y-coordinate of the image is 5 less than the y-coordinate of the point: From this example, we form a general rule: y → y 5 DEFINITION A translation of a units in the horizontal direction and b units in the vertical direction is a transformation of the plane such that the image of... |
are left to the student. (See exercise 10.) Corollary 6.7a Under a translation, angle measure is preserved. Corollary 6.7b Under a translation, collinearity is preserved. Corollary 6.7c Under a translation, midpoint is preserved. 14365C06.pgs 7/31/07 1:21 PM Page 235 Translations in the Coordinate Plane 235 We can wri... |
reflection in the line x 3, then A is the image of A under the translation (x, y) → (x 6, y). a. Do you agree with Hunter when A is a point with an x-coordinate greater than or equal to 3? Justify your answer. b. Do you agree with Hunter when A is a point with an x-coordinate greater than 0 but less than 3? Justify yo... |
C(5, 5). c. Use the rule from part b to find the coordinates of B, the image of B, and C, the image of C, under this translation. d. On the graph drawn in part a, draw and label CBC, the image of ABC. 9. The coordinates of the vertices of ABC are A(0, 1), B(2, 1), and C(3, 3). a. On graph paper, draw and label ABC. b.... |
y 0 (the x-axis), and draw DEF on the graph drawn in a. c. Find the coordinates of the vertices of DEF, the image of DEF under a reflec- tion in the line y 3, and draw DEF on the graph drawn in a. d. Find the coordinates of the vertices of RST, the image of DEF under the translation T0,6. e. What is the relationship b... |
that the position of each point is changed by a rotation of the same number of degrees. DEFINITION A rotation is a transformation of a plane about a fixed point P through an angle of d degrees such that: 1. For A, a point that is not the fixed point P, if the image of A is A, then PA = PA and mAPA d. 2. The image of t... |
a figure has rotational symmetry under a rotation of do, we can rotate the figure by do to an image that is an identical figure. Each figure shown below has rotational symmetry. 14365C06.pgs 7/12/07 2:57 PM Page 240 240 Transformations and the Coordinate Plane Any regular polygon (a polygon with all sides congruent an... |
will prove this theorem by using a rectangle with opposite vertices at the origin and at P. Note that in quadrants I and II, when b is positive, b is negative, and in quadrants III and IV, when b is negative, b is positive. 14365C06.pgs 7/12/07 2:57 PM Page 241 S(0, b) y P(a, b) x O R(a, 0) Rotations in the Coordinate... |
, the origin. Note: The symbol R is used to designate both a point reflection and a rotation. 1. When the symbol R is followed by a letter that designates a point, it repre- sents a reflection in that point. 2. When the symbol R is followed by both a letter that designates a point and the number of degrees, it represen... |
4, refer to the figure at the right. 3. What is the image of each of the given points under R90°? a. A e. H b. B f. J c. C g. K d. G h. L 4. What is the image of each of the given points under R–90°? a. A b. B c. C d. G Glide Reflections 243 y C B A O x LK J D EF G H I e. H h. L 5. The vertices of rectangle ABCD are A... |
vertices are A(1, 2), B(5, 3), and C(3, 4). Under the translation T0,–4, the image of ABC is ABC whose vertices are A(1, 2), B(5, 1), and C(3, 0). Under a glide reflection, the image of ABC is ABC. Note that the line of reflection, the y-axis, is a vertical line. The translation is in the vertical direction because th... |
3,3(3, 4) (0, 1) The vertices of PQR are P(2, 1), Q(2, 1), and R(0, 1). Answer Glide Reflections 245 y R Q x P b. T3,3(x, y) (x 3, y 3) T3,3(2, 1) (1, 2) T3,3(4, 1) (1, 2) T3,3(4, 3) (1, 0) The vertices of PQR are P (2, 1), Q(2, 1), and R(0, 1). Answer ryx(x 3, y 3) (y – 3, x 3) ryx(1, 2) (2, 1) ryx(1, 2) (2, 1) ryx(1,... |
A(1, 1), B(5, 4), and C(3, 5) under the given composition of transformations. b. Sketch ABC and its image. c. Explain why the given composition of transformations is or is not a glide reflection. d. Write the coordinates of the image of (a, b) under the given composition of transformations. 3. A reflection in the x-ax... |
a dilation. For example, in the coordinate plane, a dilation of 2 with center at the origin will stretch each ray by a factor of 2. If the image of A is A, h then A is a point on OA and OA 2OA. y O A A x DEFINITION A dilation of k is a transformation of the plane such that: 1. The image of point O, the center of dilat... |
coordinates of M and of M, the image of M. Verify that the midpoint is preserved. Solution a. D3(x, y) (3x, 3y). Therefore, E(0, 0), F(9, 0), G(12, 6), and H(3, 6). Answer b. Since E and F lie on the x-axis, EF is the absolute value of the difference of their x-coordinates. EF 3 0 3. 11 Therefore, the midpoint, M, is ... |
