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two sides of one 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 greater than the third side of the second triangle. 1. a. With a partner or in a small group, prove the Hinge Theor... |
, 5 (3) 7, 12, 20 (4) 6, 7, 12 6. Which of the following statements is true for all values of x? (1) x 5 and x 5 (2) x 5 or x 5 7. In ABC and DEF, (3) If x 5, then x 3. (4) If x 3, then x 5., and A D. In order to prove AB > DE ABC DEF using ASA, we need to prove that (1) B E BC > EF (2) C F AC > DF (3) (4) 8. Under a r... |
all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. Given: PR Prove: AP BP bisects ARB but PR is not perpendicular to ARB. 14. Given: In quadrilateral ABCD, h AC bisects DAB and h CA bisects DCB. Prove: B D Part IV Answer all questions in this part. Each correct an... |
iz (1646–1716) in Germany and Isaac Newton (1642–1727) in England each developed methods for determining the slope of a tangent to a curve at any point as well as methods for determining the area bounded by a curve or curves. Newton acknowledged the influence of the work of mathematicians and scientists who preceded hi... |
5 3 The result of both computations is the same. When we compute the slope of a line that is determined by two points, it does not matter which point is considered as the first point and which the second. 14365C08.pgs 7/10/07 8:46 AM Page 292 292 Slopes and Equations of Lines Also, when we find the slope of a line usi... |
AB (3, 3) C(1, 2) 1 x 14365C08.pgs 7/10/07 8:46 AM Page 293 Negative Slope The Slope of a Line 293. Let the Points C and D are two points on coordinates of C be (2, 3) and the coordinates of D be (5, 1). As the values of x increase, the val- g EF ues of y decrease. The graph of ward. g EF slants down- g EF 5 1 2 3 5 2... |
x2 2, y1 Then, x1 Dy Dx 5 g slope of P1P2 5 2. 4, and y2 y2 2 y1 x2 2 x1 5 2 2 4 4 2 (22) 5 22 6 5 21 3 Answer y P1(2, 4) 2 6 O 1 1 1 1 P2(4, 2) x EXAMPLE 2 Through point (1, 4), draw the line whose slope is 23 2. Solution How to Proceed (1) Graph point A(1, 4). (2) Note that, since slope Dy 3 Dx 5 2 2 5 23 2, when y ... |
5); 18. (1, 3); m 1 m 5 2 3 21. (1, 0); m 5 4 Applying Skills 13. (1, 3); m 3 16. (3, 2); m 0 m 5 23 19. (2, 3); 2 22. (0, 2); m 2 3 14. (2, 5); m 1 17. (4, 7); m 2 21 20. (1, 5); m 3 23. (2, 0); m 1 2 24. a. Graph the points A(2, 4) and B(8, 4). b. From point A, draw a line that has a slope of 2. c. From point B, dra... |
AB by slope of AB 5 slope of BP 21 2 5 23 2 6 5 26 29 5 2 3 5 3(y 2 5) 5 2(x 2 6) 3y 2 15 5 2x 2 12 3y 5 2x 1 3 y 5 2 3x 1 1 Recall that when the equation is solved for y in terms of x, the coefficient of x is the slope of the line and the constant term is the y-intercept, the y-coordinate of the point where the line ... |
y 5 22(0) 1 6 5 0 1 6 5 6 The y-intercept is 6. When the equation is solved for y, the y-intercept is the constant term. c. The x-intercept is the x-coordinate of the point at which the line intersects the x-axis, that is, the value of x when y is 0. Since (3, 0) is a given point on the line, the x-intercept is 3. Ans... |
5) Solve the equation 3 2m b for b in b 2m 3 terms of m: (6) Substitute the value of b found in (5) for b in the second equation and solve for m: (7) Substitute this value of m in either equation to find the value of b: The equation is y 2x 1. 7 3m b 7 3m (2m 3) 7 5m 3 10 5m 2 m b 2m 3 b 2(2) 3 b 1 14365C08.pgs 7/10/07... |
2 and y-intercept 4 15. a. Do the points P(3, 3), Q(5, 4), and R(1, 1) lie on the same line? 14. No slope and x-intercept 2 b. If P, Q, and R lie on the same line, find the equation of the line. If P, Q, and R do not lie on the same line, find the equations of the lines g PQ, g QR, and g.PR 14365C08.pgs 7/10/07 8:46 A... |
of x in the equation that you wrote in a represent? d. What does the constant term in the equation that you wrote in a represent? 19. Show that if the equation of the line can be written as x-axis at (a, 0) and the y-axis at (0, b). x a 1 y b 5 1, then the line intersects the 8-3 MIDPOINT OF A LINE SEGMENT The midpoin... |
