Geometry: A Complete Course

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1 Geometry: omplete ourse (with Trigonometry) Module - Student WorkText Written by: Thomas. lark Larry. ollins RRT 4/2010

2 6. In the figure below, and share the common segment. Prove the following conditional statements, using the two-column format. a) If, then. b) If, then. 7. Use the results of xercise 6,and the information given in each of the following, to draw a conclusion concerning each diagram. (Note: You are not being asked to formally prove anything.) a) b) WY XZ c) MN OP M W X Y Z N O P 8. Given: F Prove: (hint: Use the results of exercise 6) Thinking through Theorem 1: 9. re the six points shown in the figure in xercise 8 all necessarily in the same plane? xplain why or why not. 10. oes the relationship of the triangles in xercise 8 have any effect on the proof in xercise 8? xplain. 230 Unit III Fundamental Theorems

3 Lesson 1 xercises: 1. Prove Theorem 3 If you have a given ray, then there is exactly one point at a given distance from the endpoint of the ray. (Note: This is the same theorem we proved in the lesson. We are using it as an exercise to make sure you understand its proof. Use your course notes to check.) a) State the theorem. b) raw and label a diagram to accurately show the conditions of the theorem. c) List the given information. d) Write the statement we wish to prove. e) emonstrate the direct proof using the two-column format. 2. ray has how many endpoints? 3. What is the endpoint of QR? 4. o and name the same ray? Why or why not? 5. Must QT and QS be opposite rays? Why or why not? 6. Must QT and QS be opposite rays if points Q, T, and S are collinear? Why or why not? 7. raw a diagram that illustrates the information given. a) Point Q is on, but Q is not on. b) and X are opposite rays. c) RS and RT are the same ray. d) PQ and PT are not collinear. 8. For each of the following, tell whether and are opposite rays. nswer yes, no, or not enough information. a) The coordinate of is 7, the coordinate of is 8, and the coordinate of is 14. b),, and are not collinear. 9. Point is between points and. Point is between and. If = x, = y, and = z, find the lengths of and, in terms of x, y, and z. Part Theorems bout Segments and Rays 235

4 10. The length of is 6. The coordinate of point is 6. a) What are the possible coordinates of point? b) What are the possible coordinates of the midpoint of? c) Find the length of the line segment whose endpoints are the possible midpoints of. 11. Given: The coordinates of R and T are 0 and t, respectively; t > 0; T is between R and Q. (The diagram is already drawn.) Prove: The coordinate of Q is positive. 0 t q R T Q omplete the proof: STTMNT RSON 1. T is between R and Q; the 1. coordinates of R and T are zero and t, respectively. 2. Q has a coordinate, call it q. 2. Postulate 6 (Ruler) First ssumption 3. RT + TQ = RQ RT = 0 -t = -t = t 4. Postulate 6 (Ruler) Third ssumption TQ = t q RQ = 0 q = q = q 5. t + t q = q t q < q 6. efinition of >. 7. t q< q and t q> q 7. bsolute Value Property: (consider t q< q) If x < k, then x < k and x > k (See step 28) (VideoText lgebra-ii-4) 8. t q+ q< q + q t+ 0 < q + q Unit III Fundamental Theorems

5 32. t + 0 < q + q t < q + q q = q 34. efinition of bsolute Value 35. t < q+ q t < ( )( t)> ( )( ) ( 1) ( t)> t > (Note: t > 0 was given. So t q> q has no bearing on the proof. nd we have already considered t > 0 ) 12. Using the concepts in xercise 11 as a guide, write a proof for the following: Given: The coordinates of R and T are 0 and t, respectively; t >0; X is between R and T. Prove: The coordinate of X is positive. 0 x t R X T 238 Unit III Fundamental Theorems