6. (1, 9) In 7–10, find the coordinates of the image of each given point under D5. 7. (2, 2) 9. (3, 5) 8. (1, 10) 10. (0, 4) In 11–14, each given point is the image under D2. Find the coordinates of each preimage. 11. (6, 2) 13. (6, 5) 12. (4, 0) 14. (10, 7) In 15–20, find the coordinates of the image of each given po... |
3. c. Using ABC and its image ABC, show that distance is not preserved under the dilation. Hands-On Activity In this activity, we will verify that angle measure, collinearity, and midpoint are preserved under a dilation. Using geometry software, or a pencil, ruler and a protractor, draw the pentagon A(1, 2), B(4, 2), ... |
07 2:57 PM Page 251 f: A one-to-one algebraic function f(x) x 5 y f(x) x 5 1 O 1 x Transformations as Functions 251 rm: A reflection in line m A B D C E m 1. For every x in the domain there is one and only one y in the range. For example: f(3) 8 f(0) 5 f(2) 3 2. Every f(x) or y in the range corresponds to one and only ... |
, we start with the coordinates of A and move from right to left. Find the image of A under the reflection and then, using the coordinates of that image, find the coordinates under the translation. T2,0 Orientation The figures below show the images of ABC under a point reflection, a translation, and a rotation. In each... |
3) 2, 5) rx-axis(2, 5) (2, 5) c. A line reflection is an opposite d. isometry. The composition of two opposite isometries is a direct isometry. This composition is a direct isometry. + ry-axis( x, y) (x, y) rx-axis and RO(x, y) (–x, y). The composition of a reflection in the y-axis followed by a reflection in the x-ax... |
7. ry-axis T21,4 RO + rx-axis + ry 5 x + T3,22 9. 12. A reflection in the line y x followed by a rotation of 90° about the origin is equivalent to 10. 11. what single transformation? 13. A rotation of 90° about the origin followed by another rotation of 90° about the origin is equivalent to what single transformation?... |
• A point reflection in P is a transformation plane such that: 1. If point A is not point P, then the image of A is A where P the mid- point of AAr. 2. The point P is its own image. • A translation is a transformation in a plane that moves every point in the plane the same distance in the same direction. • A translati... |
differ- ence of the x-coordinates. 6.3 Two points are on the same vertical line if and only if they have the same x-coordinate. 6.4 The length of a vertical line segment is the absolute value of the difference of the y-coordinates. 6.5 Each vertical line is perpendicular to each horizontal line. Theorems and Corollari... |
• Axis of symmetry • Line symmetry 6-3 ry-axis • rx-axis • ryx 6-4 Point reflection in P • RP • Point symmetry • RO 6-5 Translation • Translation of a units in the horizontal direction and b units in the vertical direction • Ta,b • Translational symmetry 6-6 Rotation • RP,d • Positive rotation • Negative rotation • Ro... |
and write the coordinates of its vertices. d. Under what single transformation is ABC the image of ABC? 19. a. On graph paper, locate the points R(4, 1), S(1, 1), and T(1, 2). Draw RST. b. Draw RST, the image of RST under a reflection in the origin, and write the coordinates of its vertices. c. Draw RST, the image of ... |
AC (4) AB BC AD DC 4. If is the perpendicular bisector of DBE, which of the following could g ABC be false? (1) AD AE (4) AB BC 5. DEF is not congruent to LMN, DE LM, and EF MN. Which of (3) CD CE (2) DB BE the following must be true? (1) DEF and LMN are not both right triangles. (2) mD mL (3) mF mN (4) mE mM 6. Under... |
point of a line segment whose end- points are A(2, 5) and B(2, 3)? Part III Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical... |
along one side of the triangle rather than traverse the other two, to get to the food. But no matter how evident the truth of a statement may be, it is important that it be logically established in order that it may be used in the proof of theorems that follow. Many of the inequality theorems of this chapter depend on... |
:45 AM Page 264 264 Geometric Inequalities This corresponds to the following transitive property of inequality: Postulate 7.2 If a, b, and c are real numbers such that a b and b c, then a c. In arithmetic: If 12 7 and 7 3, then 12 3. In algebra: If 5x 1 2x and 2x 16, then 5x 1 16. In geometry: If BA BD and BD BC, then ... |
PLE 2 Given: Q is the midpoint of PS and RS QS. Prove: RS PQ P Q R S Proof Statements Reasons 1. Q is the midpoint of. PS 1. Given. 2. PQ > QS 3. PQ = QS. 4. RS QS 5. RS PQ 2. The midpoint of a line segment is the point that divides the segment into two congruent segments. 3. Congruent segments have equal measures. 4. ... |