is the average of the y-coordinates of C and D. C(3, 3) D(3, 1) 1 1 O N x 1 1 y-coordinate of N 5 1 1 (23) 2 5 22 2 5 21 These examples suggest the following relationships: If the endpoints of a horizontal segment are (a, c) and (b, c), then the coordinates of the midpoint are: a 1 b 2 If the endpoints of a vertical s... |
. y • Since vertical lines are perpendicular to horizontal lines, PSM and MTQ are right angles and therefore congruent. 1 O 1 M(5, 3) P(2, 1) S(5, 1) Q(8, 5) T(8, 3) R(8, 1) x • Therefore, PSM MTQ by SAS and PM > MQ because correspond- ing parts of congruent triangles are congruent. We can conclude that the coordinates... |
. is on AB, and, x1 1 x2 2 y1 1 y2 2 AM > MB (1) Show that M x1 1 x2 2 y1 1 y2 2, A B lies on AB : slope of AM 5 y1 1 y2 2 2 y1 2 2 x1 x1 1 x2 slope of MB 5 y2 2 x2 2 y1 1 y2 2 x1 1 x2 2 5 y1 1 y2 2 2y1 x1 1 x2 2 2x1 y2 2 y1 x2 2 x1 2y2 2 (y1 1 y2) 2x2 2 (x1 1 x2) y2 2 y1 x2 2 x1 Points A, M, and B lie on the same line... |
1 1 x2 2 x2 2 2x2 2 x1 2 x2 2 x2 2 x1 2 P y1 1 y2 2 2 y2 P y1 1 y2 2 2y2 2 y1 2 y2 2 P P P P. Therefore, ADM and MEB are right angles and are Vertical lines are perpendicular to horizontal lines. ME'BE congruent. ADM is the midpoint of MEB by SAS and. AM > MB AB >. Therefore, M x1 1 x2 2 y1 1 y2, 2 A B AD'MD and We gen... |
Find the coordinates of M. Let (x1, y1) be (1, 1) and (x2, y2) be (7, 3). The coordinates of M are: y C(2, 6) B(7, 3) M(4, 2) x P(x, y) 1 1 O 1 A(1, 1) 1 x1 1 x2 2 y1 1 y2 4, 2) B, 1 1 3 2 B (2) Write the equation of the line through C(2, 6) and M(4, 2). Let P(x, y) be any other point on the line. y 2 6 slope of PC 5 ... |
2, 5) 11. (3, 5), (1, 1) 14. 1 3, 9, B A 2 3, 3 B A AB. Find the coordinates of the third point when the coordinates of 16. A(3, 3), M(3, 9) 18. B(4, 2), M 20. A(0, 7), M A B 3 2, 0 A 0, 7 2 B 21. The points A(1, 1) and C(9, 7) are the vertices of rectangle ABCD and B is a point on the same horizontal line as A. a. Wha... |
90° about the origin, the image of A(1, m1) is A(m1, 1). Since AOA is a g'OAr right angle, g OA Let l2 be the line through A(m1, 1) and O(0, 0), and let the slope of l2 be m2. Then: g OAr. y 1 l2 A(m1, 1) l1 A(1, m1 m2 5 0 2 1 5 21 0 2 (2m1) 0 1 m1 5 2 1 m1 We have shown that when two lines through the origin are perp... |
b) x l1 Proof: Let l1 and l2 be two perpendicular lines that intersect at (a, b). Under the translation (x, y) → (x a, y b), the image of (a, b) is (0, 0). l2r l1r l1r and Theorem 8.2 tells us that if the slope of l1 is m, then the slope of its, is m. Since l1 and l2 are perimage, pendicular, their images,, are also p... |
of the points is equal to the slope of a segment joining another pair of these points. g AC g'AB 7. 7. Contradiction (steps 1, 6). 14365C08.pgs 7/10/07 8:46 AM Page 310 310 Slopes and Equations of Lines We can restate Theorems 8.3a and 8.3b as a biconditional. Theorem 8.3 Two non-vertical lines are perpendicular if an... |
y 9. through (0, 4) and (2, 0) 11. through (4, 4) and (4, 2) 4. y x 2 6. 2x y 3 8. through (1, 1) and (5, 3) 10. y-intercept 2 and x-intercept 4 12. parallel to the x-axis through (5, 1) In 13–16, find the equation of the line through the given point and perpendicular to the given line. 13. 21 2, 22 ; 2x 7y 15 14. (0, ... |
), then ABCD is a rectangle. 14365C08.pgs 7/10/07 8:46 AM Page 312 312 Slopes and Equations of Lines 27. The vertices of ABC are A(2, 2), B(6, 6) and C(6, 0). a. What is the slope of AB? b. Write an equation for the perpendicular bisector of AB. c. What is the slope of BC? d. Write an equation for the perpendicular bis... |
As rl1.. rl1 and rl2, in that For a–c, using the procedure above write the equations of two lines under which reflections in the two lines are equal to the given translation. Check your answers using the given coordinates. a. T4,4 c. T1,3 b. T3,2 D(0, 0), E(5, 3), F(2, 2) D(1, 2), E(5, 3), F(5, 6) D(6, 6), E(2, 5), F(1... |