6 Lesson 2 xercises: 1. Prove Theorem 4 If you have a given line segment, then that segment has exactly one midpoint. (Note: This is the same theorem we proved in the lesson. We are using it as an exercise to make sure you understand its proof. Use your course notes to check.) a) State the theorem. b) raw and label a diagram to accurately show the conditions of the theorem. c) List the given information. d) Write the statement we wish to prove. e) emonstrate the indirect proof using the two-column format. 2. For Q to be the midpoint of MN, what conditions must be satisfied? 3. If, how is related to? 4. The midpoint of PQ is point N. RS bisects PQ. What is the intersection of PQ and RS? 5. If X, Y, and Z are three collinear points, and XZ = YZ, what statement can be made about point Z? l 6. If line bisects M, what point of M does contain? l 7. If is not congruent to F, how is related to F? 8. If M and N have coordinates 15 and 183 respectively, does MN have a midpoint? For xercises 9 through 11, use the information provided and demonstrate the indicated proof. lso name the property which describes the relationship. 9. Given: is a line segment Prove: 240 Unit III Fundamental Theorems

7 5. = 0 b = b = b, and 5. b b b M = 0 = =, and b b b M = b = = M = M M M 7. b b 8. M + M = M + M = M + M = 1 i b M + M = b M + M = M is between point and point M is the midpoint of omplete the following proof, by supplying the missing reasons. Given: The coordinates of point and point on are 0 and b, respectively; b > 0; point M is the midpoint of. b Prove: The coordinate of M is 2 0 x M b Statement Reason 1. M is the midpoint of M corresponds to some real number x b > b x > 0 x Unit III Fundamental Theorems

8 Unit III Fundamental Theorems Part Theorems bout ngles Part 1 (One ngle) Lesson 1 Theorem 7: If, in a half-plane, there is a ray in the edge of the half-plane, then there is exactly one other ray through the endpoint of the given ray, such that the angle formed by the two rays has a given measure. Objective: To understand this theorem as an application of previously accepted postulates and demonstrate its proof directly. Lesson 1 xercises: 1. Prove Theorem 7 If, in a half-plane, there is a ray in the edge of the half-plane, then there is exactly one other ray through the endpoint of the given ray, such that the angle formed by the two rays has a given measure. (Note: This is the same theorem we proved in the lesson. We are using it as an exercise to make sure you understand its proof. Use your course notes to check.) a) State the theorem. b) raw and label a diagram to accurately show the conditions of the theorem. c) List the given information. d) Write the statement we wish to prove. e) emonstrate the direct proof using the two-column format. 250 Unit III Fundamental Theorems

9 Unit III Fundamental Theorems Part F Theorems bout ngles Part 2 (Two ngles) Lesson 3 Theorem 11: If you have right angles, then those right angles are congruent. Objective: To understand this theorem as an application of previously accepted postulates and demonstrate its proof directly. Lesson 3 xercises: 1. Prove Theorem 11 If you have right angles, then those right angles are congruent. (Note: This is the same theorem we proved in the lesson. We are using it as an exercise to make sure you understand its proof. Use your course notes to check.) a) State the theorem. b) raw and label a diagram to accurately show the conditions of the theorem. c) List the given information. d) Write the statement we wish to prove. e) emonstrate the direct proof using the two-column format. 2. Name all the right angles in the figure below. H l 1 l 2 G Q l 3 F l 4 Part F Theorems bout ngles Part 2 (Two ngles) 261

10 8. Given: 1 2 Prove: 9. Given: PQ MN RQ bisects MQP R P S SQ bisects NQP Prove: MQS NQR M Q N 10. Given: at point Q F Prove: 1 and 3 are complementary Q Part F Theorems bout ngles Part 2 (Two ngles) 263

11 Unit III Fundamental Theorems Part G Theorems bout ngles Part 3 (More Than Two ngles) Lesson 2 Theorem 14: If two angles are supplementary to the same angle or congruent angles, then they are congruent to each other. Objective: To understand this theorem as an application of previously accepted postulates and demonstrate its proof directly. Lesson 2 xercises: 1. Prove Theorem 14, Part 2 If two angles are supplementary to the same angle, then they are congruent to each other. (Note: This is the second part of the theorem we proved in the lesson. We are using it as an exercise to make sure you understand its proof. Use your course notes to check, but understand that the course notes refer to the first case.) a) State the theorem. b) raw and label a diagram to accurately show the conditions of the theorem. c) List the given information. d) Write the statement we wish to prove. e) emonstrate the direct proof using the two-column format. Part G Theorems bout ngles Part 3 (More Than Two ngles) 271