Addition and Subtraction 267 KLM 15. If 16. If KM KN, KN NM, and NM NL, prove that KM NL., prove that KM NM. NM and LM LK M 7-2 INEQUALITY POSTULATES INVOLVING ADDITION AND SUBTRACTION Postulates of equality and examples of inequalities involving the numbers of arithmetic can help us to understand the inequality postu... |
DC mEDA E C Proof Statements 1. mBDE mCDA 2. mBDE mEDC mEDC mCDA B D A Reasons 1. Given. 2. If equal quantities are added to unequal quantities, then the sums are unequal in the same order. 3. mBDC mBDE mEDC 3. The whole is equal to the sum of its parts. 4. mEDA mEDC mCDA 4. The whole is equal to the sum of 5. mBDC mED... |
.pgs 1/25/08 3:50 PM Page 270 270 Geometric Inequalities 7-3 INEQUALITY POSTULATES INVOLVING MULTIPLICATION AND DIVISION Since there are equality postulates for multiplication and division similar to those of addition and subtraction, we would expect that there are inequality postulates for multiplication and division ... |
: BC BA B D E A C Proof Statements Reasons 1. BE BD 2. 3BE 3BD 3. BC 3BE, BA 3BD 4. BC BA 1. Given. 2. If unequal quantities are multi- plied by positive equal quantities, then the products are unequal in the same order. 3. Given. 4. Substitution postulate for inequalities. EXAMPLE 2 Given: mABC mDEF, ABC, h EH h BG bi... |
b c 16. ac bc 11. c a c b c a. c 14. b 17. a c 18. Given: BD BE, D is the midpoint BA of Prove: BA BC, E is the midpoint of. BC B D A E C 20. Given: AB AD, AE AF 1 2AD Prove: AE AF 1 2AB, D F C A E B 19. Given: mDBA mCAB, mCBA 2mDBA, mDAB 2mCAB Prove: mCBA mDAB D C E A B 21. Given: mCAB mCBA, bisects CAB, BE AD bisect... |
means that b c a. Therefore, we need only test the longest side. (1) Is 10 4 6? No (2) Is 16 8 8? No (3) Is 16 6 8? No (4) Is 14 10 12? Yes Answer 14365C07.pgs 7/10/07 8:45 AM Page 274 274 Geometric Inequalities EXAMPLE 2 Two sides of a triangle have lengths 3 and 7. Find the range of possible lengths of the third sid... |
, 4, 5 7. 2, 2, 3 4. 5, 8, 13 8. 1, 1, 2 5. 6, 7, 10 9. 3, 4, 4 6. 3, 9, 15 10. 5, 8, 11 In 11–14, find values for r and t such that the inequality r s t best describes s, the length of the third sides of a triangle for which the lengths of the other two sides are given. 11. 2 and 4 12. 12 and 31 13. 13 2 and 13 2 14. ... |
the polygon, often called an interior angle. If, at h vertex A, we draw, AD we form BAE, an exterior angle of the polygon at vertex A., the opposite ray of h AE C D A E B DEFINITION An exterior angle of a polygon is an angle that forms a linear pair with one of the interior angles of the polygon. 14365C07.pgs 7/10/07 ... |
AM, extending the ray A C D through M to point E so only one midpoint. 2. Two points determine a line. A line segment can be extended that AM. EM to any length.. EC 3. Draw 4. mBCD mBCE mECD 3. Two points determine a line. 4. A whole is equal to the sum of its 5. BM CM EM AM 6. 7. AMB EMC 8. AMB EMC 9. B MCE parts. 5.... |
5. mBAD mBAC 180 6. 180 mBAD 7. 180 mBAD 90 8. BAD is obtuse. theorem. 3. Given. 4. Substitution postulate for inequalities. 5. If two angles form a linear pair, then they are supplementary. 6. The whole is greater than any of its parts. 7. Steps 4 and 6. 8. An obtuse angle is an angle whose degree measure is greater ... |
Given: SMR with STM extended through M to P Prove: mRMP mSRT R A C 19. Given: Point F not on › ‹ ABCDE and FC FD Prove: mABF mEDF F S T M P A CB D E 7-6 INEQUALITIES INVOLVING SIDES AND ANGLES OF A TRIANGLE A 22° We know that if the lengths of two sides of a triangle are equal, then the measures of the angles opposite... |
9. Transitive property of inequality. The converse of this theorem is also true, as can be seen in this example: Let the measures of the angles of ABC be mA 40, mB 80, and mC = 60. Write the angle measures in order: 80 60 40 Name the angles in order: mB mC mA Name the sides opposite these angles in order: AC AB BC If ... |
adjacent interior angle, so mCBA mCDA. Since angles of an isosceles triangle have equal measures, so mA mCBA. A quantity may be substituted for its equal in an inequality, so mA mCDA. A, ABC is isosceles. The base CA > CB D B 14365C07.pgs 7/10/07 8:45 AM Page 284 284 Geometric Inequalities C If the measures of two angl... |
starting with the shortest. 12. In RST, S is obtuse and mR mT. List the lengths of the sides of the triangle in order starting with the largest. 14365C07.pgs 8/2/07 5:46 PM Page 285 Applying Skills 13. Given: C is a point that is not on, ABD mABC mCBD. Prove: AC BC Chapter Summary 285 C A B D 14. Let ABC be any right ... |
than the measure of either nonadjacent interior angle. 14365C07.pgs 7/10/07 8:45 AM Page 286 286 Geometric Inequalities 7.3 If the lengths of two sides of a triangle are unequal, then the measures of the angles opposite these sides are unequal and the larger angle lies opposite the longer side. 7.4 If the measures of ... |
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