0, b) y (0, b) y (0, b) y (0, b) (a, 0) O (c, 0) x O (a, 0) (c, 0) x (0, 0) (a, 0) x (a, 0) O (a, 0) x The triangle with vertices (a, 0), (0, b), (c, 0) can be any triangle. It is convenient to place one side of the triangle on the x-axis and the vertex opposite that side on the y-axis. The triangle can be acute if a a... |
B A 5 2 1 23 2 5 5 4 9 2 (23) 4 2 (22) 5 12 is is 28 5 21 2. 6 5 2. y A(3, 5) D(4, 9) (1, 3) 1 O 1 C(2, 3) B(5, 1) x CD CD and and AB AB rocal of the slope of the other. bisect each other because they have a common midpoint, (1, 3). are perpendicular because the slope of one is the negative recip- EXAMPLE 2 Prove that... |
. Therefore, ANM and CMN are right angles. All right angles are congruent, so ANM CMN. Also, MN > MN. (6) Then, AMN CMN by SAS (steps 4 and 5). (7) AM > CM because they are corresponding parts of congruent triangles. (8) (step 1) and AM > BM AM > BM > CM or AM BM = CM. The midpoint of the hypotenuse of a right triangle... |
right angle? c. Which side is the hypotenuse? d. What are the coordinates of the midpoint of the hypotenuse? e. What is the equation of the median from the vertex of the right angle to the hypotenuse? f. What is the equation of the altitude from the vertex of the right angle to the hypotenuse? g. Is the triangle an is... |
, 0), (0, b), and (a, 0) are the vertices of an isosceles tri- angle. (Hint: A translation will let you use the results of Exercise 11.) 13. The coordinates of the vertices of ABC are A(0, 0), B(2a, 2b), and C(2c, 2d). a. Find the coordinates of E, the midpoint of AB and of F, the midpoint of AC. b. Prove that the slop... |
0) O C(c, 0) x y F E We will show that altitudes. and BO CF tudes AE and BO intersect in the same point as alti- Intersection of altitudes and AE BO Intersection of altitudes and BO CF 1. The slope of side 0 2 b c 2 0 5 2b c. BC is 2. The slope of altitude is perpendicular to BC AE, which c, is. b 1. The slope of side... |
14365C08.pgs 7/10/07 8:46 AM Page 319 Concurrence of the Altitudes of a Triangle 319 The point where the altitudes of a triangle intersect is called the orthocenter. EXAMPLE 1 The coordinates of the vertices of PQR are P(0, 0), Q(2, 6), and R(4, 0). Find the coordinates of the orthocenter of the triangle. Solution Let... |
(a, 0) P(2, 0) or a 2 B(0, b) Q(0, 6) or b 6 C(c, 0) R(6, 0) or c 6. 14365C08.pgs 7/10/07 8:46 AM Page 320 320 Slopes and Equations of Lines The coordinates of S, the point at which the altitudes of PRQ intersect, are 0, 22(6) 6 B The intersection of the altitudes of PQR is S, the preimage of S(0, 2) under the translat... |
(a, 0), B(0, b), and C(c, 0), as shown in the diagrams of the proof of Theorem 8.4. Assume that b 0. 1. Esther said that if A is to the left of the origin and C is to the right of the origin, then the point of intersection of the altitudes is above the origin. Do you agree with Esther? Explain why or why not. 2. Simon ... |
a line through A, perpendicular to CP. Write the equation of is a line through C, perpendicular to AP. Write the equation of g AB g CB.. c. Find B, the intersection of g AB and g.CB 14365C08.pgs 8/2/07 5:48 PM Page 322 322 Slopes and Equations of Lines d. Write an equation of the line that contains the altitude from B... |
also has slope m. 8.3 Two non-vertical lines are perpendicular if and only if the slope of one is the negative reciprocal of the other. 8.4 The altitudes of a triangle are concurrent. 14365C08.pgs 7/10/07 8:46 AM Page 323 Chapter Summary 323 VOCABULARY 8-1 Slope • x • y 8-2 y-intercept • x-intercept • Point-slope form... |
. Write an equation of the perpendicular bisector of each side of the tri- angle. e. Show that the three perpendicular bisectors intersect in a point and find the coordinates of that point. 14365C08.pgs 7/10/07 8:46 AM Page 324 324 Slopes and Equations of Lines 8. The vertices of ABC are A(7, 1), B(5, 3), and C(3, 5). ... |
are a horizontal and a vertical line. Find the area of these three triangles. STEP 8. Find the sum of the three areas found in step 7. Subtract this sum from the area of the rectangle. The difference is the area of the given triangle. Repeat steps 1 through 8 for each of the triangles with the given vertices. a. (2, 0... |