12 2. Prove Theorem 14, Part 1 If two angles are supplementary to the congruent angles, then they are congruent to each other. (Note: This is the first part of the theorem we proved in the lesson.) a) State the theorem. b) raw and label a diagram to accurately show the conditions of the theorem. c) List the given information. d) Write the statement we wish to prove. e) emonstrate the direct proof using the two-column format. 3. Find the measure of the supplement of each of the following angles: a) m = 42 b) m MNO = x 3 c) m RST = x 2 4. Two supplementary angles are congruent. Find their measures. 5. If and F are supplementary, find the value of x, m, and m F in each of the following: a) m = 2x; m F = x 15 b) m = x + 16; m F = 2x 16 c) m = x 2 ; m F = 12x 9 6. Find the measure of an angle that is twice as large as its supplement. 7. If, m + m = 180, and m + m = 180, then. Why? 272 Unit III Fundamental Theorems

13 2. Prove Theorem 15, Part 2 If two lines intersect, then the vertical angles formed are congruent. (Note: This is the same theorem we proved in the lesson, but you are being asked to prove the other 2 vertical angles congruent angles 2 and 4.) a) State the theorem. b) raw and label a diagram to accurately show the conditions of the theorem. c) List the given information. d) Write the statement we wish to prove. e) emonstrate the direct proof using the two-column format. For xercises 3 through 7, use the diagram to the right: 3. If m 3 = 110, find: a) m 4 n m b) m 1 c) m 2 4. If m 1 = m 3, find: a) m b) m 2 c) m 3 d) m 4 5. If 3 and 4 are congruent and supplementary, what can you conclude about line m and line n? 6. If 3 and 4 are congruent and complementary, what is m 4? 7. Suppose m 3 is increased by 10. What effect would this have on: a) m 4 b) m 1 c) m 2 Part G Theorems bout ngles Part 3 (More Than Two ngles) 275

14 In xercises 8 through 11, find the value of x and the measure of each simple angle in the diagram, for which there is no given measure. X 5x 20 4x P 9. Y R N Q 7x x + 85 T M N 10. M Q 11. X Y P 6x 9 P 4x W U T R Z x Given: S R T Prove: 1 3 P 3 Q For xercises 13 through 15, use the diagram to the right: M Q P 13. Given: PR NP R N Prove: MP PR 14. Given: PR NP Prove: MP QP 15. Given: P bisects MPR Prove: P bisects NPQ 276 Unit III Fundamental Theorems

15 5. p q, m 1 = 100, m 2 = 55 Find m p q 6. Find m 4x 40 x m n, m 6 = 75 a) Find m 3 d) Find m 4 b) Find m 7 e) Find m 1 c) Find m 2 f) Find m m n 8. m. Find x. a) m 1 = 3x + 7 m 6 = 5x 3 b) m 4 = 8x + 12 m 6 = 2x + 54 m c) m 4 = x 2 + 5x m 8 = 9x Find x and y. 63 x y x z F 68 Find x, y and z. 51 y G Part H Theorems bout Parallel Lines 283

16 11. Find x, y, m and m 80 o (2x+y) o (5x+y) o (5x y) o 12. Given: p q, s t F p Prove: GH F G I H K L J q s t 13. Given: m, p q Prove: m p q 14. Given: m 1 = m 2 2 Prove: bisects Given: bisects N N Prove: N N o 284 Unit III Fundamental Theorems

17 10. Given: LK HJ IK GI LH LK HM IK m LKI = 72 M K L N J G H I Find: a) m MHJ f) m LHM b) m MHI g) m GHL c) m HJI h) m HJK d) m JHI i) m MNK e) m LHJ j) m LNM 11. Given: RU SV RS UT RS RU m USW = 55 TU bisects SUV R S U T V W Prove: m TUV = Given: UW Y UX Prove: m Y + m = W X Y V U m XU + m UVY 290 Unit III Fundamental Theorems

18 Using the figure to the right, determine whether the information given in xercises 15 through 18 allows you to conclude that line a is parallel to line b using Theorem m 17 + m 13 = m 5 + m 6 + m 8 = m 14 + m 15 + m 16 = m 1 + m 16 + m 20 = 180 a b w x t 19. Given: R and S are supplementary angles Q S Q R Prove: QP RS P S 20. Given: F F F F F bisects FG Prove: G F G 21. Given: FH Prove: G F H G Part H Theorems bout Parallel Lines 299

19 8. In the figure to the right, m is given, and. Find m. Justify your answer In the figure to the right,. Find m and m. Justify your answers In the figure to the right, F. Find m and m. Justify your answers. G O 125 F 11. Suppose a set of points consists of a plane and a line not in the plane. State the minimum number of points that must be in the set. 12. Suppose two different planes both contain point and point. escribe the relationship between the planes. Part H Theorems bout Parallel Lines 303

20 In xercises 2 through 7, use the figure at the right and the given information to apply Theorem 23, naming two lines which are parallel to each other because these two lines are parallel to a third line. t 1 2. a b ; ; 12 and 14 are supplementary angles ; and 15 are supplementary angles; ; 7 and 17 are supplementary angles ; 16 and 18 are supplementary angles a b c d e Of the seven lines making up the figure below, which lines must be parallel? Give a reason for each answer, using theorems or corollaries we have proved in our Geometry t 1 t 2 t 3 t 4 Referring to the figure to the right, determine whether the information given in xercises 9 through 12 allows you to conclude that line a is parallel to line b using Theorem 23. Say yes or no, and give reasons for your answers and 18 are supplementary angles; b t 3 ; c t 3 ; m 18 = m 11 + m 12; 3 and 8 are supplementary angles 12. m 11 + m 12 + m 17 = 180; m 7 = m 10 + m 11 a b c t 1 t 2 t 3 Part H Theorems bout Parallel Lines 305

21 4. Recall from Unit II, Part, Lesson 6, xercise 4 (page 143), that an angle formed by two intersecting planes is a dihedral angle, and is generally named by using a point on each face and two points on the edge. In the figure at the right, two planes, P and Q, are cut by a third plane, M. Planes P and M intersect in. Planes Q and M intersect in. Points and F are on plane P. Points J and H are on plane Q. Points G, L, and K are on plane M. a) Name two pairs of alternate interior dihedral angles. b) Name two pairs of alternate exterior dihedral angles. G c) Name two pairs of corresponding dihedral angles. d) Name two pairs of vertical dihedral angles. e) Plane P is parallel to plane Q. 1) o the alternate interior dihedral angles have equal measures? H L K M F P J Q 2) o the corresponding dihedral angles have equal measures? 5. Prove: If two planes are perpendicular to the same line, then the two planes are parallel. Given: Plane H at point P Plane K at point Q P Q H K Prove: Plane H Plane K omplete the following proof: (Hint: raw the plane indicated in Statement 3) STTMNT 1. Plane H at point P 1. Given Plane K at point Q RSON 2. Line contains point P and point Q Unit III Fundamental Theorems

22 3. There exists a plane which passes 3. Postulate 5 through line and contains points P and Q. Through line pass plane M intersecting plane H in P and intersecting plane K in Q. 4. Line P 4. efinition of a plane perpendicular to a Line Q line- plane is perpendicular to a given line, if and only if, every line that intersects the given line in the plane, is perpendicular to the given line. 5. PQ is a right angle 5. QP is a right angle 6. m PQ = m QP = m PQ + m QP = m PQ + m QP = PQ and QP are supplementary P Q P and Q do not intersect 11. efinition of Parallel Lines - Lines which are coplanar and do not intersect. 12. Plane H and plane K do not intersect 12. ny given line in plane H will not intersect the corresponding line in plane K, any given line being one of the infinitely many in plane H, and having the same relationship as P and Q. 13. Plane H Plane K 13. efinition of Parallel Planes Part H Theorems bout Parallel Lines 311

23 6. Prove that if a line is perpendicular to a given plane, then any plane containing that line is perpendicular to the given plane. Given: Line plane Q Plane P contains line Q t Prove: P Q P omplete the following proof: STTMNT RSON 1. plane Q; plane P contains 1. Given 2. Line t is in plane Q and plane P t Plane P plane Q 4. efinition of Perpendicular Planes Two intersecting planes are perpendicular, if and only if, there exists a line in one of the planes, which is perpendicular to both the other plane, and the line of intersection of the two planes. 7. Prove that if two planes intersect to form a right dihedral angle, then the planes are perpendicular. Given: Plane P intersects plane M in V V--R is a right dihedral angle P R Prove: Plane P Plane M M 312 Unit III Fundamental Theorems

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