(0, 6) and (3, 0). The equation of the line through these points is (1) y 2x 6 (2) y 2x 6 1 (3) y 3 2x 1 (4) y x 3 2 8. The converse of the statement “If two angles are right angles then they are congruent” is (1) If two angles are congruent then they are right angles. (2) If two angles are not right angles then they ... |
correct numerical answer with no work shown will receive only 1 credit. 13. In the diagram, TQ'RS h TQ bisects RTS and. Prove that RST is isosceles. 14. The following statements are true: • If Evanston is not the capital of Illinois, then Chicago is not the capital. • Springfield is the capital of Illinois or Chicago ... |
postulate that would make it possible to prove Euclid’s parallel postulate. Other postulates have been proposed that appear to be simpler and which could provide the basis for a proof of the parallel postulate. The form of the parallel postulate most commonly used in the study of elementary geometry today was proposed... |
or intersecting. 14365C09.pgs 7/10/07 8:48 AM Page 330 330 Parallel Lines EXAMPLE 1 If line l is not parallel to line p, what statements can you make about these two lines? Solution Since l is not parallel to p, l and p cannot be the same line, and they have exactly one point in common. Answer Parallel Lines and Trans... |
09.pgs 7/10/07 8:48 AM Page 331 In the diagram shown on page 330, the two lines cut by the transversal are not parallel lines. However, when two lines are parallel, many statements may be postulated and proved about these angles. Proving Lines Parallel 331 Theorem 9.1a If two coplanar lines are cut by a transversal so ... |
versal so that the interior angles on the same side of the transversal are supplementary, then the lines are parallel. Given intersects g EF of 4. Prove g AB g CD g AB and g CD, and 5 is the supplement E A C 4 3 5 F B D Proof Angle 4 and angle 3 are supplementary since they form a linear pair. If two angles are supplem... |
ector of an angle divides the angle into two congruent angles. A (3) Therefore, by the transitive property of congruence, DBA D. (4) Then, DBA and D are congruent alternate interior angles when h BA are intersected by transversal and two coplanar lines are cut by a transversal so that the alternate interior angles form... |
that are congruent, then the two lines are parallel. 14365C09.pgs 7/10/07 8:48 AM Page 335 Properties of Parallel Lines 335 9-2 PROPERTIES OF PARALLEL LINES In the study of logic, we learned that a conditional and its converse do not always have the same truth value. Once a conditional statement has been proved to be ... |
assumption is false and 1 2.. This is a D C F g AB g CD 2 H A E B Note that Theorem 9.1b is the converse of Theorem 9.1a. We may state the two theorems in biconditional form: Theorem 9.1 Two coplanar lines cut by a transversal are parallel if and only if the alternate interior angles formed are congruent. Each of the ... |
g AB g CD and g AB g at G and H, CD, mBGH 3x 20, and respectively. If mGHC 2x 10: a. Find the value of x. c. Find mGHD. b. Find mGHC. A C E G B D H F Solution a. Since g AB g CD and these lines are cut by transversal g EF, the alternate inte- rior angles are congruent: mBGH mGHC 3x 20 2x 10 3x 2x 10 20 x 30 b. mGHC 2x... |
are cut by a transversal so that the corresponding angles formed are congruent, then the two lines are parallel. SUMMARY OF PROPERTIES OF PARALLEL LINES If two lines are parallel: 1. A transversal forms congruent alternate interior angles. 2. A transversal forms congruent corresponding angles. 3. A transversal forms s... |
m8 when m3 65. 9. m3 when m3 3x and m5 x 28. 10. m5 when m3 x and m4 x 20. 11. m7 when m1 x 40 and m2 5x 10. 12. m5 when m2 7x 20 and m8 x 100. as shown in the diagram. Find: 4. m2 when m6 150. 6. m7 when m1 75. 8. m5 when m2 130. 12 13. Two parallel lines are cut by a transversal. For each pair of interior angles on ... |
transversal is perpendicular to one of two parallel lines, it is per- pendicular to the other.” 19. Prove that if two parallel lines are cut by a transversal, the alternate exterior angles are con- gruent. 20. Given: ABC, h CE Prove: A B. bisects exterior BCD, and h CE AB. 21. Given: CAB DCA and DCA ECB Prove: a. g AB... |
. Theorem 9.10a If two non-vertical lines in the same plane are parallel, then they have the same slope. Given l1 l2 Prove The slope of l1 is equal to slope of l2. Proof In the coordinate plane, let the slope of l1 be m 0. Choose any point on l1. Through a given point, one and only one line can be drawn perpendicular t... |
The vertices of quadrilateral ABCD are A(2, 4), B(6, 2), C(2, 6), and D(1, 2). a. Show that two sides of the quadrilateral are parallel. b. Show that the quadrilateral has two right angles. 14365C09.pgs 7/10/07 8:48 AM Page 344 344 Parallel Lines Solution The slope of AB 5 22 2 (2422) 2 2 6 5 8 21 2 2 5 24 2 2 (21) 5 ... |
your answer. Developing Skills In 3–8, for each pair of lines whose equations are given, tell whether the lines are parallel, perpendicular, or neither parallel nor perpendicular. 3. x y = 7 x y 3 5. x 1 3y 1 2 y 3x 2 7. x = 2 x 5 4. 2x y 5 y 2x 3 6. 2x y 6 2x y = 3 8. x = 2 y 3 In 9–12, write an equation of the line ... |
parallel sides. b. Show that PQRS does not have a right angle. 16. The coordinates of the vertices of quadrilateral KLMN are K(2, 1), L(4, 3), M(2, 1), and N(1, 2). a. Show that KLMN has only one pair of parallel sides. b. Show that KLMN has two right angles. Hands-On Activity 1 In this activity, we will use a compass... |
sum of the measures of the angles of a triangle is 180°. Given ABC Prove mA mB mC 180 D A B E C Proof Statements Reasons 1. Let g DE be the line through B that is parallel to. AC 2. mDBE 180 3. mDBA mABC mCBE 180 4. A DBA and C CBE 5. mA mDBA and mC CBE 1. Through a given point not on a given line, there exists one an... |
mC 180, and then solve the resulting equation: 3mA 180 so mA 60. Corollary 9.11e The sum of the measures of the angles of a quadrilateral is 360°. Proof: In quadrilateral ABCD, we draw, forming two triangles. The sum of AC the measures of the angles of quadrilateral ABCD is the sum of the measures of the angles of the... |
degree measure of one of its angles is 90. EXAMPLE 3 B is a not a point on DCB and AC > BC g ACD. Ray h CE. Prove that bisects g CE. g AB Solution Given: h CE bisects DCB and AC > BC. Prove: g AB g CE Proof Statements h CE bisects DCB. 1. 2. DCE ECB 3. mDCE mECB AC > BC 4. 5. mCAB mCBA 6. mDCB mCAB mCBA 7. mDCB mDCE m... |
9. 90, 36 10. 65, 65 In 11–14, the measure of the vertex angle of an isosceles triangle is given. Find the measure of a base angle. 11. 20 12. 90 13. 76 14. 110 In 15–18, the measure of a base angle of an isosceles triangle is given. Find the measure of the vertex angle. 15. 80 16. 20 17. 45 18. 63 19. What is the mea... |
exterior angle of a triangle is equal to the sum of the measures of the nonadjacent interior angles.” 31. a. In the coordinate plane, graph points A(5, 2), B(2, 2), C(2, 1), D(1, 1). g AB and BDC. b. Draw c. Explain how you know that BDC is an isosceles right triangle. d. What is the measure of BDC? Justify your answe... |
4. Given. 5. ASA. Therefore, when two angles and any side in one triangle are congruent to the corresponding two angles and side of a second triangle, we may say that the triangles are congruent either by ASA or by AAS. The following corollaries can proved using AAS. Note that in every right tri- angle, the hypotenuse... |
of the vertices of ABC are A(6, 0), B(1, 0) and C(5, 2). The coordinates of DEF are D(3, 0), E(8, 0), and F(4, 2). Prove that the triangles are congruent. C(5, 2) y F(4, 2) B(1, 0) O A(6, 0) D(3, 0) E(8, 0) x Solution (1) Prove that the triangles are right triangles. In ABC: 2 2 0 The slope of AC is The slope of CB is... |
mark two points, D and E, on BC BC. (3) Draw AD (4) In ABD and ABE, and AE. AB AD > AE, AB. In these B D E C A B B, and two triangles, two sides and the angle opposite one of the sides are congruent to the corresponding parts of the other triangle. But ABD and ABE are not congruent. This counterexample proves that SSA... |
8:48 AM Page 357 The Converse of the Isosceles Triangle Theorem 357 12. Given: Quadrilateral ABCD with A C and h BD the bisector of ABC. h DB bisects ADC. Prove: 13. Given: AB CD, AB > CD, and AB'BEC. A Prove: AED and BEC bisect each other. B E C C A B D D 14. a. Use a translation to prove that ABC and DEF in Example ... |
, the bisector of 2. ACD BCD 3. A B CD 4. CD 5. ACD BCD 6. CA CB 1. Every angle has one and only one bisector. 2. An angle bisector of a triangle is a line segment that bisects an angle of the triangle. 3. Given. 4. Reflexive property of congruence. 5. AAS. 6. Corresponding parts of congruent triangles are congruent. T... |
3x, and mC 4x 30. Describe the triangle as acute, right, or obtuse, and as scalene, isosceles, or equilateral. Solution The sum of the degree measures of the angles of a triangle is 180. x 30 3x 4x 30 180 8x 60 180 8x 120 x 15 Substitute x 15 in the representations given for the three angle measures. m/A 5 x 1 30 5 15... |
Converse of the Isosceles Triangle Theorem 361 In 3–6, in each case the degree measures of two angles of a triangle are given. a. Find the degree measure of the third angle of the triangle. b. Tell whether the triangle is isosceles or is not isosceles. 3. 70, 40 4. 30, 120 6. 80, 40 5. 50, 65 7. In ABC, mA mC, AB 5x 6... |
9.13 by drawing the altitude from C. 9-7 PROVING RIGHT TRIANGLES CONGRUENT BY HYPOTENUSE, LEG We showed in Section 5 of this chapter that, when two sides and an angle opposite one of these sides in one triangle are congruent to the corresponding two sides and angle in another triangle, the two triangles may or may not... |
F since all right angles are congruent. (7) Therefore, DEF GEF by AAS. (8) Therefore, ABC DEF by the transitive property of congruence (steps 4 and 7). This theorem is called the hypotenuse-leg triangle congruence theorem, abbreviated HL. Therefore, from this point on, when the hypotenuse and a leg of one right triangl... |
three angle and and AC AB, and and BC BC A N B The point where the angle bisectors of a triangle are concurrent is called the incenter. EXAMPLE 1 Given: ABC, AD AB. BC ⊥, BD AB DC, and C Prove: DAB BCD Proof We can show that ADB and CBD are right triangles and use HL to prove them congruent. A D B 14365C09.pgs 7/10/07... |
angle of APB. c. Find the measure of each angle of BPC. d. Find the measure of each angle of CPA. e. Does the bisector of ACB also bisect APB? Explain your answer. 5. Triangle ABC is an isosceles triangle with mC 140. Let P be the incenter of ABC. a. Find the measure of each acute angle of ABC. b. Find the measure of ... |
and Exterior Angles of Polygons 367 11. Given: Quadrilateral ABCD, AB'BD, BD'DC, and AD > CB. Prove: A C and AD CB 12. In QRS, the bisector of QRS is perpendicular to QS at P. a. Prove that QRS is isosceles. b. Prove that P is the midpoint of QS. 13. Each of two lines from the midpoint of the base of an isosceles tria... |
180 degrees. Polygon ABCD is a concave polygon and a quadrilateral. In the rest of this textbook, unless otherwise stated, all polygons are convex Interior Angles of a Polygon A pair of angles whose vertices are the endpoints of a common side are called consecutive angles. And the vertices of consecutive angles are ca... |
interior angle. The interior angle and the exterior angle are supplementary. Therefore, the sum of their measures is 180°. If a polygon has n sides, the sum of the interior and exterior angles of the polygon is 180n. Therefore, in a polygon with n sides: The measures of the exterior angles 180n the measures of the int... |
D 18(n 2) x 2x 12 x 22 3x 180(4 2) 7x 10 360 7x 350 x 50 14365C09.pgs 7/10/07 8:48 AM Page 371 Interior and Exterior Angles of Polygons 371 mA x 50 mC x 22 50 22 72 mB 2x 12 2(50) 12 88 mD 3x 3(50) 150 b. Each exterior angle is the supplement of the interior angle with the same vertex. The measure of the exterior angle... |
interior angles contains: a. 90° b. 120° c. 140° d. 160° 17. Find the number of sides a polygon if the sum of the degree measures of its interior angles is: a. 180 e. 1,440 b. 360 f. 2,700 c. 540 g. 1,800 d. 900 h. 3,600 Applying Skills 18. The measure of each interior angle of a regular polygon is three times the mea... |
activity, we will study the intersection of the angle bisectors of polygons. a. Draw various polygons that are not regular of different sizes and numbers of sides. Construct the angle bisector of each interior angle. Do the angle bisectors appear to intersect in a single point? b. Draw various regular polygons of diff... |
to the other. If two of three lines in the same plane are each parallel to the third line, then they are parallel to each other. If two lines are vertical lines, then they are parallel. If two lines are horizontal lines, then they are parallel. 9.8 9.9 9.10 Two non-vertical lines in the coordinate plane are parallel i... |
n sides is 180(n 2)°. 9.17 The sum of the measures of the exterior angles of a polygon is 360°. VOCABULARY 9-1 Euclid’s parallel postulate • Playfair’s postulate • Coplanar • Parallel lines • Transversal • Interior angles • Exterior angles • Alternate interior angles • Alternate exterior angles • Interior angles on th... |
of an exterior angle of the triangle at vertex R? 11. An exterior angle at the base of an isosceles triangle measures 130°. Find the measure of the vertex angle. 12. In ABC, if 13. In DEF, if 14. In PQR, AB DE AC DF and mA 70, find mB. and mE 13, find mD. PQ is extended through Q to point T, forming exterior RQT. If m... |
from the postulates of Euclid? How can the postulates from this chapter be rewritten to fit the non-Euclidean geometry you investigated? What theorems from this chapter are not valid in the nonEuclidean geometry that you investigated? One possible non-Euclidean geometry is the geometry of the sphere suggested in the C... |
) (5, 2) x 3y 4 is (1) 3x y 1 (2) x 3y 1 (3) 3x y 1 (4) x 3y 1 7. The coordinates of the midpoint of the line segment whose endpoints are (3, 4) and (5, 6) are (1) (1, 1) (2) (4, 5) (3) (4, 5) (4) (4, 5) 8. If a, b, c, and d are real numbers and a b and c d, which of the follow- ing must be true? (1) a c b d (2) a c b ... |
mEDA.. A D E C B Part IV Answer all questions in this part. Each correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 cre... |
ercises Cumulative Review 379 14365C10.pgs 7/10/07 8:50 AM Page 380 380 Quadrilaterals 10-1 THE GENERAL QUADRILATERAL Patchwork is an authentic American craft, developed by our frugal ancestors in a time when nothing was wasted and useful parts of discarded clothing were stitched into warm and decorative quilts. Quilt ... |
1 D C Quadrilateral ABCD is a parallelogram because The symbol for parallelogram ABCD is ~ABCD. The Parallelogram 381 AB CD and BC DA. A B that are parallel in the figure. Note the use of arrowheads, pointing in the same direction, to show sides Theorem 10.1 A diagonal divides a parallelogram into two congruent triangl... |
ram bisect each other. Given ~ABCD with diagonals AC and BD intersecting at E. D C Prove AC and BD bisect each other. E A B Proof Statements Reasons 1. AB CD D C 2. BAE DCE and ABE CDE B 3. AB > CD C 4. ABE CDE E E D A A 1. Opposite sides of a parallelogram are parallel. 2. If two parallel lines are cut by a transversa... |
/07 8:50 AM Page 384 384 Quadrilaterals Developing Skills 3. Find the degree measures of the other three angles of a parallelogram if one angle measures: a. 70 b. 65 c. 90 d. 130 e. 155 f. 168 In 4–11, ABCD is a parallelogram. 4. The degree measure of A is represented by 2x 20 and the degree measure of B by 2x. Find th... |
Petrina’s bedroom has four right angles. 18. The deck that Jeremiah is building is in the shape of a quadrilateral, ABCD. The measure of the angle at A is not equal to the measure of the angle at C. Prove that the deck is not in the shape of a parallelogram. 14365C10.pgs 7/10/07 8:50 AM Page 385 Proving That a Quadril... |
D is a parallelogram. Proof Since AB is parallel to CD, BAC and DCA are a A B pair of congruent alternate interior angles. Therefore, by SAS, DCA BAC. Corresponding parts of congruent triangles are congruent, so DAC ACB. and congruent alternate interior angles DAC and ACB. Therefore, and ABCD is a parallelogram. is a t... |
gruent. 3. One pair of opposite sides is both congruent and parallel. 4. Both pairs of opposite angles are congruent. 5. The diagonals bisect each other. 14365C10.pgs 7/10/07 8:50 AM Page 387 EXAMPLE 1 Proving That a Quadrilateral Is a Parallelogram 387 Given: ABCD is a parallelogram. D F C E is on AB, F is on DC, and ... |
point on h EF such that F is the midpoint of EJ. Prove that FJGH is a parallelogram. CD 13. ABCD is a parallelogram. The midpoint of is R, and the midpoint of of a. Prove that APS CRQ and that BQP DSR. b. Prove that PQRS is a parallelogram. is S. DA AB is P, the midpoint of BC is Q, the midpoint 14. A quadrilateral ha... |
the base of the rectangle. Thus, if side, is called the is taken as the base, then either consecutive side, AD BC or AB altitude of the rectangle. Since a rectangle is a special kind of parallelogram, a rectangle has all the properties of a parallelogram. In addition, we can prove two special properties for the rectan... |
congruent, the parallelogram is a rectangle. Given Parallelogram ABCD with AC > BD Prove ABCD is a rectangle. Strategy Prove that DAB CBA. Therefore, DAB and CBA are both congruent and supplementary. D A C B The proof of Theorem 10.11 is left to the student. (See exercise 13.) SUMMARY To prove that a quadrilateral is ... |
, then the parallelogram is not a rectangle. Do you agree with Pauli? Explain why or why not. 2. Cindy said that if two congruent line segments intersect at their midpoints, then the quadrilateral formed by joining the endpoints of the line segments in order is a rectangle. Do you agree with Cindy? Explain why or why n... |
0, 2), B(2, 2), C(4, 5), and D(6, 1). Under what specific transformation is ABCD the image of ABCD? d. Prove that ABCD is a rectangle. 14365C10.pgs 7/10/07 8:50 AM Page 393 The Rhombus 393 16. The coordinates of the vertices of PQRS are P(2, 1), Q(1, 3), R(5, 1), and S(2, 5). a. Prove that PQRS is a parallelogram. b. P... |
C Prove AB > BC > CD > DA Proof By definition, rhombus ABCD is a parallelogram. It is AB > DA given that and are congruent, so transitive property of congruence, AB > CD. Opposite sides of a parallelogram BC > DA. Using the AB > BC > CD > DA. A B 14365C10.pgs 7/10/07 8:50 AM Page 394 394 Quadrilaterals Theorem 10.13 T... |
It is given that in ABCD,. Since both pairs of opposite sides are congruent, ABCD is a parallelogram. Two consecutive sides of parallelogram ABCD are congruent, so by definition, ABCD is a rhombus. A B Theorem 10.16 If the diagonals of a parallelogram are perpendicular to each other, the parallelogram is a rhombus. Gi... |
Since ABCD is a parallelogram, opposite sides are equal in length: (2) Substitute x 12 to find the : AD lengths of sides and AB A B DC AB 3x 11 2x 1 x 12 AB 2x 1 2(12) 1 25 AD x 13 12 13 25 (3) Since ABCD is a parallelogram with two congruent consecutive sides, ABCD is a rhombus. Exercises Writing About Mathematics 1.... |
perpendicular to each other. A rhombus is a parallelogram. Conclusion: In a parallelogram, diagonals are perpendicular to each other. 14. The diagonals of a rhombus bisect the angles of the rhombus. A rhombus is a parallelogram. Conclusion: The diagonals of a parallelogram bisect its angles. 15. Consecutive angles of ... |
point of intersection of the diagonals. C PD A B c. Prove that the diagonals are perpendicular. d. Is ABCD a rhombus? Justify your answer. 25. ABCD is a parallelogram. The midpoint of AB is M, the midpoint of CD is N, and AM AD. a. Prove that AMND is a rhombus. b. Prove that MBCN is a rhombus. c. Prove that AMND is co... |
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